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2.3: Some Dimensional Reasoning and its Consequences - Geosciences


Obtaining the Correct Dimensions

Like every physically correct equation, Equation 2.2.1 must represent equality not only of magnitudes but also of dimensions. So whatever the form of the term or terms on the right side of Equation 2.2.1, the variables (U), (D), ( ho), and (mu) must combine in such a way that each term has the dimensions of force, because the left side has the dimensions of force. The following list gives the dimensions of each of the five variables involved in flow past a sphere, in terms of mass (mathrm{M}), length (mathrm{L}), and time (mathrm{T}):

(egin{array}{ll}{F_{D}} & {-quad mathrm{ML} / mathrm{T}^{2}} {U} & {-quadmathrm{L} / mathrm{T}} {D} & {-quadmathrm{L}} { ho} & {-quadmathrm{M} / mathrm{L}^{3}} {mu} & {-quadmathrm{M} / mathrm{LT}}end{array})

The only variable here whose dimensions are not straightforward is (mu); the dimensions (mathrm{M} / mathrm{L} mathrm{T}) are obtained by use of Equation 1.3.6, by which (mu) is defined.

It is advantageous to rewrite equations like Equation 2.2.1 in dimensionless form. To do this, first make the left side dimensionless by dividing (F_{D}) by some product of independent variables that itself has the dimensions of force. Using the list of dimensions above, you can verify that ( ho U^{2} D^{2}) has the dimensions of force:

( ho U^{2} D^{2}: left(mathrm{M} / mathrm{L}^{3} ight)(mathrm{L} / mathrm{T})^{2}(mathrm{L})^{2}=mathrm{ML} / mathrm{T}^{2})

So dividing the left side of Equation 2.2.1 by ( ho U^{2} D^{2}) makes the left side of the equation dimensionless. The result, (F_{D} / ho U^{2} D^{2}), can be viewed as a dimensionless form of (F_{D}). That leaves the right side of Equation 2.2.1 to be made dimensionless. There is one and only one way the four variables (U), (D), ( ho), and (mu) can be combined into a dimensionless variable, namely ( ho U D / mu):

( ho U D / mu): (left(mathrm{M} / mathrm{L}^{3} ight)(mathrm{L} / mathrm{T})(mathrm{L}) /(mathrm{M} / mathrm{LT})) ... (mathrm{M}), (mathrm{L}), (mathrm{T}) cancel

(That statement is not strictly true—but all the other possibilities are just ( ho U D / mu) raised to some power, and they are not independent of ( ho U D / mu).) So whatever the form of the function (f), the right side of the dimensionless form of Equation 2.2.1 can be written using just one dimensionless variable:

[frac{F_{D}}{ ho U^{2} D^{2}}=fleft(frac{ ho U D}{mu} ight) label{dimensionless} ]

A Simplified Function

Equation ef{dimensionless} is an equivalent but dimensionless form of Equation 2.2.1. The great advantage of the dimensionless equation is that it involves only two variables—a dependent dimensionless variable (F_{D} / ho U^{2} D^{2}) and an independent dimensionless variable ( ho U D / mu)—instead of the original five. Think of the enormous saving in effort this implies for an experimental program to characterize the drag force. If you had to measure (F_{D}) as a function of each one of the four variables while holding the other three constant, you would generate mountains of data and graphs. But Equation ef{dimensionless} tells you that (U), (D), ( ho), and (mu) need only be varied so as to make ( ho U D / mu) vary. All of the experimental points for (F_{D} / ho U^{2} D^{2}) obtained by varying ( ho U D / mu) should plot as a curve in a two- dimensional graph with these two variables along the axes. Whatever the values of (U), (D), ( ho), and (mu), all possible realizations of flow past a sphere are expressed by just one curve. This curve is shown in Figure (PageIndex{1}) together with some of the experimental points that have been used to define it. The physics behind the curve is discussed in Chapter 3, after more background in the principles of fluid dynamics. And you could find the curve by varying only one of the four variables (U), (D), ( ho), and (mu)—although you may not be able to get a very wide range of values of ( ho U D / mu) by varying only one of those variables. A fairly small number of experiments involving values of the original independent variables that combined to span a wide range of ( ho U D / mu) would suffice to characterize all other possible combinations of independent variables. This is because each point in the dimensionless graph represents a great many different possible combinations of the original variables—an infinity of these, in fact. You thus gain a far-reaching predictive capability on the basis of relatively little observational effort.

A skeptic might find all this to be too good to be true. But the fact is that this is how things work, and the analysis of flow past a sphere is just one good example. A note of caution is in order, however. It is prudent to vary as many of the variables over as wide a range as possible; this does not take an enormous number of observations, and it is a check on the correctness of your analysis. You will see below in more detail that if there is a larger number of important variables than you think, your data points would form a scattered band rather than a single curve. Then if you varied just one variable to try to find the curve, you would indeed get a curve, but it would not be the curve you were after; you would be missing the scatter that would manifest itself if you varied the other variables as well.

Several notes

First, variables of the form ( ho U D / mu) are called Reynolds numbers, usually denoted by (mathrm{Re}). Whenever both density and viscosity are important in a problem and both a length variable and a velocity are involved, a Reynolds number can be formed and used. There are thus many different Reynolds numbers, with different length and velocity variables depending on the particular problem. You will encounter others in later chapters.

Second, for the steady flow we have assumed, the variables (U), (D), ( ho), and (mu) characterize not only everything about the distributions of shear stress and pressure over the entire surface of the sphere, which add up to (F_{D}), but also the distributions of shear stress, pressure, and fluid velocity at every point in the surrounding fluid. Because ( ho U D / mu) replaces these four variables on the right side of Equation ef{dimensionless}, the same can be said of the Reynolds number. Anything about forces and motions you might want to consider can be viewed as being specified completely by the Reynolds number.

Third, there is a further important consequence of the fact that each point on the curve of (F_{D} / ho U^{2} D^{2}) vs. ( ho U D / mu) represents an infinity of combinations of (U), (D), ( ho), and (mu). Suppose that you wanted to find the drag force exerted by a certain flow on a sphere that is too large to fit into your laboratory or your basement. You could work with a much smaller sphere by adjusting the values of (U), ( ho), and (mu) so that ( ho U D / mu) is the same as in the flow in question past the large sphere (Figure (PageIndex{2})). Then from the curve in Figure (PageIndex{1}) the value of (F_{D} / ho U^{2} D^{2}) is also the same, and from it you could find the drag force (F_{D}) on the large sphere by substituting the corresponding values of (U), (D), and ( ho). Or, on the other hand, you could study the flow around a very small sphere by use of a much larger sphere, with the same complete confidence in the results (Figure (PageIndex{2})). This is the essence of scale modeling: the study of one physical system by use of another at a smaller or larger physical scale but with variables adjusted so that all forces and motions in the two systems are in the same proportions. Figure (PageIndex{2}) shows how you might use flow around a small sphere with diameter (D_{m}) to model flow around a much larger sphere with diameter (D_{o}). You would have to adjust the flow velocities (U_{m}) and (U_{o}), as well as the fluid viscosities (mu_{m}) and (mu_{o}) and the fluid densities ( ho_{m}) and ( ho_{o}), so that the Reynolds number (mathrm{Re}_{m}), equal to ( ho_{m} U_{m} D_{m} / mu_{m}), in the model is the same as the Reynolds number (mathrm{Re}_{o}), equal to ( ho_{o} U_{o} D_{o} / mu_{o}), in the large-scale flow. Then all forces and motions are in the same proportion in the two flows, and, specifically, the dimensionless drag force, or the drag coefficient, is the same in the two flows. Despite the great difference in physical scale, both of the flows are represented by the same point on the graph of drag coefficient vs. Reynolds number, so anything about the two flows, provided only that it is expressed in dimensionless form, is the same in the two flows. Each point on the curve of (F_{D} / ho U^{2} D^{2}) vs. ( ho U D / mu) represents an infinite number of possible experiments, each of which is a scale model of all the others!

