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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Your Input

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

Answer

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