Calculus: Analytic Geometry and Calculus, with Vectors

580 Partial derivatives

ax/0a, ay/aa, az/aa in terms of partial derivatives of fi, f2, fa. Partial ans.:

The first equation can be put in the form

aft ax af,ay of l az_ - af,

ax as + ay as+ az as as

and the required condition is

af, afI afl

ar ay az

aft alt aft

ax ay az

afa afa af3

ax ay az

2 Supposing that x and y are differentiable functions of a and 0 for which

2x2 + 3y - 2a2 - 30 = 0

x2 + 2y3 - a - 2$2 = 0,

calculate ax,/aa and ay/aa. Ins.:

ax__8ay2 - 1

as 8xy2 - 2x'

ay 2 - 4a

as 12y2 - 3

3 Supposing that p > 0 and that p and 0 are functions of x and y for which

pcoscb=x, psin0=y,

differentiate with respect to x and then with respect to y to obtain

TX

cos 4) - p sin 4) a = 1,

axsin4)+pcos0 =0,

and solve for the derivatives to obtain

cos4)-psin4)-=0

ey 49Y

49Psin4)+pcos4)- =1

ap= h, ap=sin a4)_ _ in^4 a¢= cos

ax

cos 0'

ay ax p ay p

4 Copy the formulas at the conclusion of Problem 3 and use them in appro-

priate places in the process of deriving the following formulas that are used to

make transformations from rectangular to polar and cylindrical coordinates.

Supposing that is is a function of x and y and that x and y are functions of p and

¢ for which

(1) x=pcos0, y=psin0,

show that

(2)

(3)

au au ap au ao

ax apax+COax

au au ap au a4)

ay spay+a4ay