Q. 22 In Exercises 17鈥�24, find a pot... [FREE SOLUTION]

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Chapter 14: Q. 22 (page 1095)

In Exercises 17鈥�24, find a potential function for the given vector field.
$F (x, y, z) = e^{y^{2}} i + (2 x y e^{y^{2}} + \sin z) j + y \cos z k$

Short Answer

Expert verified

A potential function for the given vector field is $f (x, y, z) = {xe}^{y^{2}} + ysinz$ .

Step by step solution

Step 1. Given Information

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

$F (x, y, z) = e^{y^{2}} i + (2 x y e^{y^{2}} + \sin z) j + y \cos z k$

Step 2. Since F(x,y,z)=ey2i+(2xyey2+sinz)j+ycoszk

$F (x, y, z) = 鈭� e^{y^{2}} d x + B + C F (x, y, z) = e^{y^{2}} 鈭� d x + B + C F (x, y, z) = e^{y^{2}} 路 x + 伪 + B + C F (x, y, z) = x e^{y^{2}} + 伪 + B + C$

where $伪$ is an arbitrary constant and B is the integral with respect to y of the terms in role="math" localid="1650473294189" $F_{2} (x, y, z)$ in which the factor x does not appear.

Step 3. In this case, that is all of F2(x,y,z), so

$B = 鈭� (2 x y e^{y^{2}} + \sin z) d y B = 鈭� 2 x y e^{y^{2}} d y + 鈭� \sin z d y B = x 鈭� 2 y e^{y^{2}} d y + 鈭� \sin z d y Let u = y^{2} \frac{d u}{d y} = 2 y d u = 2 y d y B = x 鈭� e^{u} d u + \sin z 鈭� d y B = x e^{u} + \sin z 路 y + 尾 B = x e^{y^{2}} + y \sin z + 尾$

where尾 is an arbitrary constant.

Step 4. Now finding C=∫ycoszdz

$C = y 鈭� \cos z d z C = y \sin z + 纬$

where $纬$ is an arbitrary constant.

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

We have,

$f (x, y, z) = {xe}^{y^{2}} + ysinz$

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91影视

In Exercises 17鈥�24, find a potential function for the given vector field.
$F (x, y, z) = e^{y^{2}} i + (2 x y e^{y^{2}} + \sin z) j + y \cos z k$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Since F(x,y,z)=ey2i+(2xyey2+sinz)j+ycoszk

Step 3. In this case, that is all of F2(x,y,z), so

Step 4. Now finding C=∫ycoszdz

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91影视

In Exercises 17鈥�24, find a potential function for the given vector field.F(x,y,z)=ey2i+(2xyey2+sinz)j+ycoszk

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Since F(x,y,z)=ey2i+(2xyey2+sinz)j+ycoszk

Step 3. In this case, that is all of F2(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

Probability and Statistics

Discrete Mathematics

Mechanics Maths

Geometry

Applied Mathematics

Statistics

Study anywhere. Anytime. Across all devices.

In Exercises 17鈥�24, find a potential function for the given vector field.
$F (x, y, z) = e^{y^{2}} i + (2 x y e^{y^{2}} + \sin z) j + y \cos z k$