Chapter 14: Q. 7 (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.
$F (x, y) = <y^{3} e^{x y^{2}}, (1 + 2 x y) e^{x y^{2}}>$

Short Answer

Expert verified

Potential function is $F (x, y) = y e^{x y^{2}}$

Step by step solution

Step 1. Given Information.

The vector field is $F (x, y) = <y^{3} e^{x y^{2}}, (1 + 2 x y) e^{x y^{2}}>$ .

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 $ℝ$ .

role="math" localid="1650908697732" $\frac{\partial}{\partial y} y^{3} e^{x y^{2}} = 3 y^{2} e^{x y^{2}} + 2 x y^{4} e^{x y^{2}}$

role="math" localid="1650908704305" $\frac{\partial}{\partial x} (1 + 2 x y) e^{x y^{2}} = 3 y^{2} e^{x y^{2}} + 2 x y^{4} e^{x y^{2}}$

Partial derivatives are the same. Hence $F (x, y)$ is conservative.

Step 3. Find potential function.

Potential function is given by $g (x, y) = \int x d x + B$ .

Here, B is integral with respect to y of the terms in which the x does not appear.

So,

$\int y^{3} e^{x y^{2}} d x = y e^{x y^{2}} + C$ and $B = 0$ .

Potential function is $F (x, y) = y e^{x y^{2}}$ .

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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.
$F (x, y) = <y^{3} e^{x y^{2}}, (1 + 2 x y) e^{x y^{2}}>$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

Step 3. Find potential function.

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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=y3exy2,1+2xyexy2

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

Step 3. Find potential function.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Discrete Mathematics

Applied Mathematics

Pure Maths

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Statistics

Study anywhere. Anytime. Across all devices.

Conservative Vector Fields: Determine whether the vector fields that follow are conservative. If the field is conservative, find a potential function for it.
$F (x, y) = <y^{3} e^{x y^{2}}, (1 + 2 x y) e^{x y^{2}}>$