Useful Mathematics. 1. Multivariable Calculus. 1.1 Taylor s Theorem. Monday, 13 May 2013

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1 Useful Mathematics Monday, 13 May 013 Physics 111 In recent years I have observed a reticence among a subpopulation of students to dive into mathematics when the occasion arises in theoretical mechanics and it arises from the very beginning. Rather than merely posting this to the course page and assuming that you will take the time to go through it, I am sending it to you now so that you can consult it as often as necessary over the summer to enable you to arrive in September ready to take the plunge. I don t believe that there is anything here that you have not seen before; but it is certainly possible that did not anticipate its etreme usefulness in physics and so perhaps scrimped on the ink necessary to tattoo these ideas firmly to your brain. If so, it s time to get out that needle Multivariable Calculus 1.1 Taylor s Theorem Taylor s theorem in one dimension provides a way of approimating a sufficiently differentiable function in the neighborhood of a point with a polynomial of degree n that matches the first n derivatives of the function at the point of epansion: f() f () f ( 0 ) + f ( 0 )( 0 ) + f ( 0 ) ( 0) + f ( 0 ) ( 0) 3 +! 3! (1) In many circumstances, the linear approimation is quite sufficient to characterie the function in the neighborhood of interest, because the magnitude of the remaining terms is much smaller than the term f ( 0 )( 0 ). In this region, it is common in physics to use the shorthand differential notation, df = f ( 0 )d () Figure 1: The function sin epanded about the point 0 = 0 in Taylor series through order 1, 3, and 5. For a smooth function such as sin, the Taylor series epansion converges rapidly, so that only a few terms are needed to approimate the function quite accurately. Physics of 11 Peter N. Saeta

2 1. MULTIVARIABLE CALCULUS 1. Useful Partial Differential Relations where df = f () f ( 0 ) indicates the distance from f 0 = f ( 0 ), and d the distance from 0. Strictly speaking, the equality is eact only in the limit as the differentials tend to ero, unless f () happens to be linear. Equation assumes that the first derivative eists at 0 and is a statement that the tangent line approimation is accurate in an open neighborhood around the point at which one evaluates the derivative. Thus, let no mathematician get his panties in a wad. Taylor s theorem can be generalied for functions of more than one variable: f (, y) = f ( 0, y 0 ) + f ( 0) + f (y y 0) + 1! [ f ( 0) + f ( 0)(y y 0 ) + f (y y 0) ] + where all derivatives are evaluated at the point ( 0, y 0 ). In the region surrounding this point for which the tangent plane is a sufficient approimation, one can use the shorthand notation df = f f d + dy (4) which epresses the fact that the value of the function f can be changed either by varying the value of or by varying the value of y (or both). In the limit as all differential quantities tend to ero, Eq. (4) epresses the eistence of the derivative of f at the point ( 0, y 0 ). (3) 1. Useful Partial Differential Relations It frequently happens that the independent variables of the function f (, y) of the previous section can be thought of as functions of one or more other independent variables. For eample, if and y represent spatial coordinates that vary in time, we may wish to know how f varies with time. If f has no eplicit time dependence, then we note that d = d dt dt, d y = dy dt dt since and y are independent. Substituting into Eq. (4) gives or more succinctly df = f d f dy dt + dt dt dt df dt = f d dt + f dy dt If f does depend on t eplicitly, then we must add this eplicit dependence to the above epression for the total derivative of f : df dt = f t + f d dt + f dy dt (5) (6) Physics 111 of 11 Peter N. Saeta

