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Divergence Calculator

Calculate divergence step by step

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

Related calculators: Partial Derivative Calculator, Dot Product Calculator

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Your Input

Calculate div⁡⟨sin⁡(xy),cos⁡(xy),ez⟩\operatorname{div} \left\langle \sin{\left(x y \right)}, \cos{\left(x y \right)}, e^{z}\right\rangle.

Solution

By definition, div⁡⟨sin⁡(xy),cos⁡(xy),ez⟩=∇⋅⟨sin⁡(xy),cos⁡(xy),ez⟩\operatorname{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, div⁡⟨sin⁡(xy),cos⁡(xy),ez⟩=⟨∂∂x,∂∂y,∂∂z⟩⋅⟨sin⁡(xy),cos⁡(xy),ez⟩\operatorname{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, div⁡⟨sin⁡(xy),cos⁡(xy),ez⟩=∂∂x(sin⁡(xy))+∂∂y(cos⁡(xy))+∂∂z(ez).\operatorname{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 xx: ∂∂x(sin⁡(xy))=ycos⁡(xy)\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 yy: ∂∂y(cos⁡(xy))=−xsin⁡(xy)\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 zz: ∂∂z(ez)=ez\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: div⁡⟨sin⁡(xy),cos⁡(xy),ez⟩=−xsin⁡(xy)+ycos⁡(xy)+ez.\operatorname{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

div⁡⟨sin⁡(xy),cos⁡(xy),ez⟩=−xsin⁡(xy)+ycos⁡(xy)+ez\operatorname{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