Chapter 14: Q. 24 (page 1096)

In Exercises 17–24, find a potential function for the given vector field.
$G (x, y, z) = (y e^{x y + z}, x e^{x y + z}, e^{x y + z})$

Short Answer

Expert verified

A potential function for the given vector field is $g (x, y, z) = e^{x y + z}$ .

Step by step solution

Step 1. Given Information

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

$G (x, y, z) = (y e^{x y + z}, x e^{x y + z}, e^{x y + z})$

Step 2. Since G(x,y,z)=(yexy+z, xexy+z, exy+z)

$g (x, y, z) = \int y e^{x y + z} d x + B + C g (x, y, z) = \int y e^{x y} \cdot e^{z} d x + B + C Let u = x y \frac{d u}{d x} = y d u = y d x g (x, y, z) = e^{z} \int e^{u} d x + B + C g (x, y, z) = e^{z} e^{u} + α + B + C g (x, y, z) = e^{z} e^{x y} + α + B + C g (x, y, z) = e^{x y + z} + α + B + C$

where $α$ is an arbitrary constant and B is the integral with respect to y of the terms in localid="1650474562962" $G_{2} (x, y, z)$ in which the factor x does not appear.

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

$B = \int x e^{x y + z} d y B = \int x e^{x y} \cdot e^{z} d y Let u = x y \frac{d u}{d y} = x d u = x d y B = e^{z} \int e^{u} d y B = e^{z} e^{u} + β B = e^{z} e^{x y} + β B = e^{x y + z} + β$

whereβ is an arbitrary constant.

Step 4. Now finding C=∫exy+zdz

$C = \int e^{x y} \cdot e^{z} d z C = e^{x y} \int e^{z} d z C = e^{x y} e^{z} + γ C = e^{x y + z} + γ$

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,

$g (x, y, z) = e^{x y + z}$

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In Exercises 17–24, find a potential function for the given vector field.
$G (x, y, z) = (y e^{x y + z}, x e^{x y + z}, e^{x y + z})$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Since G(x,y,z)=(yexy+z, xexy+z, exy+z)

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

Step 4. Now finding C=∫exy+zdz

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In Exercises 17–24, find a potential function for the given vector field.G(x,y,z)=(yexy+z,xexy+z,exy+z)

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Since G(x,y,z)=(yexy+z, xexy+z, exy+z)

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

Step 4. Now finding C=∫exy+zdz

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Applied Mathematics

Pure Maths

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Calculus

Study anywhere. Anytime. Across all devices.

In Exercises 17–24, find a potential function for the given vector field.
$G (x, y, z) = (y e^{x y + z}, x e^{x y + z}, e^{x y + z})$