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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Let R(s, t) = G(u(s, t), v(s, t)), where G, u, and v are differentiable, u(1,2) = 5, us(1, 2) = 4, ut(l, 2) = −3, v(l, 2) = 7, vs (1, 2) = 2, vt (1, 2) = 6, Gu (5, 7) = 9, Gv (5, 7) = −2. Find Rs(1, 2) and Rt(1, 2).

To determine

To find: The value of   Rs(1,2)andRt(1,2) using Chain Rule where R(s,t)=G(u(s,t),v(s,t)) and if u(1,2)=5,us(1,2)=4,ut(1,2)=3,v(1,2)=7, vs(1,2)=2,vt(1,2)=6,Gu(5,7)=9andGv(5,7)=2 .

Explanation

Chain Rule:

“Suppose that z=f(x,y) is a differentiable function of x and y , where x=g(t)andy=h(t) are both differentiable functions of t then, z is differentiable function of t and dzdt=zxdxdt+zydydt ”.

Calculation:

The value of Rs(s,t) using chain rule is as follows,

Rs=Ruus+Rvvs (1)

The equation (1) at s=1 and t=2 can be expressed as,

Rs(1,2)=Gu(5,7)us(1,2)+Gv(5,7)vs(1,2) (2)

Substitute the respective values in the equation (2),

Rs(1,2)=Gu(5,7)us(1,2)+Gv(5,7)vs(1,2)=9(4)+(2)(2)=364=32

Thus, the value of Rs(1,2) is 32

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P-14RECh-14 P-15RECh-14 P-16RECh-14 P-17RECh-14 P-18RECh-14 P-19RECh-14 P-20RECh-14 P-21RECh-14 P-22RECh-14 P-23RECh-14 P-24RECh-14 P-25RECh-14 P-26RECh-14 P-27RECh-14 P-28RECh-14 P-29RECh-14 P-30RECh-14 P-31RECh-14 P-32RECh-14 P-33RECh-14 P-34RECh-14 P-35RECh-14 P-36RECh-14 P-37RECh-14 P-38RECh-14 P-39RECh-14 P-40RECh-14 P-41RECh-14 P-42RECh-14 P-43RECh-14 P-44RECh-14 P-45RECh-14 P-46RECh-14 P-47RECh-14 P-48RECh-14 P-49RECh-14 P-50RECh-14 P-51RECh-14 P-52RECh-14 P-53RECh-14 P-54RECh-14 P-55RECh-14 P-56RECh-14 P-57RECh-14 P-58RECh-14 P-59RECh-14 P-60RECh-14 P-61RECh-14 P-62RECh-14 P-63RECh-14 P-64RECh-14 P-65RECh-14 P-1PCh-14 P-2PCh-14 P-3PCh-14 P-4PCh-14 P-5PCh-14 P-6PCh-14 P-7PCh-14 P-8P

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