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.
$F (x, y) = \frac{2 x}{y} i + \frac{x^{2}}{y^{2}} j$ .

Short Answer

Expert verified

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

Step by step solution

Step 1. Given Information.

The vector fields is $F (x, y) = \frac{2 x}{y} i + \frac{x^{2}}{y^{2}} j$ .

Step 2. Explanation.

A vector field $F (x, y) = <F_{1} (x, y), F_{2} (x, y)>$ is conservative if and only if role="math" localid="1650905948212" $\frac{\partial F_{1}}{\partial y} = \frac{\partial F_{2}}{\partial x}$ . When role="math" localid="1650905502572" $\frac{\partial F_{1}}{\partial y}$ and role="math" localid="1650905627701" $\frac{\partial F_{2}}{\partial x}$ are both continuous on some open subset of $ℝ$ .

role="math" localid="1650905908680" $\frac{\partial}{\partial y} (\frac{2 x}{y}) = - \frac{2 x}{y^{2}}$

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

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.
$F (x, y) = \frac{2 x}{y} i + \frac{x^{2}}{y^{2}} j$ .

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=2xyi+x2y2j.

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.
$F (x, y) = \frac{2 x}{y} i + \frac{x^{2}}{y^{2}} j$ .