Quick question on partial derivative of a gradient vector field,

Question

Given a scalar field $x_1(u,v,w,t)$ that maps $R^4$ to $R$,

Do I have this correct:

$$\frac {\partial}{\partial t} \nabla x_1$$ $$=\frac {\partial}{\partial t}(\frac {\partial x_1}{\partial u},\frac {\partial x_1}{\partial v},\frac {\partial x_1}{\partial w},\frac {\partial x_1}{\partial t}) $$ $$=(\frac {\partial}{\partial t}\frac {\partial x_1}{\partial u},\frac {\partial}{\partial t}\frac {\partial x_1}{\partial v},\frac {\partial}{\partial t}\frac {\partial x_1}{\partial w},\frac {\partial}{\partial t}\frac {\partial x_1}{\partial t})? $$

Thanks,

Answer 1

Yes, you do have this correct.

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