PARTIAL DERIVATIVES
383
42.64
42.65
If u = x
2 -2y
2 + z
3
and * = sinf, y = e', z = 3f, find duldt.
If u = /(jc,,..., x n ) and x l = h t (t),..., x n = h n (t), state the chain rule for duldt.
By the formula of Problem 42.64,
= 2x cos t-4ye' + 3z2(3) = 2 sin (cos t - 4e'(e') + 9(3f)2 = sin 2t - 4e2' + Sir.
42.66
If w = <&(x, y, z) and x=f(u, v), y = g(u, v), z = h(u, v), state the chain rule for dwldu and dw/dv.
42.67
Let z = r
2 + s
2 + t
2 and t = rsu. Clarify the two possible values of dzldr and find both values.
(1) If z=f(r,s,t) = r
2 + s
2 + t
2 , then dzldr means f r (r, s, t) = 2r. (2) If z = r
2 + s
2 + t
2 = r
2 +
s
2 + (rsu)
2 = r
2 + s
2 + rVu
2 = g(r, s, u), then dzldr means g r (r, s, u) = 2r + 2rs
2 u
2 . To distinguish the two
possible values, one often uses
r
42.68
How fast is the volume V of a rectangular box changing when its length / is 10 feet and increasing at the rate of
2 ft/s, its width w is 5 feet and decreasing at the rate of 1 ft/s, and its height h is 3 feet and increasing at the rate of
2 ft/s?
42.69
If w = x
2 + 3xy - 2y
2 , and x = r cos 9, y = r sin 0, find dwldr and dwld6.
42.70
Let u = f(x, y), x = r cos 0, y = r sin 6. Show that
and
By the chain rule,
42.71 Let u=f(x, y), x = rcos0, y = rsm0. Show that urr = fxx cos2 0 + 2fxy cos 0 sin 6 + fyy sin2 0.
ur = L cos e + f, sin e- Hence, urr = cos 0 (/„ cos 0 + f,y sin 0) + sin 0 (fyt cos 0 + fyy sin2 0) = /„ cos2 0 +
2^,, cos 0 sin 0 + /^ sin2 0.
42.72
Let M = f(x, y), x = rcosO, y = rsin 0. Show that
for the second value.
for the first value, and