Chapter 14: Q. 38 (page 1096)

Show that the vector fields in Exercises 33–40 are not conservative.
$F (x, y, z) = \tan (y z) i + (x z s e c^{2} (y z) - 2) j + 4 z^{3} k$

Short Answer

Expert verified

The vector fields is not conservative because $\frac{\partial F_{1}}{\partial z} \neq \frac{\partial F_{2}}{\partial y} .$ .

Step by step solution

Step 1. Given Information

We have to show that the vector fields in the given exercise is not conservative.
$F (x, y, z) = \tan (y z) i + (x z s e c^{2} (y z) - 2) j + 4 z^{3} k$

Step 2. A vector field F(x,y)=(F1(x,y),F2(x,y)) is not conservative if and only if 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j4yPC90ZXh0Pjx0ZXh0IGZvbnQtZmFtaWx5PSJtYXRoMTRlM2NkNDkyNTllNmY2MDQ1NWUyNTUyNzgzIiBmb250LXNpemU9IjE2IiB0ZXh0LWFuY2hvcj0ibWlkZGxlIiB4PSI3MS41IiB5PSI0NCI+JiN4MjIwMjs8L3RleHQ+PHRleHQgZm9udC1mYW1pbHk9IkFyaWFsIiBmb250LXNpemU9IjE2IiBmb250LXN0eWxlPSJpdGFsaWMiIHRleHQtYW5jaG9yPSJtaWRkbGUiIHg9IjgyLjUiIHk9IjQ0Ij55PC90ZXh0Pjx0ZXh0IGZvbnQtZmFtaWx5PSJtYXRoMTRlM2NkNDkyNTllNmY2MDQ1NWUyNTUyNzgzIiBmb250LXNpemU9IjE2IiB0ZXh0LWFuY2hvcj0ibWlkZGxlIiB4PSI5Ny41IiB5PSIzMiI+LjwvdGV4dD48L3N2Zz4=" localid="1650549830535" ∂F1∂z≠∂F2∂y.

For the vector field $F (x, y, z) = \tan (y z) i + (x z s e c^{2} (y z) - 2) j + 4 z^{3} k$

$\frac{\partial F_{1}}{\partial z} = \frac{\partial}{\partial z} \tan (y z) \frac{\partial F_{1}}{\partial z} = y \tan (y z)$

Step 3. Now finding ∂F2∂x

$\frac{\partial F_{2}}{\partial x} = \frac{\partial}{\partial x} 4 z^{3} \frac{\partial F_{2}}{\partial x} = 0$

Hence, $\frac{\partial F_{1}}{\partial z} \neq \frac{\partial F_{2}}{\partial y} .$

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Show that the vector fields in Exercises 33–40 are not conservative.
$F (x, y, z) = \tan (y z) i + (x z s e c^{2} (y z) - 2) j + 4 z^{3} k$

Short Answer

Step by step solution

Step 1. Given Information

Step 3. Now finding ∂F2∂x

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Show that the vector fields in Exercises 33–40 are not conservative.F(x,y,z)=tan(yz)i+(xzsec2(yz)−2)j+4z3k

Short Answer

Step by step solution

Step 1. Given Information

Step 3. Now finding ∂F2∂x

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Calculus

Probability and Statistics

Theoretical and Mathematical Physics

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Pure Maths

Study anywhere. Anytime. Across all devices.

Show that the vector fields in Exercises 33–40 are not conservative.
$F (x, y, z) = \tan (y z) i + (x z s e c^{2} (y z) - 2) j + 4 z^{3} k$