   Chapter 16.1, Problem 32E

Chapter
Section
Textbook Problem

Match the functions f with the plots of their gradient vector fields labeled I-IV. Give reasons for your choices.32. f(x, y) = sin x 2 + y 2 To determine

To match: The function f(x,y)=sinx2+y2 with the plots labeled as I-IV.

Explanation

Given data:

f(x,y)=sinx2+y2

Formula used:

Write the expression for gradient vector field of two dimensional vector.

f(x,y)=fxi+fyj (1)

Write the required differentiation formulae with respect to x as follows.

x(sinx)=cosxx(x)=12xx(x2)=2xx(y2)=0

Write the required differentiation formulae with respect to y as follows.

y(siny)=cosyy(y)=12yy(y2)=2yy(x2)=0

Differentiate the term sinx2+y2 with respect to x .

x(sinx2+y2)=(cosx2+y2)xx2+y2=(cosx2+y2)12x2+y2x(x2+y2)=(cosx2+y2)12x2+y2(2x+0)=xcosx2+y2x2+y2

Differentiate the term sinx2+y2 with respect to y

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