Fourth, in Figure (PageIndex{1}) the dimensionless drag force is written in a conventional form that is slightly different from that derived above: (F_{D} /left( ho U^{2} / 2 ight) A), where (A) is the cross-sectional area of the sphere, equal to (pi D^{2} / 4). This differs from (F_{D} / ho U^{2} D^{2}) by the factor (pi / 8), but its dimensions are exactly the same. It is usually called a drag coefficient, denoted by (C_{D}); you can see why that term came about by writing

[F_{D}=C_{D} frac{ ho U^{2}}{2} A label{2.3} ]

where the factor (left( ho U^{2} / 2 ight) A) on the right side has dimensions of force. The functional relationship between dimensionless drag force and Reynolds number in Equation ef{dimensionless} can be written in an entirely equivalent form using (C_{D}):

[C_{D}=frac{F D}{frac{ ho U^{2}}{2} A}=fleft(frac{ ho U D}{mu} ight) label{2.4} ]

Fifth, there are alternative versions of the dependent dimensionless variable. Dividing by ( ho U^{2} D^{2}) is not the only way to nondimensionalize (F_{D}). You can check for yourself that (F_{D} / mu U D), ( ho F_{D} / mu^{2}), and (F_{D} / mu U) are other possibilities, obtained by combining (F_{D}) with the four variables ( ho), (mu), (U), and (D) taken three at a time. (You will see in the next section how to derive such variables.) Sometimes, as in the last two cases, one of the variables drops out; this happens when (mathrm{M}) or (mathrm{L}) or (mathrm{T}) appears in only one of the four variables chosen. Any of these three alternative dependent dimensionless variables would serve just as well as (F_{D} / ho U^{2} D^{2}) to represent the data. You will see below, however, that sometimes one is more revealing than the others.


Multi-dimensional iterative reasoning in action: The case of the Colonel Blotto game

We introduce a novel decision procedure involving multi-dimensional iterative reasoning, in which a player decides separately on the various features of his strategy using an iterative process. This type of strategic reasoning fits a range of complicated situations in which a player faces a large and non-ordered strategy space. In this paper, the procedure is used to explain the results of a large web-based experiment of a tournament version of the Colonel Blotto game. The interpretation of the participants’ choices as reflecting multi-dimensional iterative reasoning is supported by an analysis of their response times and the relation between the participants’ behavior in this game and their choices in another game which triggers standard k-level reasoning. Finally, we reveal the most successful strategies in the tournament, which appear to reflect 2–3 levels of reasoning in the two main “dimensions”.

Highlights

► We introduce a novel decision procedure involving multi-dimensional iterative reasoning. ► The procedure explains the results of a large web-based experiment of a version of the Blotto game. ► The interpretation of choices is supported by subjects’ response times and behavior in another game.


Contents

Many parameters and measurements in the physical sciences and engineering are expressed as a concrete number—a numerical quantity and a corresponding dimensional unit. Often a quantity is expressed in terms of several other quantities for example, speed is a combination of length and time, e.g. 60 kilometres per hour or 1.4 kilometres per second. Compound relations with "per" are expressed with division, e.g. 60 km/1 h. Other relations can involve multiplication (often shown with a centered dot or juxtaposition), powers (like m 2 for square metres), or combinations thereof.

A set of base units for a system of measurement is a conventionally chosen set of units, none of which can be expressed as a combination of the others and in terms of which all the remaining units of the system can be expressed. [5] For example, units for length and time are normally chosen as base units. Units for volume, however, can be factored into the base units of length (m 3 ), thus they are considered derived or compound units.

Sometimes the names of units obscure the fact that they are derived units. For example, a newton (N) is a unit of force, which has units of mass (kg) times units of acceleration (m⋅s −2 ). The newton is defined as 1 N = 1 kg⋅m⋅s −2 .

Percentages, derivatives and integrals Edit

Percentages are dimensionless quantities, since they are ratios of two quantities with the same dimensions. In other words, the % sign can be read as "hundredths", since 1% = 1/100 .

Taking a derivative with respect to a quantity adds the dimension of the variable one is differentiating with respect to, in the denominator. Thus:

  • position (x) has the dimension L (length)
  • derivative of position with respect to time (dx/dt, velocity) has dimension T −1 L—length from position, time due to the gradient
  • the second derivative (d 2 x/dt 2 = d(dx/dt) / dt, acceleration) has dimension T −2 L.

Likewise, taking an integral adds the dimension of the variable one is integrating with respect to, but in the numerator.

    has the dimension T −2 L M (mass multiplied by acceleration)
  • the integral of force with respect to the distance (s) the object has travelled ( ∫ F d s , work) has dimension T −2 L 2 M .

In economics, one distinguishes between stocks and flows: a stock has units of "units" (say, widgets or dollars), while a flow is a derivative of a stock, and has units of "units/time" (say, dollars/year).

In some contexts, dimensional quantities are expressed as dimensionless quantities or percentages by omitting some dimensions. For example, debt-to-GDP ratios are generally expressed as percentages: total debt outstanding (dimension of currency) divided by annual GDP (dimension of currency)—but one may argue that, in comparing a stock to a flow, annual GDP should have dimensions of currency/time (dollars/year, for instance) and thus Debt-to-GDP should have units of years, which indicates that Debt-to-GDP is the number of years needed for a constant GDP to pay the debt, if all GDP is spent on the debt and the debt is otherwise unchanged.

In dimensional analysis, a ratio which converts one unit of measure into another without changing the quantity is called a conversion factor. For example, kPa and bar are both units of pressure, and 100 kPa = 1 bar . The rules of algebra allow both sides of an equation to be divided by the same expression, so this is equivalent to 100 kPa / 1 bar = 1 . Since any quantity can be multiplied by 1 without changing it, the expression " 100 kPa / 1 bar " can be used to convert from bars to kPa by multiplying it with the quantity to be converted, including units. For example, 5 bar × 100 kPa / 1 bar = 500 kPa because 5 × 100 / 1 = 500 , and bar/bar cancels out, so 5 bar = 500 kPa .

The most basic rule of dimensional analysis is that of dimensional homogeneity. [6]

Only commensurable quantities (physical quantities having the same dimension) may be compared, equated, added, or subtracted.

However, the dimensions form an abelian group under multiplication, so:

One may take ratios of incommensurable quantities (quantities with different dimensions), and multiply or divide them.

For example, it makes no sense to ask whether 1 hour is more, the same, or less than 1 kilometre, as these have different dimensions, nor to add 1 hour to 1 kilometre. However, it makes perfect sense to ask whether 1 mile is more, the same, or less than 1 kilometre being the same dimension of physical quantity even though the units are different. On the other hand, if an object travels 100 km in 2 hours, one may divide these and conclude that the object's average speed was 50 km/h.

The rule implies that in a physically meaningful expression only quantities of the same dimension can be added, subtracted, or compared. For example, if mman, mrat and Lman denote, respectively, the mass of some man, the mass of a rat and the length of that man, the dimensionally homogeneous expression mman + mrat is meaningful, but the heterogeneous expression mman + Lman is meaningless. However, mman/L 2 man is fine. Thus, dimensional analysis may be used as a sanity check of physical equations: the two sides of any equation must be commensurable or have the same dimensions.

This has the implication that most mathematical functions, particularly the transcendental functions, must have a dimensionless quantity, a pure number, as the argument and must return a dimensionless number as a result. This is clear because many transcendental functions can be expressed as an infinite power series with dimensionless coefficients.

All powers of x must have the same dimension for the terms to be commensurable. But if x is not dimensionless, then the different powers of x will have different, incommensurable dimensions. However, power functions including root functions may have a dimensional argument and will return a result having dimension that is the same power applied to the argument dimension. This is because power functions and root functions are, loosely, just an expression of multiplication of quantities.

Even when two physical quantities have identical dimensions, it may nevertheless be meaningless to compare or add them. For example, although torque and energy share the dimension T −2 L 2 M , they are fundamentally different physical quantities.

To compare, add, or subtract quantities with the same dimensions but expressed in different units, the standard procedure is first to convert them all to the same units. For example, to compare 32 metres with 35 yards, use 1 yard = 0.9144 m to convert 35 yards to 32.004 m.

A related principle is that any physical law that accurately describes the real world must be independent of the units used to measure the physical variables. [7] For example, Newton's laws of motion must hold true whether distance is measured in miles or kilometres. This principle gives rise to the form that conversion factors must take between units that measure the same dimension: multiplication by a simple constant. It also ensures equivalence for example, if two buildings are the same height in feet, then they must be the same height in metres.

