VC3.41

From Exampleproblems

We shall use the identity $\mathrm{grad}(uv)=u\mathrm{grad}v+v\mathrm{grad}u\,$ --(1)

Now, $\mathrm{grad}u=i\frac{\partial}{\partial x}(e^{2x}+x^2z)+j\frac{\partial}{\partial y}(e^{2x}+x^2z)+k\frac{\partial}{\partial z}(e^{2x}+x^2z)=i(2e^{2x}+2xz)+j(0)+k(x^2)\,$ --(2)

and $\mathrm{grad}v=i\frac{\partial}{\partial x}(2z^2y-xy^2)+j\frac{\partial}{\partial y}(2z^2y-xy^2)+k\frac{\partial}{\partial z}(2z^2y-xy^2)=i(-y^2)+j(2z^2-2xy)+k(4zy)\,$ --(3)