1MA6 Partial Differentiation and Multiple Integrals: I
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1 1MA6/1 1MA6 Partial Differentiation and Multiple Integrals: I Dr D W Murray Michaelmas Term Total differential. (a) State the conditions for the expression P (x, y)dx+q(x, y)dy to be the perfect differential of a function z = z(x, y). erify that this condition is satisfied when P = 2xy, Q = x 2 y 2 and find by integration the corresponding function z(x, y). (b) A particle constrained to lie in the x y plane executes the elliptical orbit shown in the Figure. It is then subject to a force F with components F x = 4xy in x-direction and F y = 2(x 2 y 2 ) in the y direction. What is the net work done by the force F in moving the particle through one complete orbit? y 2 2. Function of a function x 1 2 (a) Find the first partial derivatives of ( ) y 1 f(x, y) = (x + y) tan 1 (x y) x xy by finding a function u = u(x, y) such that f is a function of u alone. Does the function u help you to find the second derivatives? (b) The variable z is a function of x and y, but it is found that y is a function of x only. Obtain an expression for dz/dx. Hence find x c, the value of x at which the curve y = ax + b/x approaches closest to the origin in the positive quadrant. (Assume a and b are positive.)
2 1MA6/2 3. The Chain Rule. (a) The transformation between Cartesian and spherical polar coordinates is x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ. Use the chain rule for partials to find f/ r, f/ θ, and f/ φ, when f = x 2 + y 2 z 2. Check your results by substituting for x, y, z to obtain an explicit function of r, θ, φ. (b) The quantity f depends on x, y and z as f = 1 x + y + e xy + z, where it is known that x = sin t and y = sin 2t. i. Assuming z may or may not be a function of t, decide whether the partial or a total derivative of f with respect to t exists, and use the chain rule to find it. ii. Repeat the exercise, assuming that z = 2t 2. (c) Prove using the chain rule that if w = lnr where r 2 = x 2 + y 2 + z 2 then x w x + y w y + z w z = 1, and x 2 w y z = y 2 w z x = z 2 w x y. 4. Implicit Functions. Show that when a function of x and y is such that y = y(x), then the operator d/dx is [ d dx = x + dy ] dx y Also show that when y = y(x) is defined implicitly from φ(x, y) = 0, dy dx = φ x/φ y. Hence show that d 2 y dx = 1 [ φxx φ 2 2 φ 3 y 2φ xy φ x φ y + φ yy φx] 2 y. Find the values of dy/dx and d 2 y/dx 2 using these formula when e y cos x = 1. erify your results by performing the differentiations explicitly, ie by writing y = f(x).
3 1MA6/3 5. More implicit functions (a) Show that two expressions of the form f(x 1, x 2, x 3 ) = 0 g(x 1, x 2, x 3 ) = 0 enable one to derive dx i /dx j (i, j = 1, 2, 3). (b) Describe geometrically, and sketch, the locus defined by satisfying both f = x 2 + y 2 + z 2 1 = 0 and g = x + y + z = 0. There are two points on this locus for which x = y. Find these points and, using the derivation of part (a), find dz/dx and dz/dy at them. Explain your results geometrically. 6. Transformations. The independent coordinates x, y are related to another pair u, v by the transformation x = u cos v, y = v cos u. (a) Find x/ u and so on and evaluate the Jacobian J 1 = (x, y) (u, v). (b) Can you easily invert the transformation to find u = u(x, y) and v = v(x, y)? (c) If not, how can you evaluate the Jacobian J 2 = (u, v) (x, y)? (d) Notice anything about the product of the Jacobians? 7. Implicit functions and transformations. (From Honour Mods 1991: there are hints at the back) (a) If x 2 + 2y 2 3z 2 = 0 and x 2 4y 2 + z 2 = 0, find dy/dx in terms of x and y. (b) If z = f(u, v) where u = 1 2 ln(x2 + y 2 ) and v = tan 1 (y/x) prove that ( 2 ) ( z x + 2 z 1 2 ) f = 2 y 2 x 2 + y 2 u + 2 f. 2 v 2
4 1MA6/4 8. Application to BJT characteristics. Note that you only require Ohm s law to deal with this and don t get confused by the the subscripts they are just labels, not partial derivatives! The circuit behaviour of a bipolar junction transistor is described by the voltages between the base and emitter ( BE ) and collector and emitter ( CE ), along with the base and collector currents i B and i C (Figure A). The collector current is a function i C = i C ( CE, i B ). Write down an expression for the change in i C when CE and i B change slightly. A particular power transistor has the i C ( CE, i B ) dependence shown in Figure B, where slices through the function are sketched at various values of i B. Use the sketch to estimate ( i C / i B ) and ( i C / CE ) when CE = 10 and i C = 250mA. The transistor is connected into the circuit shown in figure C. Use Ohm s Law to write down a relationship between i C and CE. How would this appear on the sketch B? Illustrate using the graph how i C changes when i B changes with the transistor in this circuit. i C i C B B E BE CE (A) i B i C 40Ω +20 BE CE (C)
5 1MA6/5 9. Application to thermodynamics. Don t immediately rush for books on thermodynamics. You can reach these results using your knowledge of partial differentiation. (a) For a hydrostatic system the change in internal energy U is related to the temperature T, pressure p, change in entropy ds and change in volume d by du = T ds pd. Hence derive the Maxwell relationship ( ) ( ) T p = S S. (b) Quantities U, T, p,, S are such that only two are independent, so that any quantity can be expressed as a function of any other two eg U = U(p, ), p = p(u, ), p = p(u, T ) and so on. Write expressions for the total differentials of S = S(T, ), S = S(T, p) and = (p, T ). By finding two equations relating ds to dt and dp show that ( ) S T p ( ) S T = ( ) ( ) S T T p [Multiplying the lhs by what gives C p C v the difference of the specific heats at constant pressure and volume?]
6 1MA6/6 Some answers and hints 2(a) f x = x2 +y 2 yx 2 tan 1 (y/x) x2 y 2 x(x 2 +y 2 ) ; f y = x2 +y 2 xy 2 tan 1 (y/x) + x2 y 2 y(x 2 +y 2 ) 2(b) x c = b(1 + a 2 ) 1/4. 3(a) 2r cos 2θ, 2r 2 sin 2θ, 0. 4(b) d 2 y/dx 2 = sec 2 x. 5(b) x = y = ±1/ 6, z = 2/ 6. 6(b) J 1 = cos v cos u uv sin u sin v and J 1 J 2 = 1. 7(a) Do this (i) explicitly by elimination (a technique which is not always available) and (ii) by generating simultaneous equations in dy/dx and dz/dx. Both times you should get dy/dx = 2x/5y. 7(b) Obtain expressions for / x and / y in terms of x, y, / u and / v. Then use as simultaneous equations to find / u and / v in terms of x, y, / x and / y. Then find 2 / u 2 and so on. 8 circa 100, and 4mS.
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