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Chapter 14: Q 5 (page 1153)

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

F(x,y)=2xyi+x2y2j

Short Answer

Expert verified

The vector field is non conservative

Step by step solution

01

Step 1:Given information 

The given expression isF(x,y)=2xyi+x2y2j

02

Step 2:Simplification 

Consider the vector field,

F(x,y)=2xyi+x2y2j

A vector field F(x,y)=P(x,y)i+Q(x,y)jis conservative if and only if 鈭�P鈭�y=鈭�Q鈭�x

now comparing,

F(x,y)=2xyi+x2y2j

with F(x,y)=P(x,y)i+Q(x,y)j

Then,

P=2xy

Q=x2y2

now calculating

鈭�P鈭�y,鈭�Q鈭�x

鈭�P鈭�y=-2xy2and 鈭�Q鈭�x=2xy2

here,鈭�P鈭�y鈮�鈭�Q鈭�x

Therefore the vector field is non conservative

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Most popular questions from this chapter

Use the curl form of Green鈥檚 Theorem to write the line integral of F(x, y) about the unit circle as a double integral. Do not evaluate the integral.

Use what you know about average value from previous sections to propose a formula for the average value of a multivariate function f(x, y, z) on a smooth surface S.

How would you show that a given vector field in 鈩�2 is not conservative?

F(x,y,z)=zcosyzj+zsinyzk, where S is the portion of the plane with equation 2x鈭�8y鈭�10z=42 that lies on the positive side of the rectangle with corners0,-蟺,0(0,蟺,0),(0,蟺,蟺),(0,鈭�蟺,蟺)in theyz-plane.

F(x,y,z)=鈭�yi+xj鈭�eyzk, where S is the cylinder with equation x2+y2=9from z=2,4, with n pointing outwards.
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