Chapter 14: Q. 23 (page 1096)

In Exercises 17–24, find a potential function for the given vector field.
$G (x, y, z) = \cos y i + (\sin z - x \sin y) j + y \cos z k$

Short Answer

Expert verified

A potential function for the given vector field is $g (x, y, z) = x \cos y + y \sin z$ .

Step by step solution

Step 1. Given Information

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

$G (x, y, z) = \cos y i + (\sin z - x \sin y) j + y \cos z k$

Step 2. Since G(x,y,z)=cosyi+(sinz−xsiny)j+ycoszk

$G (x, y, z) = \int \cos y d x + B + C G (x, y, z) = \cos y \int d x + B + C G (x, y, z) = \cos y \cdot x + B + α + C G (x, y, z) = x \cos y + α + B + C$

where $α$ is an arbitrary constant and B is the integral with respect to y of the terms in role="math" localid="1650474027433" $G_{2} (x, y, z)$ in which the factor x does not appear.

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

$B = \int (\sin z - x \sin y) d y B = \int \sin z d y - x \int \sin y d y B = \sin z \int d y - x (- \cos y) + β B = \sin z \cdot y + x \cos y + β B = y \sin z + x \cos y + β$

whereβ is an arbitrary constant.

Step 4. Now finding C=∫ycoszdz

$C = y \int \cos z d z C = y \sin z + γ$

where $γ$ is an arbitrary constant.

Setting the constants equal to zero since they do not affect the gradient of $g (x, y, z)$

We have,

$g (x, y, z) = x \cos y + y \sin z$

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In Exercises 17–24, find a potential function for the given vector field.
$G (x, y, z) = \cos y i + (\sin z - x \sin y) j + y \cos z k$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Since G(x,y,z)=cosyi+(sinz−xsiny)j+ycoszk

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

Step 4. Now finding C=∫ycoszdz

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In Exercises 17–24, find a potential function for the given vector field.G(x,y,z)=cosyi+(sinz−xsiny)j+ycoszk

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Since G(x,y,z)=cosyi+(sinz−xsiny)j+ycoszk

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

Step 4. Now finding C=∫ycoszdz

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Logic and Functions

Probability and Statistics

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Discrete Mathematics

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Study anywhere. Anytime. Across all devices.

In Exercises 17–24, find a potential function for the given vector field.
$G (x, y, z) = \cos y i + (\sin z - x \sin y) j + y \cos z k$