The factor-label method is the sequential application of conversion factors expressed as fractions and arranged so that any dimensional unit appearing in both the numerator and denominator of any of the fractions can be cancelled out until only the desired set of dimensional units is obtained. For example, 10 miles per hour can be converted to meters per second by using a sequence of conversion factors as shown below:

NOx concentration = 10 parts per million by volume = 10 ppmv = 10 volumes/10 6 volumes NOx molar mass = 46 kg/kmol = 46 g/mol Flow rate of flue gas = 20 cubic meters per minute = 20 m 3 /min The flue gas exits the furnace at 0 °C temperature and 101.325 kPa absolute pressure. The molar volume of a gas at 0 °C temperature and 101.325 kPa is 22.414 m 3 /kmol. 1000 g NO x 1 kg NO x × 46 kg NO x 1 kmol NO x × 1 kmol NO x 22.414 m 3 NO x × 10 m 3 NO x 10 6 m 3 gas × 20 m 3 gas 1 minute × 60 minute 1 hour = 24.63 g NO x hour >_><1>_>>>> imes >_>>><1 >_>>>> imes >_>>><22.414 >^<3> >_>>>> imes >^<3> >_>>><10^<6> >^<3> >>>>> imes >^<3> >>>><1 >>>> imes >>><1 >>>=24.63 >_>>>>

After canceling out any dimensional units that appear both in the numerators and denominators of the fractions in the above equation, the NOx concentration of 10 ppmv converts to mass flow rate of 24.63 grams per hour.

Checking equations that involve dimensions Edit

The factor-label method can also be used on any mathematical equation to check whether or not the dimensional units on the left hand side of the equation are the same as the dimensional units on the right hand side of the equation. Having the same units on both sides of an equation does not ensure that the equation is correct, but having different units on the two sides (when expressed in terms of base units) of an equation implies that the equation is wrong.

For example, check the Universal Gas Law equation of PV = nRT , when:

  • the pressure P is in pascals (Pa)
  • the volume V is in cubic meters (m 3 )
  • the amount of substance n is in moles (mol)
  • the universal gas law constant R is 8.3145 Pa⋅m 3 /(mol⋅K)
  • the temperature T is in kelvins (K)

As can be seen, when the dimensional units appearing in the numerator and denominator of the equation's right hand side are cancelled out, both sides of the equation have the same dimensional units. Dimensional analysis can be used as a tool to construct equations that relate non-associated physico-chemical properties. The equations may reveal hitherto unknown or overlooked properties of matter, in the form of left-over dimensions – dimensional adjusters – that can then be assigned physical significance. It is important to point out that such 'mathematical manipulation' is neither without prior precedent, nor without considerable scientific significance. Indeed, the Planck constant, a fundamental constant of the universe, was 'discovered' as a purely mathematical abstraction or representation that built on the Rayleigh–Jeans law for preventing the ultraviolet catastrophe. It was assigned and ascended to its quantum physical significance either in tandem or post mathematical dimensional adjustment – not earlier.

Limitations Edit

The factor-label method can convert only unit quantities for which the units are in a linear relationship intersecting at 0. (Ratio scale in Stevens's typology) Most units fit this paradigm. An example for which it cannot be used is the conversion between degrees Celsius and kelvins (or degrees Fahrenheit). Between degrees Celsius and kelvins, there is a constant difference rather than a constant ratio, while between degrees Celsius and degrees Fahrenheit there is neither a constant difference nor a constant ratio. There is, however, an affine transform ( x ↦ a x + b , rather than a linear transform x ↦ a x ) between them.

For example, the freezing point of water is 0 °C and 32 °F (0 °C), and a 5 °C change is the same as a 9 °F (−13 °C) change. Thus, to convert from units of Fahrenheit to units of Celsius, one subtracts 32 °F (the offset from the point of reference), divides by 9 °F (−13 °C) and multiplies by 5 °C (scales by the ratio of units), and adds 0 °C (the offset from the point of reference). Reversing this yields the formula for obtaining a quantity in units of Celsius from units of Fahrenheit one could have started with the equivalence between 100 °C and 212 °F (100 °C), though this would yield the same formula at the end.

Hence, to convert the numerical quantity value of a temperature T[F] in degrees Fahrenheit to a numerical quantity value T[C] in degrees Celsius, this formula may be used:

T[C] = (T[F] − 32) × 5/9.

To convert T[C] in degrees Celsius to T[F] in degrees Fahrenheit, this formula may be used:

T[F] = (T[C] × 9/5) + 32.

Dimensional analysis is most often used in physics and chemistry – and in the mathematics thereof – but finds some applications outside of those fields as well.

Mathematics Edit

Finance, economics, and accounting Edit

In finance, economics, and accounting, dimensional analysis is most commonly referred to in terms of the distinction between stocks and flows. More generally, dimensional analysis is used in interpreting various financial ratios, economics ratios, and accounting ratios.

  • For example, the P/E ratio has dimensions of time (units of years), and can be interpreted as "years of earnings to earn the price paid".
  • In economics, debt-to-GDP ratio also has units of years (debt has units of currency, GDP has units of currency/year).
  • In financial analysis, some bond duration types also have dimension of time (unit of years) and can be interpreted as "years to balance point between interest payments and nominal repayment". has units of 1/years (GDP/money supply has units of currency/year over currency): how often a unit of currency circulates per year.
  • Interest rates are often expressed as a percentage, but more properly percent per annum, which has dimensions of 1/years.

Fluid mechanics Edit

In fluid mechanics, dimensional analysis is performed to obtain dimensionless pi terms or groups. According to the principles of dimensional analysis, any prototype can be described by a series of these terms or groups that describe the behaviour of the system. Using suitable pi terms or groups, it is possible to develop a similar set of pi terms for a model that has the same dimensional relationships. [8] In other words, pi terms provide a shortcut to developing a model representing a certain prototype. Common dimensionless groups in fluid mechanics include:

The origins of dimensional analysis have been disputed by historians. [9] [10]

The first written application of dimensional analysis has been credited to an article of François Daviet at the Turin Academy of Science. Daviet had the master Lagrange as teacher. His fundamental works are contained in acta of the Academy dated 1799. [10]

This led to the conclusion that meaningful laws must be homogeneous equations in their various units of measurement, a result which was eventually later formalized in the Buckingham π theorem. Simeon Poisson also treated the same problem of the parallelogram law by Daviet, in his treatise of 1811 and 1833 (vol I, p. 39). [11] In the second edition of 1833, Poisson explicitly introduces the term dimension instead of the Daviet homogeneity.

In 1822, the important Napoleonic scientist Joseph Fourier made the first credited important contributions [12] based on the idea that physical laws like F = ma should be independent of the units employed to measure the physical variables.

James Clerk Maxwell played a major role in establishing modern use of dimensional analysis by distinguishing mass, length, and time as fundamental units, while referring to other units as derived. [13] Although Maxwell defined length, time and mass to be "the three fundamental units", he also noted that gravitational mass can be derived from length and time by assuming a form of Newton's law of universal gravitation in which the gravitational constant G is taken as unity, thereby defining M = T −2 L 3 . [14] By assuming a form of Coulomb's law in which Coulomb's constant ke is taken as unity, Maxwell then determined that the dimensions of an electrostatic unit of charge were Q = T −1 L 3/2 M 1/2 , [15] which, after substituting his M = T −2 L 3 equation for mass, results in charge having the same dimensions as mass, viz. Q = T −2 L 3 .

Dimensional analysis is also used to derive relationships between the physical quantities that are involved in a particular phenomenon that one wishes to understand and characterize. It was used for the first time (Pesic 2005) in this way in 1872 by Lord Rayleigh, who was trying to understand why the sky is blue. Rayleigh first published the technique in his 1877 book The Theory of Sound. [16]

The original meaning of the word dimension, in Fourier's Theorie de la Chaleur, was the numerical value of the exponents of the base units. For example, acceleration was considered to have the dimension 1 with respect to the unit of length, and the dimension −2 with respect to the unit of time. [17] This was slightly changed by Maxwell, who said the dimensions of acceleration are T −2 L, instead of just the exponents. [18]

The Buckingham π theorem describes how every physically meaningful equation involving n variables can be equivalently rewritten as an equation of nm dimensionless parameters, where m is the rank of the dimensional matrix. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables.

A dimensional equation can have the dimensions reduced or eliminated through nondimensionalization, which begins with dimensional analysis, and involves scaling quantities by characteristic units of a system or natural units of nature. This gives insight into the fundamental properties of the system, as illustrated in the examples below.

Definition Edit

The dimension of a physical quantity can be expressed as a product of the basic physical dimensions such as length, mass and time, each raised to a rational power. The dimension of a physical quantity is more fundamental than some scale unit used to express the amount of that physical quantity. For example, mass is a dimension, while the kilogram is a particular scale unit chosen to express a quantity of mass. Except for natural units, the choice of scale is cultural and arbitrary.

