Answer
$$a_{\mathrm{x}}=x$$
$$a_{\mathrm{y}}=2 x^{2} z+x^{2} y z$$
$$a_{\mathrm{z}}=x^{2} z^{2}+y z^{2}$$
Work Step by Step
we know that acceleration of a fluid particle can be expressed as and in component form; In task we have velocity:
$$V=x \hat{i}+x^{2} z \hat{j}+y z \hat{k}$$$$a=\frac{D \vec{V}}{D t}=\frac{\partial V}{\partial t}+U \frac{\partial V}{\partial x}+V \frac{\partial V}{\partial y}+W \frac{\partial V}{\partial z}$$$$U=x$$$$V=x^{2} z$$$$W=y z$$$$ a_{\mathrm{x}} =U \frac{\partial U}{\partial x}+V \frac{\partial U}{\partial y}+W \frac{\partial U}{\partial z} $$$$ a_{\mathrm{x}} =x \frac{\partial x}{\partial x}+x^{2} z \frac{\partial x}{\partial y}+y z \frac{\partial x}{\partial z} $$$$ a_{\mathrm{x}} =x $$ $$ a_{\mathrm{y}} =U \frac{\partial V}{\partial x}+V \frac{\partial V}{\partial y}+W \frac{\partial V}{\partial z} $$ $$ a_{\mathrm{y}} =x \frac{\partial x^{2} z}{\partial x}+x^{2} z \frac{\partial x^{2} z}{\partial y}+y z \frac{\partial x^{2} z}{\partial z} $$ $$ a_{y} =2 x^{2} z+x^{2} u z $$ $$ a_{z} =U \frac{\partial W}{\partial x}+V \frac{\partial W}{\partial y}+W \frac{\partial W}{\partial z} $$ $$ a_{z} =x \frac{\partial y z}{\partial x}+x^{2} z \frac{\partial y z}{\partial y}+y z \frac{\partial y z}{\partial z} $$ $$a_{z} =x^{2} z^{2}+y z^{2} $$