3 1. MULTIVARIABLE CALCULUS 1. Useful Partial Differential Relations We can generalie these epressions for the case of multiple independent variables in a straightforward manner. Let us suppose that both and y are really functions of u and v. We then substitute d = du + dv, dy = du + dv u v u v into Eq. (4) to obtain v [ f df = u + f u u ] [ f du + v + f v v u ] dv (7) Besides the basic relations developed above, there are a few more general properties of the derivatives of multivariable functions that are useful: ( ) ( ) ( ) [( ) = = ( ) / ( ) = t t ( ) / ( ) = ] 1 Reciprocity the reciprocal of a derivative is equal to the derivative with the parameters interchanged y Equality of mied partials when the firstorder partial derivatives eists and are continuous and differentiable in a neighborhood surrounding the point in question, the mied second-order partial derivatives are equal Parametric derivative the ratio of the variations of and y with t, holding fied, is just the variation of with y, holding fied Cycle rule to find out how varies with y when we hold fied, calculate the ratio of the changes of with variations of y and, and throw in a negative sign! Ecept for the cycle rule, these are reasonably straightforward. To derive the cycle rule, consider a quantity S(E,V ) that is a function of the two independent variables E and V. The total differential of S is thus S S ds = de + dv E V E This statement means that in the neighborhood ( ) of a point (E,V ) the function S( has) a tangent plane with slope in the E direction of S E and slope in the V direction of S V. E We can always find a direction in the plane for which S does not vary; this is the line of intersection between the tangent plane and the horiontal plane, S = constant. In this Physics of 11 Peter N. Saeta

4 . JACOBIAN DETERMINANTS direction, ds = 0, so that S S 0 = de + E V S S de = dv E V E E S / ( ) S = E S E V E dv The final partial derivative E the derivation. must be taken at fied S, since we have assumed ds = 0 in The cycle rule can be written in a variety of forms, including ( ) = 1 y ( ) = The latter form bears a passing (deceptive) resemblance to the chain rule for a function of a single variable, df [(t)] = df d dt d dt which has no minus sign, of course. Take care not to confuse them! y. Jacobian Determinants The Jacobian determinant transforms the generalied volume element from one set of coordinates to another. As an eample, consider an integral over the entire Cartesian plane of some function f (, y), A = f (, y)d dy The double integral epresses the limit of a sum of the values of the function f over the entire plane, with a weighting factor epressed by 1 d dy; i.e., weighting each little range of each coordinate equally. If we were to perform the same integral in plane polar coordinates, we would of course have A = f (r,θ)r dr dθ where the weighting factor is now r dr dθ. We need the etra factor of r to give an arc length r dθ. In other words d dy = r dr dθ Physics of 11 Peter N. Saeta

5 . JACOBIAN DETERMINANTS This result is most readily obtained geometrically from Fig., but to please Prof. Lagrange, we can also take a more algebraic approach by considering the variations in the y coordinates caused by variations in the r θ coordinates. Suppose we start at a point (, y) (r,θ) and consider the displacement that arises by a small variation in r. This is the vector ( ) dr, r r dr and similarly for a variation in θ, ( θ ) dθ, θ dθ For arbitrary coordinate transformations, these are not guaranteed to be orthogonal, although they are for the Cartesian-polar transformation we are using as an eample. In any case, the area of the patch swept out as r and θ are varied infinitesimally is given by the product of the lengths of these two vectors times the sine of the included angle. This is the just the magnitude of the cross product of the two vectors, or d dy = r θ θ r dr dθ Using the coordinate transformation epressions = r cosθ and y = r sinθ gives as epected. d dy = cosθ r cosθ ( r sinθ sinθ) dr dθ = r dr dθ The generaliation of this procedure to n dimensions yields the determinant of an n n matri of partial derivatives, which I illustrate here in 3 dimensions ξ η ζ d dy d = (, y, ) (ξ,η,ζ) dξdηdζ = ξ η ζ dξdηdζ (8) where the inner bars indicate the determinant and the outer bars indicate the absolute value. The generaliation to n dimensions is straightforward. ξ η ζ r d! dr Figure : The familiar area element in polar coordinates has sides dr and r dθ. Physics of 11 Peter N. Saeta

