Test the vector field F to determine if it is conservative.
-$$\mathbf { F } = \left( \frac { 8 \mathrm { x } ^ { 7 } \mathrm { y } ^ { 8 } } { \mathrm { z } ^ { 7 } } \right) \mathrm { i } + \left( \frac { 8 \mathrm { x } ^ { 8 } \mathrm { y } ^ { 7 } } { \mathrm { z } ^ { 7 } } \right) \mathrm { j } - \left( \frac { 7 \mathrm { x } ^ { 8 } \mathrm { y } ^ { 8 } } { \mathrm { z } ^ { 8 } } \right) \mathbf { k }$$
A) Not conservative
B) Conservative

Question

Quizplus · Accepted Answer

This vector field is conservative because
$$\frac{\partial M}{\partial y} = 64x^7y^7z^{-7} = \frac{\partial N}{\partial x}$$
and
$$\frac{\partial N}{\partial z} = -56x^8y^7z^{-8} = \frac{\partial P}{\partial y}$$
and
$$\frac{\partial P}{\partial x} = 56x^7y^8z^{-8} = \frac{\partial M}{\partial z}.$$

Test the Vector Field F to Determine If It Is