If-F-y-f-z-z-f-y-i-z-f-x-x-f-z-j-x-f-y-y-f-x-k-prove-that-F-r-f-

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Question Number 4 by user1 last updated on 25/Jan/15

$$\mathrm{If}$$$$\mathrm{F}(y\frac{\partial f}{\partial z}-z\frac{\partial f}{\partial y})\mathbf{i}+(z\frac{\partial f}{\partial x}-x\frac{\partial f}{\partial z})\mathbf{j}+(x\frac{\partial f}{\partial y}-y\frac{\partial f}{\partial x})\mathbf{k}$$$$\mathrm{prove}\mathrm{that}\mathbf{F}=\mathbf{r}\times ▽f.$$

Answered by user1 last updated on 29/Oct/14

$$\mathbf{r}\times ▽\mathbf{f}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ x& y& z\\ \frac{\partial f}{\partial x}& \frac{\partial f}{\partial y}& \frac{\partial f}{\partial z}\end{array}\right|$$$$=(y\frac{\partial f}{\partial z}-z\frac{\partial f}{\partial y})\mathbf{i}+(z\frac{\partial f}{\partial x}-x\frac{\partial f}{\partial z})\mathbf{j}+(x\frac{\partial f}{\partial y}-y\frac{\partial f}{\partial x})\mathbf{k}$$$$=\mathbf{F}$$

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