6 3. HYPERBOLIC FUNCTIONS 3. Hyperbolic Functions The symmetric and antisymmetric combinations of the eponential functions occur frequently in mathematical physics, and share an intimate connection with the more familiar trigonometric functions. From Euler s formula, 4 3 e iθ = cosθ + i sinθ cosh sinh it is possible to epress the trigonometric functions in terms of comple eponentials. Very important note: this is not scary! tanh sin = ei e i i cos = ei + e i tan = sin cos sinh = e e cosh = e + e tanh = sinh cosh sin = i sinhi cos = coshi tan = i tanhi Strictly speaking, the function cosh is the hyperbolic cosine of, but we typically call it the cosh of. By a somewhat twisted analogy, we pronounce sinh sinch and tanh tanch. The reciprocal functions are defined analogously to the normal trigonometric functions, as well: csch = 1 sinh Don t even try to pronounce csch. 1 sech = cosh coth = cosh sinh 3.1 Derivatives The hyperbolic trigonometric functions have derivatives very similar to regular trigonometric functions, only slightly simpler. They are dsinh = cosh d dtanh = sech d dcsch = coth csch d dcosh = sinh d dcoth = csch d dsech = sech tanh d Physics of 11 Peter N. Saeta

7 4. LAGRANGE S METHOD OF UNDETERMINED MULTIPLIERS 3. Pythagorean Theorem 3. Pythagorean Theorem The familiar epression of the Pythagorean theorem in trigonometric functions, sin + cos = 1, has a slightly different form in hyperbolic functions: cosh sinh = 1 You can work out others as well, typically by dividing this primary identity by something. For eample, tanh + sech = Taylor Series Epansions We will frequently require the behavior of these functions for small or large values of the argument. Here are their Taylor (or Laurent) epansions: sinh = + 3 3! + 5 5! csch = cosh = 1 +! ! sech = tanh = coth = Lagrange s Method of Undetermined Multipliers Lagrange s method of undetermined multipliers can be used to maimie (or minimie) a multivariable function subject to one or more constraints. It is very commonly used in mechanics, where surfaces, tracks, and other devices impose constraints on motion. To illustrate the method, consider a function f (, y) which we wish to maimie subject to the constraint g (, y) = 0. If g (, y) can be solved for y as a function of, it is possible to substitute this epression into f to obtain a function of a single variable. Sometimes this is the simplest way to proceed. But when g (, y) is complicated, or when there are more independent variables, Lagrange s method is often much simpler. Generally speaking, maimiing f will not be consistent with the constraint g (, y) = 0; the direction of greatest increase in f will walk us off the curve g (, y) = 0. The whole trick is to find the maimum value of f as we walk along the curve defined by g (, y) = 0. Suppose that the point (, y) happens to lie on the constraint curve. The direction of greatest change of the function g lies along g ; in the perpendicular direction g remains constant. If f is parallel to g, then it too remains constant along the curve of constraint. Hence, our constrained maimum satisfies the condition f g Physics of 11 Peter N. Saeta

8 4. LAGRANGE S METHOD OF UNDETERMINED MULTIPLIERS If the two gradients are parallel, then one is a scaled copy of the other. Let h(, y) = f (, y) λg (, y), where λ is an unknown constant which just allows us to scale the constraint derivative so it can match both the magnitude and direction of f. Then at the constrained maimum, (f λg ) = 0 (9) The method generalies easily to greater numbers of dimensions, and multiple constraints. For each constraint, one introduces an unknown multiplier and then requires f where all the constraint equations have the form g i ( ) = 0. n λ i g i = 0 (10) i=1 Eample 1: Sliced igloo The surface of an igloo is defined by + y + = 1 0 What are the coordinates (, y, ) of the highest point on the igloo (at greatest ) consistent with the constraint y = a? In particular, find the maimum value of subject to the constraint for a = 1/4. We first cast the equations in the standard form: f (, y) = 1 y (11) g (, y) = y a (1) Setting the gradient of f λg to ero requires that each partial derivative of this quantity vanish. This gives the equations = λ y (13) y = λ (14) Dividing these equations to eliminate λ gives a relation between and y that must be satisfied at the maimum, y = y = = y Substituting this condition into the constraint equation, Eq. (1), determines the coordinates of the constrained maimum, y 3 = a = y = (a/) 1/3 and = y/ Physics of 11 Peter N. Saeta

