   Chapter 14.5, Problem 14E

Chapter
Section
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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