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

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

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