Find the curl of the vector field $\mathbf { F }$.
$$\mathbf { F } ( x , y , x ) = 7 y z ^ { 2 } \mathbf { i } + 4 x ^ { 5 } y ^ { 5 } \mathbf { j } + 8 x \mathbf { k }$$
A) $20 x ^ { 5 } y ^ { 4 }$
B) $14 y z + 20 x ^ { 4 } y ^ { 5 } - 7 z ^ { 2 } - 8$
C) $( 14 y z - 8 ) \mathbf { j } + \left( 20 x ^ { 4 } y ^ { 5 } - 7 z ^ { 2 } \right) \mathbf { k }$
D) $20 x ^ { 5 } y ^ { 4 } \mathbf { j }$

Question

Quizplus · Accepted Answer

The curl of a vector field $\mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}$ is given by $\nabla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k}$. Calculating these partial derivatives for the given $\mathbf{F}$, we find the curl to be $(14yz - 8)\mathbf{j} + (20x^4y^5 - 7z^2)\mathbf{k}$.

Find the Curl of the Vector Field F\mathbf { F }F

Find the Curl of the Vector Field $\mathbf { F }$