There are many possible choices of basic physical dimensions. The SI standard recommends the usage of the following dimensions and corresponding symbols: time (T), length (L), mass (M), electric current (I), absolute temperature (Θ), amount of substance (N) and luminous intensity (J). The symbols are by convention usually written in roman sans serif typeface. [19] Mathematically, the dimension of the quantity Q is given by

where a, b, c, d, e, f, g are the dimensional exponents. Other physical quantities could be defined as the base quantities, as long as they form a linearly independent basis – for instance, one could replace the dimension (I) of electric current of the SI basis with a dimension (Q) of electric charge, since Q = TI.

As examples, the dimension of the physical quantity speed v is

and the dimension of the physical quantity force F is

The unit chosen to express a physical quantity and its dimension are related, but not identical concepts. The units of a physical quantity are defined by convention and related to some standard e.g., length may have units of metres, feet, inches, miles or micrometres but any length always has a dimension of L, no matter what units of length are chosen to express it. Two different units of the same physical quantity have conversion factors that relate them. For example, 1 in = 2.54 cm in this case 2.54 cm/in is the conversion factor, which is itself dimensionless. Therefore, multiplying by that conversion factor does not change the dimensions of a physical quantity.

There are also physicists that have cast doubt on the very existence of incompatible fundamental dimensions of physical quantity, [20] although this does not invalidate the usefulness of dimensional analysis.

Mathematical properties Edit

The dimensions that can be formed from a given collection of basic physical dimensions, such as T, L, and M, form an abelian group: The identity is written as 1 [ citation needed ] L 0 = 1 , and the inverse of L is 1/L or L −1 . L raised to any rational power p is a member of the group, having an inverse of L −p or 1/L p . The operation of the group is multiplication, having the usual rules for handling exponents ( L n × L m = L n+m ).

This group can be described as a vector space over the rational numbers, with the dimensional symbol T i L j M k corresponding to the vector (i, j, k) . When physical measured quantities (be they like-dimensioned or unlike-dimensioned) are multiplied or divided by one other, their dimensional units are likewise multiplied or divided this corresponds to addition or subtraction in the vector space. When measurable quantities are raised to a rational power, the same is done to the dimensional symbols attached to those quantities this corresponds to scalar multiplication in the vector space.

A basis for such a vector space of dimensional symbols is called a set of base quantities, and all other vectors are called derived units. As in any vector space, one may choose different bases, which yields different systems of units (e.g., choosing whether the unit for charge is derived from the unit for current, or vice versa).

The group identity, the dimension of dimensionless quantities, corresponds to the origin in this vector space.

The set of units of the physical quantities involved in a problem correspond to a set of vectors (or a matrix). The nullity describes some number (e.g., m) of ways in which these vectors can be combined to produce a zero vector. These correspond to producing (from the measurements) a number of dimensionless quantities, <>1, . πm>. (In fact these ways completely span the null subspace of another different space, of powers of the measurements.) Every possible way of multiplying (and exponentiating) together the measured quantities to produce something with the same units as some derived quantity X can be expressed in the general form

Consequently, every possible commensurate equation for the physics of the system can be rewritten in the form

Knowing this restriction can be a powerful tool for obtaining new insight into the system.

Mechanics Edit

The dimension of physical quantities of interest in mechanics can be expressed in terms of base dimensions T, L, and M – these form a 3-dimensional vector space. This is not the only valid choice of base dimensions, but it is the one most commonly used. For example, one might choose force, length and mass as the base dimensions (as some have done), with associated dimensions F, L, M this corresponds to a different basis, and one may convert between these representations by a change of basis. The choice of the base set of dimensions is thus a convention, with the benefit of increased utility and familiarity. The choice of base dimensions is not entirely arbitrary, because they must form a basis: they must span the space, and be linearly independent.

For example, F, L, M form a set of fundamental dimensions because they form a basis that is equivalent to T, L, M: the former can be expressed as [F = LM/T 2 ], L, M, while the latter can be expressed as [T = (LM/F) 1/2 ], L, M.

On the other hand, length, velocity and time (T, L, V) do not form a set of base dimensions for mechanics, for two reasons:

  • There is no way to obtain mass – or anything derived from it, such as force – without introducing another base dimension (thus, they do not span the space).
  • Velocity, being expressible in terms of length and time (V = L/T), is redundant (the set is not linearly independent).

Other fields of physics and chemistry Edit

Depending on the field of physics, it may be advantageous to choose one or another extended set of dimensional symbols. In electromagnetism, for example, it may be useful to use dimensions of T, L, M and Q, where Q represents the dimension of electric charge. In thermodynamics, the base set of dimensions is often extended to include a dimension for temperature, Θ. In chemistry, the amount of substance (the number of molecules divided by the Avogadro constant, ≈ 6.02 × 10 23 mol −1 ) is also defined as a base dimension, N. In the interaction of relativistic plasma with strong laser pulses, a dimensionless relativistic similarity parameter, connected with the symmetry properties of the collisionless Vlasov equation, is constructed from the plasma-, electron- and critical-densities in addition to the electromagnetic vector potential. The choice of the dimensions or even the number of dimensions to be used in different fields of physics is to some extent arbitrary, but consistency in use and ease of communications are common and necessary features.

Polynomials and transcendental functions Edit

Scalar arguments to transcendental functions such as exponential, trigonometric and logarithmic functions, or to inhomogeneous polynomials, must be dimensionless quantities. (Note: this requirement is somewhat relaxed in Siano's orientational analysis described below, in which the square of certain dimensioned quantities are dimensionless.)

While most mathematical identities about dimensionless numbers translate in a straightforward manner to dimensional quantities, care must be taken with logarithms of ratios: the identity log(a/b) = log a − log b, where the logarithm is taken in any base, holds for dimensionless numbers a and b, but it does not hold if a and b are dimensional, because in this case the left-hand side is well-defined but the right-hand side is not.

Similarly, while one can evaluate monomials (x n ) of dimensional quantities, one cannot evaluate polynomials of mixed degree with dimensionless coefficients on dimensional quantities: for x 2 , the expression (3 m) 2 = 9 m 2 makes sense (as an area), while for x 2 + x, the expression (3 m) 2 + 3 m = 9 m 2 + 3 m does not make sense.

However, polynomials of mixed degree can make sense if the coefficients are suitably chosen physical quantities that are not dimensionless. For example,

This is the height to which an object rises in time t if the acceleration of gravity is 9.8 meters per second per second and the initial upward speed is 500 meters per second . It is not necessary for t to be in seconds. For example, suppose t = 0.01 minutes. Then the first term would be

Incorporating units Edit

The value of a dimensional physical quantity Z is written as the product of a unit [Z] within the dimension and a dimensionless numerical factor, n. [21]

When like-dimensioned quantities are added or subtracted or compared, it is convenient to express them in consistent units so that the numerical values of these quantities may be directly added or subtracted. But, in concept, there is no problem adding quantities of the same dimension expressed in different units. For example, 1 meter added to 1 foot is a length, but one cannot derive that length by simply adding 1 and 1. A conversion factor, which is a ratio of like-dimensioned quantities and is equal to the dimensionless unity, is needed:

Only in this manner is it meaningful to speak of adding like-dimensioned quantities of differing units.

Position vs displacement Edit

Some discussions of dimensional analysis implicitly describe all quantities as mathematical vectors. (In mathematics scalars are considered a special case of vectors [ citation needed ] vectors can be added to or subtracted from other vectors, and, inter alia, multiplied or divided by scalars. If a vector is used to define a position, this assumes an implicit point of reference: an origin. While this is useful and often perfectly adequate, allowing many important errors to be caught, it can fail to model certain aspects of physics. A more rigorous approach requires distinguishing between position and displacement (or moment in time versus duration, or absolute temperature versus temperature change).

Consider points on a line, each with a position with respect to a given origin, and distances among them. Positions and displacements all have units of length, but their meaning is not interchangeable:

  • adding two displacements should yield a new displacement (walking ten paces then twenty paces gets you thirty paces forward),
  • adding a displacement to a position should yield a new position (walking one block down the street from an intersection gets you to the next intersection),
  • subtracting two positions should yield a displacement,
  • but one may not add two positions.

This illustrates the subtle distinction between affine quantities (ones modeled by an affine space, such as position) and vector quantities (ones modeled by a vector space, such as displacement).

