Find the curl of the vector field. $\mathbf { F } ( x , y , z ) = 10 x y \mathbf { i } + 10 y z \mathbf { j } + 7 x z \mathbf { k }$
A) $10 y \mathbf { i } - 7 z \mathbf { j } - 10 x \mathbf { k }$
B) $- 10 x \mathbf { i } + 7 z \mathbf { j } - 10 y \mathbf { k }$
C) $- 10 x \mathbf { i } + 7 y \mathbf { j } + 10 z \mathbf { k }$
D) $- 10 y \mathbf { i } - 7 z \mathbf { j } - 10 x \mathbf { k }$
E) None of these

Question

Quizplus · Accepted Answer

The curl of a vector field is given by the determinant of the matrix 
$$
\begin{pmatrix}\boldsymbol{\mathrm{i}}&\boldsymbol{\mathrm{j}}&\boldsymbol{\mathrm{k}}\\frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\F_x&F_y&F_z\end{pmatrix}.
$$ 
Computing the determinant of this matrix for the given vector field, we get
\begin{align*}
\begin{vmatrix}\boldsymbol{\mathrm{i}}&\boldsymbol{\mathrm{j}}&\boldsymbol{\mathrm{k}}\\frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\10xy&10yz&7xz\end{vmatrix} &= \left( \frac{\partial}{\partial y} (7xz) - \frac{\partial}{\partial z} (10yz) \right) \boldsymbol{\mathrm{i}} - \left( \frac{\partial}{\partial x} (7xz) - \frac{\partial}{\partial z} (10xy) \right) \boldsymbol{\mathrm{j}} + \left( \frac{\partial}{\partial x} (10yz) - \frac{\partial}{\partial y} (10xy) \right) \boldsymbol{\mathrm{k}} \
&= (0 - 10y)\boldsymbol{\mathrm{i}} - (7z - 0)\boldsymbol{\mathrm{j}} + (10z - 10x)\boldsymbol{\mathrm{k}} \
&= -10y\boldsymbol{\mathrm{i}} - 7z\boldsymbol{\mathrm{j}} - 10x\boldsymbol{\mathrm{k}}.
\end{align*}
Therefore, the answer is (D).

Find the Curl of the Vector Field F(x,y,z)=10xyi+10yzj+7xzk\mathbf { F } ( x , y , z ) = 10 x y \mathbf { i } + 10 y z \mathbf { j } + 7 x z \mathbf { k }F(x,y,z)=10xyi+10yzj+7xzk

Find the Curl of the Vector Field $\mathbf { F } ( x , y , z ) = 10 x y \mathbf { i } + 10 y z \mathbf { j } + 7 x z \mathbf { k }$