Chapter 14: Q. 46 (page 1096)

Determine whether or not each of the vector fields in Exercises 41–48 is conservative. If the vector field is conservative, find a potential function for the field.
$F (x, y, z) = i + 2 j - 3 k$

Short Answer

Expert verified

Since, $\frac{\partial F_{3}}{\partial y} = \frac{\partial F_{2}}{\partial z}$ so the given vector is conservative.

A potential function for the field is $f (x, y, z) = x + 2 y - 3 z$ .

Step by step solution

Step 1. Given Information

We have to determine whether or not each of the vector fields in the given exercises is conservative. If the vector field is conservative, find a potential function for the field.

$F (x, y, z) = i + 2 j - 3 k$

Step 2. Firstly finding the given vector field is conservative or not.

A vector field $F (x, y, z) = (F_{1} (x, y, z), F_{2} (x, y, z), F_{3} (x, y, z))$ is conservative if and only if localid="1650559372319" $\frac{\partial F_{3}}{\partial y} = \frac{\partial F_{2}}{\partial z}, \frac{\partial F_{1}}{\partial z} = \frac{\partial F_{3}}{\partial x}, \frac{\partial F_{2}}{\partial x} = \frac{\partial F_{1}}{\partial y}$

$\frac{\partial F_{3}}{\partial y} = \frac{\partial}{\partial y} 3 \frac{\partial F_{2}}{\partial z} = \frac{\partial}{\partial z} 2 \frac{\partial F_{3}}{\partial y} = 0 \frac{\partial F_{2}}{\partial z} = 0$

Hence, $\frac{\partial F_{3}}{\partial y} = \frac{\partial F_{2}}{\partial z}$

Step 3. Now finding the potential function for the field.

Since, $F (x, y, z) = i + 2 j - 3 k$

$f (x, y, z) = \int 1 d x + B + C f (x, y, z) = x + α + B + C$

where $α$ is an arbitrary constant and B is the integral with respect to y of the terms in $F_{2} (x, y, z)$ in which the factorx does not appear.

Step 4. In this case, that is all of F2(x,y,z), so

$B = \int 2 d y B = 2 y + β$

where β is an arbitrary constant.

$C = - \int 3 d z C = - 3 z + γ$

where $γ$ is an arbitrary constant.

Step 5. Setting the constants equal to zero since they do not affect the gradient of f(x,y,z)

We have,

$f (x, y, z) = x + 2 y - 3 z$

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Determine whether or not each of the vector fields in Exercises 41–48 is conservative. If the vector field is conservative, find a potential function for the field.
$F (x, y, z) = i + 2 j - 3 k$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Firstly finding the given vector field is conservative or not.

Step 3. Now finding the potential function for the field.

Step 4. In this case, that is all of F2(x,y,z), so

Step 5. Setting the constants equal to zero since they do not affect the gradient of f(x,y,z)

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Determine whether or not each of the vector fields in Exercises 41–48 is conservative. If the vector field is conservative, find a potential function for the field.F(x,y,z)=i+2j−3k

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Firstly finding the given vector field is conservative or not.

Step 3. Now finding the potential function for the field.

Step 4. In this case, that is all of F2(x,y,z), so

Step 5. Setting the constants equal to zero since they do not affect the gradient of f(x,y,z)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Discrete Mathematics

Pure Maths

Decision Maths

Applied Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

Determine whether or not each of the vector fields in Exercises 41–48 is conservative. If the vector field is conservative, find a potential function for the field.
$F (x, y, z) = i + 2 j - 3 k$