  • Vector quantities may be added to each other, yielding a new vector quantity, and a vector quantity may be added to a suitable affine quantity (a vector space acts on an affine space), yielding a new affine quantity.
  • Affine quantities cannot be added, but may be subtracted, yielding relative quantities which are vectors, and these relative differences may then be added to each other or to an affine quantity.

Properly then, positions have dimension of affine length, while displacements have dimension of vector length. To assign a number to an affine unit, one must not only choose a unit of measurement, but also a point of reference, while to assign a number to a vector unit only requires a unit of measurement.

Thus some physical quantities are better modeled by vectorial quantities while others tend to require affine representation, and the distinction is reflected in their dimensional analysis.

This distinction is particularly important in the case of temperature, for which the numeric value of absolute zero is not the origin 0 in some scales. For absolute zero,

−273.15 °C ≘ 0 K = 0 °R ≘ −459.67 °F,

where the symbol ≘ means corresponds to, since although these values on the respective temperature scales correspond, they represent distinct quantities in the same way that the distances from distinct starting points to the same end point are distinct quantities, and cannot in general be equated.

For temperature differences,

(Here °R refers to the Rankine scale, not the Réaumur scale). Unit conversion for temperature differences is simply a matter of multiplying by, e.g., 1 °F / 1 K (although the ratio is not a constant value). But because some of these scales have origins that do not correspond to absolute zero, conversion from one temperature scale to another requires accounting for that. As a result, simple dimensional analysis can lead to errors if it is ambiguous whether 1 K means the absolute temperature equal to −272.15 °C, or the temperature difference equal to 1 °C.

Orientation and frame of reference Edit

Similar to the issue of a point of reference is the issue of orientation: a displacement in 2 or 3 dimensions is not just a length, but is a length together with a direction. (This issue does not arise in 1 dimension, or rather is equivalent to the distinction between positive and negative.) Thus, to compare or combine two dimensional quantities in a multi-dimensional space, one also needs an orientation: they need to be compared to a frame of reference.

This leads to the extensions discussed below, namely Huntley's directed dimensions and Siano's orientational analysis.

A simple example: period of a harmonic oscillator Edit

What is the period of oscillation T of a mass m attached to an ideal linear spring with spring constant k suspended in gravity of strength g ? That period is the solution for T of some dimensionless equation in the variables T , m , k , and g . The four quantities have the following dimensions: T [T] m [M] k [M/T 2 ] and g [L/T 2 ]. From these we can form only one dimensionless product of powers of our chosen variables, G 1 > = T 2 k / m k/m> [T 2 · M/T 2 / M = 1] , and putting G 1 = C =C> for some dimensionless constant C gives the dimensionless equation sought. The dimensionless product of powers of variables is sometimes referred to as a dimensionless group of variables here the term "group" means "collection" rather than mathematical group. They are often called dimensionless numbers as well.

Note that the variable g does not occur in the group. It is easy to see that it is impossible to form a dimensionless product of powers that combines g with k , m , and T , because g is the only quantity that involves the dimension L. This implies that in this problem the g is irrelevant. Dimensional analysis can sometimes yield strong statements about the irrelevance of some quantities in a problem, or the need for additional parameters. If we have chosen enough variables to properly describe the problem, then from this argument we can conclude that the period of the mass on the spring is independent of g : it is the same on the earth or the moon. The equation demonstrating the existence of a product of powers for our problem can be written in an entirely equivalent way: T = κ m k >>> , for some dimensionless constant κ (equal to C >> from the original dimensionless equation).

When faced with a case where dimensional analysis rejects a variable ( g , here) that one intuitively expects to belong in a physical description of the situation, another possibility is that the rejected variable is in fact relevant, but that some other relevant variable has been omitted, which might combine with the rejected variable to form a dimensionless quantity. That is, however, not the case here.

When dimensional analysis yields only one dimensionless group, as here, there are no unknown functions, and the solution is said to be "complete" – although it still may involve unknown dimensionless constants, such as κ .

A more complex example: energy of a vibrating wire Edit

Consider the case of a vibrating wire of length (L) vibrating with an amplitude A (L). The wire has a linear density ρ (M/L) and is under tension s (LM/T 2 ), and we want to know the energy E (L 2 M/T 2 ) in the wire. Let π1 and π2 be two dimensionless products of powers of the variables chosen, given by

The linear density of the wire is not involved. The two groups found can be combined into an equivalent form as an equation

where F is some unknown function, or, equivalently as

where f is some other unknown function. Here the unknown function implies that our solution is now incomplete, but dimensional analysis has given us something that may not have been obvious: the energy is proportional to the first power of the tension. Barring further analytical analysis, we might proceed to experiments to discover the form for the unknown function f. But our experiments are simpler than in the absence of dimensional analysis. We'd perform none to verify that the energy is proportional to the tension. Or perhaps we might guess that the energy is proportional to , and so infer that E = ℓs . The power of dimensional analysis as an aid to experiment and forming hypotheses becomes evident.

The power of dimensional analysis really becomes apparent when it is applied to situations, unlike those given above, that are more complicated, the set of variables involved are not apparent, and the underlying equations hopelessly complex. Consider, for example, a small pebble sitting on the bed of a river. If the river flows fast enough, it will actually raise the pebble and cause it to flow along with the water. At what critical velocity will this occur? Sorting out the guessed variables is not so easy as before. But dimensional analysis can be a powerful aid in understanding problems like this, and is usually the very first tool to be applied to complex problems where the underlying equations and constraints are poorly understood. In such cases, the answer may depend on a dimensionless number such as the Reynolds number, which may be interpreted by dimensional analysis.

A third example: demand versus capacity for a rotating disc Edit

Consider the case of a thin, solid, parallel-sided rotating disc of axial thickness t (L) and radius R (L). The disc has a density ρ (M/L 3 ), rotates at an angular velocity ω (T −1 ) and this leads to a stress S (T −2 L −1 M) in the material. There is a theoretical linear elastic solution, given by Lame, to this problem when the disc is thin relative to its radius, the faces of the disc are free to move axially, and the plane stress constitutive relations can be assumed to be valid. As the disc becomes thicker relative to the radius then the plane stress solution breaks down. If the disc is restrained axially on its free faces then a state of plane strain will occur. However, if this is not the case then the state of stress may only be determined though consideration of three-dimensional elasticity and there is no known theoretical solution for this case. An engineer might, therefore, be interested in establishing a relationship between the five variables. Dimensional analysis for this case leads to the following (5 − 3 = 2) non-dimensional groups:

demand/capacity = ρR 2 ω 2 /S thickness/radius or aspect ratio = t/R

Through the use of numerical experiments using, for example, the finite element method, the nature of the relationship between the two non-dimensional groups can be obtained as shown in the figure. As this problem only involves two non-dimensional groups, the complete picture is provided in a single plot and this can be used as a design/assessment chart for rotating discs [22]

Huntley's extension: directed dimensions and quantity of matter Edit

Huntley (Huntley 1967) has pointed out that a dimensional analysis can become more powerful by discovering new independent dimensions in the quantities under consideration, thus increasing the rank m of the dimensional matrix. He introduced two approaches to doing so:

  • The magnitudes of the components of a vector are to be considered dimensionally independent. For example, rather than an undifferentiated length dimension L, we may have Lx represent dimension in the x-direction, and so forth. This requirement stems ultimately from the requirement that each component of a physically meaningful equation (scalar, vector, or tensor) must be dimensionally consistent.
  • Mass as a measure of the quantity of matter is to be considered dimensionally independent from mass as a measure of inertia.

With these four quantities, we may conclude that the equation for the range R may be written:

In his second approach, Huntley holds that it is sometimes useful (e.g., in fluid mechanics and thermodynamics) to distinguish between mass as a measure of inertia (inertial mass), and mass as a measure of the quantity of matter. Quantity of matter is defined by Huntley as a quantity (a) proportional to inertial mass, but (b) not implicating inertial properties. No further restrictions are added to its definition.

For example, consider the derivation of Poiseuille's Law. We wish to find the rate of mass flow of a viscous fluid through a circular pipe. Without drawing distinctions between inertial and substantial mass we may choose as the relevant variables

There are three fundamental variables so the above five equations will yield two dimensionless variables which we may take to be π 1 = m ˙ / η r =>/eta r> and π 2 = p x ρ r 5 / m ˙ 2 =p_ > ho r^<5>/>^<2>> and we may express the dimensional equation as

Huntley's recognition of quantity of matter as an independent quantity dimension is evidently successful in the problems where it is applicable, but his definition of quantity of matter is open to interpretation, as it lacks specificity beyond the two requirements (a) and (b) he postulated for it. For a given substance, the SI dimension amount of substance, with unit mole, does satisfy Huntley's two requirements as a measure of quantity of matter, and could be used as a quantity of matter in any problem of dimensional analysis where Huntley's concept is applicable.

