Chapter 14: Q. 19 (page 1095)

In Exercises 17–24, find a potential function for the given vector field.
$G (x, y) = (5 x^{4} + y, x - 12 y^{3})$

Short Answer

Expert verified

A potential function for the given vector field is $g (x, y) = x^{5} + x y - 3 y^{4}$ .

Step by step solution

Step 1. Given Information

In given exercises we have to find a potential function for the given vector field.

$G (x, y) = (5 x^{4} + y, x - 12 y^{3})$

Step 2. Since G(x,y)=(5x4+y,x−12y3)

$G (x, y) = \int (5 x^{4} + y) d x + B g (x, y) = 5 \int x^{4} d x + y \int d x + B g (x, y) = 5 [\frac{x^{4 + 1}}{4 + 1}] + y x + α + B g (x, y) = 5 [\frac{x^{5}}{5}] + x y + α + B g (x, y) = x^{5} + x y + α + B$

where $α$ is an arbitrary constant and B is the integral with respect to y of the terms in localid="1650472400181" $G_{2} (x, y)$ in which the factor x does not appear.

Step 3. In this case, that is all of G2(x,y), so

$B = \int (x - 12 y^{3}) d y B = x \int d y - 12 \int y^{3} d y B = xy - 12 [\frac{y^{3 + 1}}{3 + 1}] + β B = xy - 12 [\frac{y^{4}}{4}] + β B = xy - 3 y^{4} + β$

whereβ is an arbitrary constant.

Step 4. Setting the constants equal to zero since they do not affect the gradient of g(x,y)

We have,

$g (x, y) = x^{5} + x y - 3 y^{4}$

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In Exercises 17–24, find a potential function for the given vector field.
$G (x, y) = (5 x^{4} + y, x - 12 y^{3})$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Since G(x,y)=(5x4+y,x−12y3)

Step 3. In this case, that is all of G2(x,y), so

Step 4. Setting the constants equal to zero since they do not affect the gradient of g(x,y)

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In Exercises 17–24, find a potential function for the given vector field.G(x,y)=(5x4+y,x−12y3)

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Since G(x,y)=(5x4+y,x−12y3)

Step 3. In this case, that is all of G2(x,y), so

Step 4. Setting the constants equal to zero since they do not affect the gradient of g(x,y)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Mechanics Maths

Statistics

Decision Maths

Applied Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

In Exercises 17–24, find a potential function for the given vector field.
$G (x, y) = (5 x^{4} + y, x - 12 y^{3})$