VC3.6

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\nabla e^{r^2}=i\frac{\partial e^{r^2}}{\partial x}+j\frac{\partial e^{r^2}}{\partial y}+k\frac{\partial e^{r^2}}{\partial z}\,

=[2re^{r^2}\frac{\partial r}{\partial x}]i+[2re^{r^2}\frac{\partial r}{\partial y}]j+[2re^{r^2}\frac{\partial r}{\partial z}]k\,

=2re^{r^2}[\frac{\partial r}{\partial x}i+\frac{\partial r}{\partial y}j+\frac{\partial r}{\partial z}k]\, [Equation 1]

Here r=xi+yj+zk\, and r^2=x^2+y^2+z^2\, [Equation 2]

From (2) \frac{\partial r}{\partial x}=\frac{x}{r},\frac{\partial r}{\partial y}=\frac{y}{r},\frac{\partial r}{\partial z}=\frac{z}{r}\,

Therefore \nabla e^{r^2}=2re^{r^2}[\frac{x}{r}i+\frac{y}{r}j+\frac{z}{r}k]\,

=2re^{r^2}(xi+yj+zk)=2e^{r^2}r \, by (2)


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