   Chapter 15.1, Problem 39E ### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

#### Solutions

Chapter
Section ### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

# In Exercises 37–-44, determine whether the vector field is conservative. If it is, find a potentialfunction for the vector field. F ( x , y ) = x e x 2 y ( 2 y i ^ + x j ^ ) .

To determine
Whether the vector field F(x,y) = xex2y(2yi^+xj^).is conservative. If it is, find a potentialfunction for the vector field.

Explanation

Given: The function F(x,y) = xex2y(2yi^+xj^).

Explanation: A vector field F(x,y) = Mi^+Nj^ is said to be conservative if and only if

My=Nx, where M and N are differentiable functions.

Here, M = 2xyex2y

Now, partial differentiating both sides with respect to ‘y’

=>My=y(2xyex2y)=2x[yyex2y+ex2yyy]=2x[x2yex2y+ex2y]

and, N=x2ex2y

Now, partial differentiating both sides with respect to ‘x’

=>Nx=x(x2ex2y)=x2x(ex2y)+ex2yx(x2)=2x3yex2y+2xex2y

Clearly, My=Nx

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