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

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

Solution

By definition, $$$\text{div} \left\langle \sin{\left(x y \right)}, \cos{\left(x y \right)}, e^{z}\right\rangle = \nabla\cdot \left\langle \sin{\left(x y \right)}, \cos{\left(x y \right)}, e^{z}\right\rangle$$$, or, equivalently, $$$\text{div} \left\langle \sin{\left(x y \right)}, \cos{\left(x y \right)}, e^{z}\right\rangle = \left\langle \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right\rangle\cdot \left\langle \sin{\left(x y \right)}, \cos{\left(x y \right)}, e^{z}\right\rangle$$$, where $$$\cdot$$$ is the dot product operator.

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

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

Answer

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