Find the curl of the vector field $\overrightarrow { \mathbf { F } } ( x , y , z ) = ( 8 y - z ) \hat { \mathbf { i } } + x y z \hat { j } + 4 e ^ { z } \hat { \mathbf { k } }$
A) $\operatorname { curl } ( \overrightarrow { \mathbf { F } } ) = - x y \hat { \mathbf { i } } - \hat { \mathbf { j } } + ( y z - 8 ) \hat { \mathbf { k } }$
B) $\operatorname { curl } ( \overrightarrow { \mathbf { F } } ) = - x y \hat { \mathbf { i } } - \hat { \mathbf { j } } + ( 8 - y z ) \hat { \mathbf { k } }$
C) $\operatorname { curl } ( \overrightarrow { \mathbf { F } } ) = - x y \hat { \mathbf { i } } + \hat { \mathbf { j } } + ( 8 - y z ) \hat { \mathbf { k } }$
D) $\operatorname { curl } ( \overrightarrow { \mathbf { F } } ) = x y \hat { \mathbf { i } } + \hat { \mathbf { j } } + ( y z - 8 ) \hat { \mathbf { k } }$
E) $\operatorname { curl } ( \overrightarrow { \mathrm { F } } ) = x y \hat { \mathbf { i } } - \hat { \mathbf { j } } + ( y z - 8 ) \hat { \mathbf { k } }$

Question

Find the curl of the vector field $\overrightarrow { \mathbf { F } } ( x , y , z ) = ( 8 y - z ) \hat { \mathbf { i } } + x y z \hat { j } + 4 e ^ { z } \hat { \mathbf { k } }$ 
A) $\operatorname { curl } ( \overrightarrow { \mathbf { F } } ) = - x y \hat { \mathbf { i } } - \hat { \mathbf { j } } + ( y z - 8 ) \hat { \mathbf { k } }$
B) $\operatorname { curl } ( \overrightarrow { \mathbf { F } } ) = - x y \hat { \mathbf { i } } - \hat { \mathbf { j } } + ( 8 - y z ) \hat { \mathbf { k } }$
C) $\operatorname { curl } ( \overrightarrow { \mathbf { F } } ) = - x y \hat { \mathbf { i } } + \hat { \mathbf { j } } + ( 8 - y z ) \hat { \mathbf { k } }$
D) $\operatorname { curl } ( \overrightarrow { \mathbf { F } } ) = x y \hat { \mathbf { i } } + \hat { \mathbf { j } } + ( y z - 8 ) \hat { \mathbf { k } }$
E) $\operatorname { curl } ( \overrightarrow { \mathrm { F } } ) = x y \hat { \mathbf { i } } - \hat { \mathbf { j } } + ( y z - 8 ) \hat { \mathbf { k } }$

Quizplus · Accepted Answer

The curl of a vector field $\overrightarrow{\mathbf{F}} = P\hat{\mathbf{i}} + Q\hat{\mathbf{j}} + R\hat{\mathbf{k}}$ is given by $\nabla \times \overrightarrow{\mathbf{F}} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right)\hat{\mathbf{i}} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right)\hat{\mathbf{j}} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)\hat{\mathbf{k}}$. For $\overrightarrow{\mathbf{F}} = (8y - z)\hat{\mathbf{i}} + xyz\hat{\mathbf{j}} + 4e^{z}\hat{\mathbf{k}}$, calculating the partial derivatives and substituting them into the formula for curl gives $\operatorname{curl}(\overrightarrow{\mathbf{F}}) = -xy\hat{\mathbf{i}} - \hat{\mathbf{j}} + (yz - 8)\hat{\mathbf{k}}$, which matches choice A.

Find the Curl of the Vector Field F→(x,y,z)=(8y−z)i^+xyzj^+4ezk^\overrightarrow { \mathbf { F } } ( x , y , z ) = ( 8 y - z ) \hat { \mathbf { i } } + x y z \hat { j } + 4 e ^ { z } \hat { \mathbf { k } }F(x,y,z)=(8y−z)i^+xyzj^​+4ezk^

Find the Curl of the Vector Field $\overrightarrow { \mathbf { F } } ( x , y , z ) = ( 8 y - z ) \hat { \mathbf { i } } + x y z \hat { j } + 4 e ^ { z } \hat { \mathbf { k } }$