# Divergence of $\left\langle x^{2} y, x y z, y z^{2}\right\rangle$

The calculator will find the divergence of the vector field $\left\langle x^{2} y, x y z, y z^{2}\right\rangle$, with steps shown.

Related calculators: Partial Derivative Calculator, Dot Product Calculator

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Calculate $\text{div} \left\langle x^{2} y, x y z, y z^{2}\right\rangle$.

### Solution

By definition, $\text{div} \left\langle x^{2} y, x y z, y z^{2}\right\rangle = \nabla\cdot \left\langle x^{2} y, x y z, y z^{2}\right\rangle$, or, equivalently, $\text{div} \left\langle x^{2} y, x y z, y z^{2}\right\rangle = \left\langle \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right\rangle\cdot \left\langle x^{2} y, x y z, y z^{2}\right\rangle$, where $\cdot$ is the dot product operator.

Thus, $\text{div} \left\langle x^{2} y, x y z, y z^{2}\right\rangle = \frac{\partial}{\partial x} \left(x^{2} y\right) + \frac{\partial}{\partial y} \left(x y z\right) + \frac{\partial}{\partial z} \left(y z^{2}\right).$

Find the partial derivative of component 1 with respect to $x$: $\frac{\partial}{\partial x} \left(x^{2} y\right) = 2 x y$ (for steps, see derivative calculator).

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

Find the partial derivative of component 3 with respect to $z$: $\frac{\partial}{\partial z} \left(y z^{2}\right) = 2 y z$ (for steps, see derivative calculator).

Now, just sum up the above expressions to get the divergence: $\text{div} \left\langle x^{2} y, x y z, y z^{2}\right\rangle = 2 x y + x z + 2 y z$.

$\text{div} \left\langle x^{2} y, x y z, y z^{2}\right\rangle = 2 x y + x z + 2 y z$A