Find the curl of the vector field. $\mathbf { F } ( x , y , z ) = 8 x y \mathbf { i } + 10 y z \mathbf { j } + 5 x z \mathbf { k }$
Select the correct answer.
A) $- 10 y \mathbf { i } - 5 z \mathbf { j } - 8 x \mathbf { k }$
B) $- 10 x \mathbf { i } + 5 y \mathbf { j } + 8 z \mathbf { k }$
C) $- 10 x \mathbf { i } + 5 z \mathbf { j } - 8 y \mathbf { k }$
D) $10 y \mathbf { i } - 5 z \mathbf { j } - 8 x \mathbf { k }$
E) None of these

Question

Quizplus · Accepted Answer

The curl of a vector field is given by the cross product of the del operator and the vector field. In this case, we have:$$
abla 	imes \mathbf{F} = \left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}ight) 	imes (8xy, 10yz, 5xz)$$$$= \left(\frac{\partial}{\partial y}(5xz) - \frac{\partial}{\partial z}(10yz), \frac{\partial}{\partial z}(8xy) - \frac{\partial}{\partial x}(5xz), \frac{\partial}{\partial x}(10yz) - \frac{\partial}{\partial y}(8xy)ight)$$$$= (-10y, 8x, -8y)$$$$= -10y \mathbf{i} - 5z \mathbf{j} - 8x \mathbf{k}$$Therefore, the correct answer is A.

Find the Curl of the Vector Field F(x,y,z)=8xyi+10yzj+5xzk\mathbf { F } ( x , y , z ) = 8 x y \mathbf { i } + 10 y z \mathbf { j } + 5 x z \mathbf { k }F(x,y,z)=8xyi+10yzj+5xzk

Find the Curl of the Vector Field $\mathbf { F } ( x , y , z ) = 8 x y \mathbf { i } + 10 y z \mathbf { j } + 5 x z \mathbf { k }$