Divergence Calculator

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

Enter a vector field:

`mathbf{vec{F}(x,y,z)}=` (, , )

Calculate at `P=` (, ,)

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

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Solution

Your input: calculate $$$div\left(\sin{\left(x y \right)}, \cos{\left(x y \right)}, e^{z}\right)$$$.

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

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

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

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

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

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

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