Chapter 14: Q. 6 (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.
$F (x, y) = <(1 + x y) e^{x y}, e^{x y}>$ .

Short Answer

Expert verified

$F (x, y)$ is not conservative.

Step by step solution

Step 1. Given Information.

The vector field is $F (x, y) = <(1 + x y) e^{x y}, e^{x y}>$ .

Step 2. Explanation.

A vector field $F (x, y) = <F_{1} (x, y), F_{2} (x, y)>$ is conservative if and only if $\frac{\partial F_{1}}{\partial y} = \frac{\partial F_{2}}{\partial x}$ , When $\frac{\partial F_{1}}{\partial y}$ and $\frac{\partial F_{2}}{\partial x}$ are both continuous on some open subset of $ℝ$ .

$\frac{\partial}{\partial y} ((1 + x y) e^{x y}) = 2 x e^{x y} + x^{2} y e^{x y}$

$\frac{\partial}{\partial x} (e^{x y}) = y e^{x y}$

so, $\frac{\partial F_{1}}{\partial y} \neq \frac{\partial F_{2}}{\partial x}$ ,

and partial derivatives are continuous on every open subset of $ℝ$ not containing the origin, $F (x, y)$ is not conservative.

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Conservative Vector Fields: Determine whether the vector fields that follow are conservative. If the field is conservative, find a potential function for it.
$F (x, y) = <(1 + x y) e^{x y}, e^{x y}>$ .

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

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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=1+xyexy,exy.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

One App. One Place for Learning.

Most popular questions from this chapter

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Conservative Vector Fields: Determine whether the vector fields that follow are conservative. If the field is conservative, find a potential function for it.
$F (x, y) = <(1 + x y) e^{x y}, e^{x y}>$ .