Test the vector field F to determine if it is conservative.
-$$\mathrm { F } = 7 \mathrm { x } ^ { 7 } \mathrm { y } ^ { 5 } \mathrm { i } + \left( 5 \mathrm { x } ^ { 7 } \mathrm { y } ^ { 4 } + \frac { \mathrm { z } ^ { 4 } } { \mathrm { y } ^ { 2 } } \right) \mathrm { j } - \frac { 4 \mathrm { z } ^ { 3 } } { \mathrm { y } } \mathbf { k }$$
A) Not conservative
B) Conservative

Question

Quizplus · Accepted Answer

A vector field is conservative if and only if it is the gradient of some scalar potential function. To test for conservativeness, we can use the property that a vector field is conservative if and only if its curl is zero. Therefore, we need to compute the curl of F and check if it is zero.

$$\nabla \times \mathrm { F } = \left( \frac { \partial Q } { \partial y } - \frac { \partial P } { \partial z } \right) \mathrm { i } + \left( \frac { \partial R } { \partial z } - \frac { \partial P } { \partial x } \right) \mathrm { j } + \left( \frac { \partial P } { \partial y } - \frac { \partial Q } { \partial x } \right) \mathrm { k } $$

Here, P = 7x^7y^5, Q = 5x^7y^4 + z^4/y^2, and R = -4z^3/y. Computing the partial derivatives and simplifying, we get:

$$\nabla \times \mathrm { F } = \left( \frac { 4z } { y^2 } \right) \mathrm { i } + \left( \frac { -35x^6y^3 - 4z^3 } { y^2 } \right) \mathrm { j } + \left( \frac { 0 } { y } \right) \mathrm { k } $$

This curl is not identically zero, and so the vector field is not conservative.

Test the Vector Field F to Determine If It Is