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Chapter 14: Q. 5 (page 1153)

Conservative Vector Fields: Determine whether the vector fields that follow are conservative. If the field is conservative, find a potential function for it.

Fx,y=2xyi+x2y2j.

Short Answer

Expert verified

Fx,yis not conservative.

Step by step solution

01

Step 1. Given Information.

The vector fields isFx,y=2xyi+x2y2j.

02

Step 2. Explanation.

A vector field Fx,y=F1x,y,F2x,yis conservative if and only if role="math" localid="1650905948212" 鈭�F1鈭�y=鈭�F2鈭�x. When role="math" localid="1650905502572" 鈭�F1鈭�yand role="math" localid="1650905627701" 鈭�F2鈭�xare both continuous on some open subset of鈩�.

role="math" localid="1650905908680" 鈭�鈭�y2xy=-2xy2

鈭�鈭�xx2y2=2xy2

so, 鈭�F1鈭�y鈮�鈭�F2鈭�x.

and partial derivatives are continuous on every open subset of 鈩� not containing the origin,Fx,y is not conservative.

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Calculus of vector-valued functions: Calculate each of the following.

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