Huntley's concept of directed length dimensions however has some serious limitations:

  • It does not deal well with vector equations involving the cross product,
  • nor does it handle well the use of angles as physical variables.

It also is often quite difficult to assign the L, L x , L y , L z , symbols to the physical variables involved in the problem of interest. He invokes a procedure that involves the "symmetry" of the physical problem. This is often very difficult to apply reliably: It is unclear as to what parts of the problem that the notion of "symmetry" is being invoked. Is it the symmetry of the physical body that forces are acting upon, or to the points, lines or areas at which forces are being applied? What if more than one body is involved with different symmetries?

Consider the spherical bubble attached to a cylindrical tube, where one wants the flow rate of air as a function of the pressure difference in the two parts. What are the Huntley extended dimensions of the viscosity of the air contained in the connected parts? What are the extended dimensions of the pressure of the two parts? Are they the same or different? These difficulties are responsible for the limited application of Huntley's directed length dimensions to real problems.

Siano's extension: orientational analysis Edit

Angles are, by convention, considered to be dimensionless quantities. As an example, consider again the projectile problem in which a point mass is launched from the origin (x, y) = (0, 0) at a speed v and angle θ above the x-axis, with the force of gravity directed along the negative y-axis. It is desired to find the range R , at which point the mass returns to the x-axis. Conventional analysis will yield the dimensionless variable π = R g/v 2 , but offers no insight into the relationship between R and θ .

Siano (1985-I, 1985-II) has suggested that the directed dimensions of Huntley be replaced by using orientational symbols 1x 1y 1z to denote vector directions, and an orientationless symbol 10. Thus, Huntley's L x becomes L1 x with L specifying the dimension of length, and 1x specifying the orientation. Siano further shows that the orientational symbols have an algebra of their own. Along with the requirement that 1i −1 = 1i , the following multiplication table for the orientation symbols results:

Note that the orientational symbols form a group (the Klein four-group or "Viergruppe"). In this system, scalars always have the same orientation as the identity element, independent of the "symmetry of the problem". Physical quantities that are vectors have the orientation expected: a force or a velocity in the z-direction has the orientation of 1z . For angles, consider an angle θ that lies in the z-plane. Form a right triangle in the z-plane with θ being one of the acute angles. The side of the right triangle adjacent to the angle then has an orientation 1x and the side opposite has an orientation 1y . Since (using

to indicate orientational equivalence) tan(θ) = θ + .

1y/1x we conclude that an angle in the xy-plane must have an orientation 1y/1x = 1z , which is not unreasonable. Analogous reasoning forces the conclusion that sin(θ) has orientation 1z while cos(θ) has orientation 10. These are different, so one concludes (correctly), for example, that there are no solutions of physical equations that are of the form a cos(θ) + b sin(θ) , where a and b are real scalars. Note that an expression such as sin ⁡ ( θ + π / 2 ) = cos ⁡ ( θ ) is not dimensionally inconsistent since it is a special case of the sum of angles formula and should properly be written:

The assignment of orientational symbols to physical quantities and the requirement that physical equations be orientationally homogeneous can actually be used in a way that is similar to dimensional analysis to derive a little more information about acceptable solutions of physical problems. In this approach one sets up the dimensional equation and solves it as far as one can. If the lowest power of a physical variable is fractional, both sides of the solution is raised to a power such that all powers are integral. This puts it into "normal form". The orientational equation is then solved to give a more restrictive condition on the unknown powers of the orientational symbols, arriving at a solution that is more complete than the one that dimensional analysis alone gives. Often the added information is that one of the powers of a certain variable is even or odd.

As an example, for the projectile problem, using orientational symbols, θ , being in the xy-plane will thus have dimension 1z and the range of the projectile R will be of the form:

It is seen that the Taylor series of sin(θ) and cos(θ) are orientationally homogeneous using the above multiplication table, while expressions like cos(θ) + sin(θ) and exp(θ) are not, and are (correctly) deemed unphysical.

Siano's orientational analysis is compatible with the conventional conception of angular quantities as being dimensionless, and within orientational analysis, the radian may still be considered a dimensionless unit. The orientational analysis of a quantity equation is carried out separately from the ordinary dimensional analysis, yielding information that supplements the dimensional analysis.

Constants Edit

Formalisms Edit

Paradoxically, dimensional analysis can be a useful tool even if all the parameters in the underlying theory are dimensionless, e.g., lattice models such as the Ising model can be used to study phase transitions and critical phenomena. Such models can be formulated in a purely dimensionless way. As we approach the critical point closer and closer, the distance over which the variables in the lattice model are correlated (the so-called correlation length, ξ ) becomes larger and larger. Now, the correlation length is the relevant length scale related to critical phenomena, so one can, e.g., surmise on "dimensional grounds" that the non-analytical part of the free energy per lattice site should be ∼ 1 / ξ d > where d is the dimension of the lattice.

It has been argued by some physicists, e.g., M. J. Duff, [20] [23] that the laws of physics are inherently dimensionless. The fact that we have assigned incompatible dimensions to Length, Time and Mass is, according to this point of view, just a matter of convention, borne out of the fact that before the advent of modern physics, there was no way to relate mass, length, and time to each other. The three independent dimensionful constants: c, ħ, and G, in the fundamental equations of physics must then be seen as mere conversion factors to convert Mass, Time and Length into each other.

Just as in the case of critical properties of lattice models, one can recover the results of dimensional analysis in the appropriate scaling limit e.g., dimensional analysis in mechanics can be derived by reinserting the constants ħ, c, and G (but we can now consider them to be dimensionless) and demanding that a nonsingular relation between quantities exists in the limit c → ∞ , ℏ → 0 and G → 0 . In problems involving a gravitational field the latter limit should be taken such that the field stays finite.

Following are tables of commonly occurring expressions in physics, related to the dimensions of energy, momentum, and force. [24] [25] [26]

SI units Edit

Natural units Edit

If c = ħ = 1 , where c is the speed of light and ħ is the reduced Planck constant, and a suitable fixed unit of energy is chosen, then all quantities of time T, length L and mass M can be expressed (dimensionally) as a power of energy E, because length, mass and time can be expressed using speed v, action S, and energy E: [26]

though speed and action are dimensionless ( v = c = 1 and S = ħ = 1 ) – so the only remaining quantity with dimension is energy. In terms of powers of dimensions:

This is particularly useful in particle physics and high energy physics, in which case the energy unit is the electron volt (eV). Dimensional checks and estimates become very simple in this system.

However, if electric charges and currents are involved, another unit to be fixed is for electric charge, normally the electron charge e though other choices are possible.

Quantity p, q, r powers of energy n
power of energy
p q r n
Action, S −1 2 1 0
Speed, v −1 1 0 0
Mass, M 0 0 1 1
Length, L 0 1 0 −1
Time, t 1 0 0 −1
Momentum, p −1 1 1 1
Energy, E −2 2 1 1

Related areas of math Edit

Programming languages Edit

Dimensional correctness as part of type checking has been studied since 1977. [27] Implementations for Ada [28] and C++ [29] were described in 1985 and 1988. Kennedy's 1996 thesis describes an implementation in Standard ML, [30] and later in F#. [31] There are implementations for Haskell, [32] OCaml, [33] and Rust, [34] Python, [35] and a code checker for Fortran. [36]
Griffioen's 2019 thesis extended Kennedy's Hindley–Milner type system to support Hart's matrices. [37] [38]


LONG-TERM EFFECTS OF MUSIC ON THE BRAIN

The original experiments on adults exposed to Mozart's music were of short duration only. In related experiments 15 , long-term effects of music were studied in groups of pre-school children aged 3-4 years who were given keyboard music lessons for six months, during which time they studied pitch intervals, fingering techniques, sight reading, musical notation and playing from memory. At the end of training all the children were able to perform simple melodies by Beethoven and Mozart. When they did they were then subjected to spatial-temporal reasoning tests calibrated for age, and their performance was more than 30% better than that of children of similar age given either computer lessons for 6 months or no special training (P π.001). The improvement was limited to spatial-temporal reasoning there was no effect on spatial recognition. The effect lasted unchanged for 24 hours after the end of the music lessons but the precise duration of the enhancement was not further explored. The longer duration of the effects than in previous reports was attributed to the length of exposure to music and the greater plasticity of the young brain. In further experiments of this kind it has been claimed that the enhancement of spatial-temporal reasoning in children after piano training has resulted in significantly greater scores in higher mathematics 16 .


