VC3.3

From Example Problems
Jump to: navigation, search

i).\nabla f(r)=[i{\frac  {\partial }{\partial x}}+j{\frac  {\partial }{\partial y}}+k{\frac  {\partial }{\partial z}}]f(r)\,

=i{\frac  {\partial }{\partial x}}f(r)+j{\frac  {\partial }{\partial y}}f(r)+k{\frac  {\partial }{\partial z}}f(r)\,

=if'(r){\frac  {\partial r}{\partial x}}+jf'(r){\frac  {\partial r}{\partial y}}+kf'(r){\frac  {\partial r}{\partial z}}\,

=f'(r)[i{\frac  {\partial r}{\partial x}}+j{\frac  {\partial r}{\partial y}}+k{\frac  {\partial r}{\partial z}}]\,

=f'(r)\nabla r\,

ii). we have \nabla r=i{\frac  {\partial r}{\partial x}}+j{\frac  {\partial r}{\partial y}}+k{\frac  {\partial r}{\partial z}}\,

Since r=xi+yj+zk\, So,r^{2}=|r|^{2}=x^{2}+y^{2}+z^{2}\,

Differentiating the above,we get

2r{\frac  {\partial r}{\partial x}}=2x,{\frac  {\partial r}{\partial x}}={\frac  {x}{r}}\,

Similarly,{\frac  {\partial r}{\partial y}}={\frac  {y}{r}},{\frac  {\partial r}{\partial z}}={\frac  {z}{r}}\,

Therefore,by using the above,\nabla r={\frac  {x}{r}}i+{\frac  {y}{r}}j+{\frac  {z}{r}}k\,

={\frac  {1}{r}}(xi+yj+zk)={\frac  {1}{r}}r\,

Main Page