9 5. COORDINATE SYSTEMS where I have assumed that (, y) lies in the first quadrant. Substituting the epressions for and y into Eq. (11) gives the value of at the maimum, ( a ) /3 ( a ) /3 ( a ) /3 = 1 = 1 3 ( ) In particular, at a = 1/4 we obtain (, y, ) = 1, 1, Coordinate Systems Other things being equal, the simplest coordinate system is Cartesian coordinates, in which the position vector is given by r = ˆ + yŷ + ẑ. Sometimes the unit vectors are written i,j,k, or e i, where i = 1,,3, corresponding to, y,. The great advantage of Cartesian coordinates is that the basis unit vectors are independent of position. In a system with cylindrical symmetry we use polar coordinates (ρ,φ) in the y plane, and along the symmetry ais. The aimuthal angle φ (sometimes written θ) is measured from the ais, as indicated in Fig. 3. In terms of the Cartesian coordinates, ρ = + y φ = tan 1 (y/) = ρ cosφ y = ρ sinφ The third important system we will sometimes use is spherical coordinates. We will stick with the conventional physics definitions for spherical polar coordinates, which differs from that favored more recently by mathematicians. 1 Since the physics literature is written with θ being the angle from the north pole, and φ being the aimuthal angle from the ais, that s what we will use. r = + y + = r sinθ cosφ θ = cos 1 (/r ) φ = tan 1 (y/) y = r sinθ sinφ = r cosθ 1 For better or worse, physicists and mathematicians diverged a very long time ago in their notation of the spherical angles. Physicists have been using the conventions I describe here for at least 130 years, as judged by papers by Lord Rayleigh and other late nineteenth century physicists. Physics of 11 Peter N. Saeta

10 5. COORDINATE SYSTEMS r ϕ ρ y r cylindrical y Cartesian θ ϕ r r y spherical Figure 3: Coordinate systems Quantity Cartesians cylindricals sphericals dr d ˆ + dy ŷ + d ẑ dρ ˆρ + ρ dφ ˆφ + d ẑ dr ˆr + r dθ ˆθ + r sinθ dφ ˆφ ṙ ẋ ˆ + ẏ ŷ + ż ẑ ρ ˆρ + ρ φ ˆφ + ż ẑ ṙ ˆr + r θ ˆθ + r sinθ φ ˆφ V ˆ + ŷ + ẑ ρ ˆρ + 1 ρ φ ˆφ + ẑ r ˆr + 1 r θ ˆθ + 1 r sinθ φ ˆφ A A + A y + A A ρ ρ + 1 ρ A φ φ + A A r r + 1 A θ r θ + 1 A φ r sinθ φ Physics of 11 Peter N. Saeta

11 6. EXERCISES 6. Eercises Problem 1 Derive the Taylor series epansion of tanh through fifth order. Problem Confirm by eplicit differentiation that dcsch = coth csch d Problem 3 Derive the binomial series, (1 + ) n n(n 1) = 1 + n + n(n 1)(n ) + 3 +! 3! Are there any restrictions on the value of n? Under what conditions does the series terminate? Problem 4 Series inversion One way to calculate the Taylor (Laurent) series for csch = 1 sinh is by series inversion. We start with csch = 1 sinh = ! + 5 5! + = 1 1 ( ) 1 + 3! + 4 5! + We now treat the term in parentheses as a small quantity (ɛ) and use the binomial approimation to epress as its own power series. The trick is to keep all terms of the ɛ same order in in the epansion. Use this approach to confirm that csch = Problem 5 A particle s position in spherical coordinates is given by r = r ˆr. Eplain why there are no θ and φ components. What is the time derivative of r? Problem 6 Euler s identity is handy for deriving trigonometric identities. Starting from e i (+y) = e i e i y use Euler s identity on both sides to derive formulas for sin( + y) and cos( + y). Problem 7 Use the Jacobian determinant approach to show that the volume element in spherical coordinates is r sinθ dr dθ dφ. Physics of 11 Peter N. Saeta