Locating and quantifying geological uncertainty in three-dimensional models: Analysis of the Gippsland Basin, southeastern Australia

Geological three-dimensional (3D) models are constructed to reliably represent a given geological target. The reliability of a model is heavily dependent on the input data and is sensitive to uncertainty. This study examines the uncertainty introduced by geological orientation data by producing a suite of implicit 3d models generated from orientation measurements subjected to uncertainty simulations. The resulting uncertainty associated with different regions of the geological model can be located, quantified and visualised, providing a useful method to assess model reliability. The method is tested on a natural geological setting in the Gippsland Basin, southeastern Australia, where modelled geological surfaces are assessed for uncertainty. The concept of stratigraphic variability is introduced and analysis of the input data is performed using two uncertainty visualisation methods. Uncertainty visualisation through stratigraphic variability is designed to convey the complex concept of 3D model uncertainty to the geoscientist in an effective manner.

Uncertainty analysis determined that additional seismic information provides an effective means of constraining modelled geology and reducing uncertainty in regions proximal to the seismic sections. Improvements to the reliability of high uncertainty regions achieved using information gathered from uncertainty visualisations are quantified in a comparative case study. Uncertainty in specific model locations is identified and attributed to possible disagreements between seismic and isopach data. Further improvements to and additional sources of data for the model are proposed based on this information. Finally, a method of introducing stratigraphic variability values as geological constraints for geophysical inversion is presented.

Highlights

► We examine the effects of data input error on three-dimensional geological model reliability. ► We demonstrate how model uncertainty can be identified, quantified and visualised. ► We apply a visualisation technique to a natural geological case study in southeastern Australia. ► We show how the appropriate additional data can improve model reliability. ► We suggest how further improvements can be made to the Gippsland Basin model.


Cube and Dice Reasoning Sample Questions

Question 1: What number will be opposite to 2?

Solution: It is a standard dice as no of any adjacent sides are 7. As, standard dice, opposite no. of 2 will be

Ans is 5, (sum of opposite side is 7)

Question 2: What no will be opposite to 4?

Solution : It is an ordinary dice as the sum of right and left side is 7. So, opposite no. of 4 can be – 1, 3 or 6.

So, the answer is → can’t be determined.

Question 3: What no will be opposite to 3?

Solution : We have to check the possibility.

Here, the no of dice is1, 2, 3, 4, 5, 6 As per above diagram 3, 2 and 4. Can’t be any of the opposite faces of 2.

So, there are all eliminated only 1, 5 or 6 are possible numbers of opposite faces of 3.

Then option b is correct i.e. 1/5/6.

Question 4: What is the example of a standard dice?

Solution : As per definition of standard dice, any of the two opposite faces of dice must be 7.

So, only in dice A the sum of two adjacent faces is 7.

Hence, the correct answer is A.

Question 5: What is the opposite face of “Red”?

Solution: Blue is common in both dices, so putting blue as a constant term we have to rotate two dices by clockwise and anticlockwise direction, so we get the opposite side of red is yellow.

Hence, the correct answer is yellow.


Context for Use

Unit 2 is similar to a lab activity that might be given to a high school science class or an introductory college-level geoscience course. The science content from this unit targets those students who have a limited understanding of climate change, Thus, there is no need to introduce science content prior to implementing this unit. The context of Unit 2 incorporates societal issues (e.g., costs of rebuilding coastal communities) to engage students in learning about climate change. This unit is most effective when students work in groups of two to three. Computers should be available for students so they can collect, organize, and analyze online climate data. This page provides an overview of the activity, and student handouts are available and can be modified to address individual instructor needs.

Unit 2 also aligns with the K–12 Framework for Science Education and the Next Generation Science Standards (NGSS) as indicated in the table below. Please click on the image to enlarge.

In addition, Unit 2 also aligns with the following high school Common Core of State Standards (CCSS) for Reading and Writing in Science and Technical subjects, as indicated in the table below. Please click on the image to enlarge.


Other Common Calculus Errors

Jumping to conclusions about infinity. Some problems involving infinity can be solved using "the elementary arithmetic of infinity". Some students jump to the conclusion that all problems involving infinity can be solved by this sort of "elementary arithmetic," and so they guess all sorts of incorrect answers (mainly 0 or infinity) to such problems.

Here is an example of the "elementary arithmetic": If we use the equation cautiously, we can say (informally) that -- though perhaps it would be less misleading to write instead . (My thanks to Hans Aberg for this suggestion and for several other suggestions on this web page.) What this rule really means is that if you take a medium-sized number and divide it by an enormous number, you get a number very close to 0. For instance, without doing any real work, we can use this rule to conclude at a glance that

Thus, the problem has the answer 0. The problem does not have an answer in any analogous fashion we might say that is undefined. This does not mean that "Undefined" is the answer to any problem of the form . What it means, rather, is that each problem involving requires a separate analysis different problems of this type have different answers. For instance,

Those first two problems are fairly obvious the last problem takes more sophisticated analysis. Just guessing would not get you an answer of 1/2. (If you don't understand what is going on in the last problem, try graphing the functions and x on one display screen on your graphing calculator. That may provide a lot of insight, though it's not a proof.)

In a similar fashion, do not have quick and easy answers they too require more specialized and sophisticated analyses.

Here is a common error mentioned by Stuart Price: Some students seem to think that Their reasoning is this: "When , then Now compute Of course, this reasoning is just a bit too simplistic. You have to deal with both of the n's in the expression at the same time -- i.e., they both go to infinity simultaneously you can't figure that one goes to infinity and then the other goes to infinity. And in fact, if you let the other one go to infinity first, you'd get a different answer: So evidently the answer lies somewhere between 1 and ∞. That doesn't tell us much my point here is that easy methods do not work on this problem. The correct answer is a number that is near 2.718. (It's an important constant, known to mathematicians as "e".) There's no way you could get that by an easy method.

That reminds me of a related question that seems to bother many students: What is

The reason that a question arises at all is because is discontinuous at (0,0). Indeed, we have x 0 =1 for all x>0, and we have 0 y =0 for all y>0. And doesn't exist, because that expression means the limit of x y as the point (x,y) approaches (0,0) along all paths where x y is defined.

Nevertheless, many (most?) mathematicians will define 0 0 = 1, just for convenience, because that makes the most formulas work (and then they will note exceptions for formulas that require a different definition).

For instance, if we're working with polynomials or power series, p(x) = a0x 0 + a1x 1 + a2x 2 + a3x 3 + . + anx n + .
-- perhaps the most common place for 0 0 to arise -- then it's convenient to have 0 0 = 1, since a0x 0 needs to be equal to a0. The Binomial Theorem would be more complicated to write if we defined 0 0 any other way.

Problems with series. Sean Raleigh reports that the most common series error he has seen is this: If is a sequence converging to 0, then many students conclude (erroneously) that the series must be convergent (i.e., must add up to a finite number). Perhaps they hold that belief because it is true for most of the examples that they have seen. Most counterexamples are too advanced to be included in an elementary textbook. Of course, every calculus book gives the simple example of the harmonic series: 1 + (1/2) + (1/3) + (1/4) + . = ∞ but one single example of divergence does not seem to outweigh in the students' minds the many examples of convergence that they have seen.

Loss or misuse of constants of integration. The indefinite integral of a function involves an "arbitrary constant", and this causes confusion for many students, because the notation doesn't convey the concept very well. An expression such as really is supposed to represent an infinite collection of functions -- it represents all of the functions

Here is an example. The formula for Integration By Parts, in its briefest form, is that can be understood more easily as

∫u(x)v'(x)dx = u(x)v(x) - ∫u'(x)v(x)dx.

Now, that formula is correct, but it can easily be mishandled and can lead to errors. Here is one particularly amusing error: Plug and into the formula above. We get

∫(1/x)(1)dx = (1/x)(x) - ∫(-1/x 2 )(x)dx

Now, regardless of what you think is the value of you just have to subtract that amount from both sides of the preceding equation, to obtain Wait, how can that be. Well, if we're very careful, we realize that the two on the two sides of the last equation are not actually the same. What that last equation really says is

That is a true equation, if we choose the constants and appropriately -- i.e., if we choose them so that Thus, the two constants are not independent of each other -- they are not completely "arbitrary". Perhaps a more accurate explanation is this: The two expressions and do not actually represent individual functions rather, each of those expressions represents a set of functions.

