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[CHAP. 6
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@2 w @2 w @2 w @2 w @2 w @2 w 2xz 2yz 0: y2 2 z2 2 2xy 2 @x @y @x @z @y @z @x @y @z
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Indicate possible exceptional points. DIFFERENTIALS 6.71. If z x3 xy 3y2 , compute (a) z and (b) dz where x 5, y 4, x 0:2, y 0:1. Explain why z and dz are approximately equal. (c) Find z and dz if x 5, y 4, x 2, y 1. Ans. (a) 11:658; b 12:3; c z 66; dz 123 q 5 Computer 3:8 2 2 2:1 3 approximately, using di erentials. Ans. 2.01 Find dF and dG if (a) F x; y x3 y 4xy2 8y3 ; b G x; y; z 8xy2 z3 3x2 yz, ln y=x . Ans. a 3x2 y 4y2 dx x3 8xy 24y2 dy b 8y2 z3 6xyz dx 16xyz3 3x2 z dy 24xy2 z2 3x2 y dz c f y2 ln y=x y2 g dx f2xy ln y=x xyg dy (c) F x; y xy2
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6.72. 6.73.
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6.74. 6.75.
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Prove that (a) d UV U dV V dU; b d U=V V dU U dV =V 2 ; c d ln U dU =U, 1 (d) d tan V dV = 1 v2 where U and V are di erentiable functions of two or more variables. Determine whether each of the following are exact di erentials of a function and if so, nd the function. a 2xy2 3y cos 3x dx 2x2 y sin 3x dy b 6xy y2 dx 2xey x2 dy c z3 3y dx 12y2 3x dy 3xz2 dz Ans. a x2 y2 y sin 3x c; b not exact, c xz2 4y3 3xy c
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DIFFERENTIATION OF COMPOSITE FUNCTIONS 6.76. (a) If U x; y; z 2x2 yz xz2 , x 2 sin t, y t2 t 1, z 3e t , nd dU=dt at t 0. H x; y sin 3x y , x3 2y 2t3 , x y2 t2 3t, nd dH=dt. ! 36t2 y 12t 9x2 6t2 6x2 t 18 Ans: a 24; b cos 3x y 6x2 y 2 If F x; y 2x y = y 2x , x 2u 3v, y u 2v, nd (d) @2 F=@v2 ; e @2 F=@u @v, where u 2, v 1. Ans. (a) 7, (b) 14; c 21; d 112; e 49 (a) @F=@u; b @F=@v; (b) if
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c @2 F=@u2 ,
6.78. 6.79.
If U x2 F y=x , show that under suitable restrictions on F, x @U=@x y @U=@y 2U. If x u cos v sin and y u sin v cos , where is a constant, show that @V=@x 2 @V=@y 2 @V=@u 2 @V=@v 2
Show that if x  cos , y  sin , the equations @u @v @u @v ; @x @y @y @x becomes @u 1 @v @v 1 @u ; @  @ @  @
Use Problem 6.80 to show that under the transformation x  cos , y  sin , the equation @2 u @2 u 0 @x2 @y2 becomes @2 u 1 @u 1 @2 u 0 @2  @ 2 @2
IMPLICIT FUNCTIONS AND JACOBIANS 6.82. If F x; y 0, prove that dy=dx Fx =Fy .
CHAP. 6]
PARTIAL DERIVATIVES
Find (a) dy=dx and (b) d 2 y=dx2 if x3 y3 3xy 0. Ans. (a) y x2 = y2 x ; b 2xy= y2 x 3 If xu2 v y3 , 2yu xv3 4x, nd (a) @u ; @x b @v . @y Ans: a v3 3xu2 v2 4 ; 6x2 uv2 2y b 2xu2 3y3 3x2 uv2 y
6.84. 6.85.
If u f x; y , v g x; y are di erentiable, prove that are considered independent in each partial derivative.
@u @x @v @x 1. Explain clearly which variables @x @u @x @v
@y @r @y @s 0, explaining which variables are independent. @r @x @s @x What notation could you use to indicate the independent variables considered If f x; y; r; s 0, g x; y; r; s 0, prove that
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