Answer to Question #289162 in Calculus for Arpan

Question

If A=2xz^2i-yzj+3xz^3k then what is the value of curlA?

Answer 1

We have

$A=2xz^{2}i-yzj+3xz^{3}k$

To get the value for Curl $A$

We shall find the cross product $\nabla$ X $A$

So, we have

$\nabla$ X $A$ $\displaystyle =\begin{vmatrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ 2xz^{2} & -yzj & 3xz^{3} \end{vmatrix}$

$\displaystyle\\=i\left\{\frac{\partial}{\partial y}(3xz^{3})-\frac{\partial}{\partial z}(-yz)\right\}-j\left\{\frac{\partial}{\partial x}(3xz^{3})-\frac{\partial}{\partial z}(2xz^{2})\right\}+k\left\{\frac{\partial}{\partial x}(-yz)-\frac{\partial}{\partial y}(2xz^{2})\right\}$

$=i\left\{-(-y)\right\}-j\left\{3z^{3}-4xz\right\}+k\left\{0-0\right\}$

$=yi-(3z^{3}-4xz)j+0k\\=yi-z(3z^{2}-4x)j$

Which is the value of Curl A.