# Divergence Calculator

The calculator will find the divergence of the given vector field, with steps shown.

Related calculators: Partial Derivative Calculator, Vector Dot (Inner) Product Calculator

(
,
,
)
(
,
,
)
Leave empty, if you don't need the divergence at a specific point.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Calculate $\operatorname{div}{\left(\sin{\left(x y \right)},\cos{\left(x y \right)},e^{z} \right)}$.

## Solution

By definition, $\operatorname{div}{\left(\sin{\left(x y \right)},\cos{\left(x y \right)},e^{z} \right)} = \nabla\cdot \left(\sin{\left(x y \right)}, \cos{\left(x y \right)}, e^{z}\right)$, or, equivalently, $\operatorname{div}{\left(\sin{\left(x y \right)},\cos{\left(x y \right)},e^{z} \right)} = \left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right)\cdot \left(\sin{\left(x y \right)}, \cos{\left(x y \right)}, e^{z}\right).$

Thus, $\operatorname{div}{\left(\sin{\left(x y \right)},\cos{\left(x y \right)},e^{z} \right)} = \frac{\partial}{\partial x} \left(\sin{\left(x y \right)}\right) + \frac{\partial}{\partial y} \left(\cos{\left(x y \right)}\right) + \frac{\partial}{\partial z} \left(e^{z}\right).$

Find the partial derivative of component 1 with respect to $x$: $\frac{\partial}{\partial x} \left(\sin{\left(x y \right)}\right) = y \cos{\left(x y \right)}$ (for steps, see derivative calculator).

Find the partial derivative of component 2 with respect to $y$: $\frac{\partial}{\partial y} \left(\cos{\left(x y \right)}\right) = - x \sin{\left(x y \right)}$ (for steps, see derivative calculator).

Find the partial derivative of component 3 with respect to $z$: $\frac{\partial}{\partial z} \left(e^{z}\right) = e^{z}$ (for steps, see derivative calculator).

Now, just sum up the above expressions to get the divergence: $\operatorname{div}{\left(\sin{\left(x y \right)},\cos{\left(x y \right)},e^{z} \right)} = - x \sin{\left(x y \right)} + y \cos{\left(x y \right)} + e^{z}.$

$\operatorname{div}{\left(\sin{\left(x y \right)},\cos{\left(x y \right)},e^{z} \right)} = - x \sin{\left(x y \right)} + y \cos{\left(x y \right)} + e^{z}$A