   Chapter 16.1, Problem 21E

Chapter
Section
Textbook Problem

Find the gradient vector field of f.21. f(x, y) = y sin(xy)

To determine

To find: The gradient vector field for equation f(x,y)=ysin(xy) .

Explanation

Given data:

f(x,y)=ysin(xy)

Formula used:

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

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

Write the required differentiation formulae as follows.

t[u(t)v(t)]=u(t)t[v(t)]+v(t)t[u(t)] (2)

Differentiate terms with respect to x .

xy=0x(sinx)=(cosx)xx=1

Differentiate terms with respect to y .

y(siny)=cosyyy=1yx=0

Find the gradient vector field of f(x,y)=ysin(xy) using equation (1).

Modify equation (1) as follows.

f(x,y)=x[ysin(xy)]i+y[ysin(xy)]j (3)

Modify equation (2) as follows.

x[ysin(xy)]=yx[sin(xy)]+sin(xy)x[y]=y[cos(xy)x(xy)]+sin(xy)(0)=y[cos(xy)x(xy)]+0=y[cos(xy)yx(x)]

Differentiate and rewrite the equation as follows.

x[ysin(xy)]=y[cos(xy)y(1)]=y2cos(xy)

Modify equation (2) as follows

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