    The expression [ ln|x| + C1 ] represents the set of all the functions of x that can be obtained by starting with the function and then adding a constant.

Some students manage to make this kind of error even with definite integrals. They start from the formula = which is correct but then when they "switch to definite integrals", they get the formula = which is not correct. If you really want to "switch to definite integrals", you need to think of that constant 1 as a special sort of function. When you switch to definite integrals, any function p(x) gets replaced by In particular, the constant function 1 is the function given by p(x)=1 for all x. So becomes or 0.

Some students may understand this better if we do the whole thing with definite integrals, right from the start. Let's use the formula

a b u(x)v′(x)dx = u(b)v(b) – u(a)v(a) – ∫ a b u′(x)v(x)dx.

Note that this formula has one more term than my previous boxed formula -- when we convert to the definite integral version, we replace it with Now plug in and We get

which (assuming 0 is not in the interval simplifies to

which is true -- i.e., there is no contradiction here.

Some students may be puzzled by the differences between the two versions of the Integration by Parts formula (in boxes, in the last few paragraphs). I will describe in a little more detail how you get from the definite integral formula (in the last box) to the indefinite integral formula (in the first box in this section). Think of a as a constant and b as a variable, and you'll get something like this:

Note that the term gets replaced by and the term "disappears" because it is constant. Finally, we can "absorb" the arbitrary constants into the indefinite integrals -- i.e., we don't need to write because any indefinite integral is only determined up to adding or subtracting a constant anyway. Thus, we arrive at the briefer formula = u(x)v(x) –

Handling constants of integration gets even more complicated in the first course on differential equations, and there are even more kinds of errors possible. I won't try to list all of them here, but here is the simplest and most common error that I've seen: In calculus, some students get the idea that you can just omit the "+C" in your intermediate computations, and then tack it on at the end of your answer, if you know which kinds of problems require an arbitrary constant. That will usually work in calculus, but it doesn't work in differential equations, because in differential equations the "C" can show up anywhere -- not necessarily as a "+C" at the end of the answer.

Here's a simple example: Let's solve the differential equation xy′+7=y (where y′ means dy/dx). One way to solve it is by the following steps:

    Rewrite the problem as y′ – (1/x)y = – 7/x, to show that it is linear.

Loss of differentials. This shows up both in differentiation and in integration. The "loss of differentials" is much like the "loss of invisible parentheses" discussed earlier in this document it is a type of sloppy writing in intermediate steps which leads to actual errors in the final answer.

When students first begin to learn to differentiate, they are always differentiating with respect to the same variable, and so they see no reason to mention that variable. Thus, in differentiating the function y = f(x) = 7x 3 +5x, they may correctly write

or they may incorrectly write "dy = 21x 2 +5." The omission of the "dx" from this last equation makes no real difference in the student's mind, and this slovenly omission may become a habit. But it will cause difficulties later in the course. In fact, I am starting to think that we could avoid a lot of difficulty if we discourage beginning calculus students from using the notations or Dy. If we require them to use the notation dy/dx , and penalize them for writing it as dy, we might save them a lot of headaches later.

The difficulty, of course, shows up when we arrive at the Chain Rule. Suddenly, the question is no longer "What is the derivative of y", but rather, "What is the derivative of y with respect to x? with respect to u? How are those two derivatives related?" The student who does not make a habit of distinguishing between dy/dx and dy/du in writing may also have difficulty distinguishing between them conceptually, and thus will have difficulty understanding the Chain Rule.

This also leads to difficulties with the "u-substitutions" rule, which is just the Chain Rule turned into a rule about integrals. For instance:

For the first three problems, the student is attempting to use the formula (which is a correct formula, but not directly applicable). However, the student has learned it incorrectly as Substitute or or into that formula to get the first three erroneous answers in the table above. The expressions and have very different meanings, but you're likely to confuse them if you write them both as

For the last problem in the table above, the student is attempting to use the formula = , which is a correct formula, but not relevant to the present problem. The student has probaby memorized that formula in the incorrect form = The expressions and have very different meanings, but you're likely to confuse them if you write them both as

Another correct way to write the rule about logarithms is . Since this expresses everything in terms of the variable x, it may make errors less likely. Admittedly, it is a complicated looking formula, but it is preferable to a wrong formula. The first, third, and fourth problems in the preceding table all require more complicated methods just using logarithms won't solve the problems for you. The problem of integrating actually requires a less complicated method -- i.e., without logarithms.

We should prohibit students from writing an integral sign without a matching differential. Just as any "(" must be matched with a ")", so too any integral sign must be matched with a "dx" or "du" or "dt" or whatever. The expression is unbalanced, and should be prohibited. If we're considering a substitution of u = 1+x 2 , then is very different from and so the expression is ambiguous and meaningless. If you write in one of your intermediate steps, you may forget whether it represents or and you may inadvertently switch from one to the other -- thus replacing one mathematical quantity with another to which it is not equal.

By the way, some students get confused about whether should be or Here is an answer. is always equal to ln|u|+C, but sometimes that answer can be simplified and sometimes it can't. In math, we generally prefer to write our answers in simplest form (and we sometimes insist on it). In those situations where we know that u will only take positive values (e.g., when or when the domain is restricted so that u can't be negative), then should be written as In those situations where we don't know whether u will be positive, we should write the answer as (But sometimes we omit the absolute value sign out of sheer laziness, justifying this with the excuse that we can make the domain smaller.)

These loss-of-differentials errors in differentiation and in integration can be caught easily by a bit of "dimensional analysis" (discussed earlier). To do that, it is useful to think in terms of " infinitesimals " -- i.e., numbers that are "infinitely small" but still not zero. Newton and Leibniz had infinitesimals in mind when they invented calculus 300 years ago, but they didn't know how to explain infinitesimals rigorously. Infinitesimals became unfashionable a century or two later, when rigorous epsilon-delta proofs were invented. If we use the real number system that most mathematicians use nowadays, there are no infinitesimals except 0. But in 1960 a logician named Abraham Robinson invented another kind of real number system that includes nonzero infinitesimals he found a way to back up the Newton-Leibniz intuition with rigorous proofs.

With the Newton-Leibniz-Robinson viewpoint, think of dx and dy as infinitesimals. Now, dy/dx is a quotient of two infinitely small numbers, so it could be a medium-sized number. Thus an equation such as dy/dx = 6x 2 could make sense. An equation such as dy = 6x 2 cannot possibly be correct -- the left side is infinitely small, and the right side is medium-sized.

The summation sign ∑ means add together finitely or countably many things -- for instance,

but ∑ generally is not used for adding uncountably many things.

(Occasionally it is so used: The sum of an arbitrary collection of nonnegative real numbers is the sup of the sums of finitely many members of that collection. But all the interesting action is happening on a countable set. It can be proved that if more than countably many of those numbers being added are nonzero, the sum must be infinity. Also, there may be other, more esoteric uses for the symbol ∑. But this web page is intended for undergraduates.)

However, in some sense we add together uncountably many things when we use an integral. An equation such as = says that we add together uncountably many infinitesimals, and we get a medium-sized number. An equation such as ∫ 3x 2 = x 3 +C couldn't possibly be right -- it says we add together uncountably many medium-sized numbers and get a medium-sized number.

A related difficulty is in trying to understand what "differentials" are. Most recent calculus books have a few pages on this topic, shortly before or after the Chain Rule. I am very sorry that the authors of calculus books have chosen to cover this topic at this point in the book. I think they are making a big mistake in doing so. When I teach calculus, I skip that section, with the intention of covering it in a later semester. Here is why:

When y=f(x), then dy=f′(x)dx is really a function of two variables-- it is a function of both x and dx. But in many calculus textbooks, that fact is not confronted directly it is swept under the rug and hidden. Several hundred pages later in most calculus textbooks, we are introduced to functions of two variables, and given a decent notation for them -- e.g., we may have z = h(u,v). At this point the student may begin to understand functions of two variables, and we have partial derivatives etc. But before this point, we are not given any good notations for a function of two variables. Our beginning math students have difficulty enough with abstractions even when they are provided with decent notation how can we expect them to think abstractly without the notation? Thus, when I teach calculus, I describe "dx" and "dy" as "pieces of the notation with no independent meanings of their own. I think that this approach is much kinder to the beginning students.

This web page was selected as the "cool math web page of the week", for the week of May 22, 2002, by KaBoL.