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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