VC5.17

From Example Problems
Jump to: navigation, search

Since r=xi+yj+zk\, and dr=dxi+dyj+dzk\,

Therefore,\oint _{C}(e^{x}dx+2ydy-dz)=\oint _{C}(e^{x}i+2yj-k)\cdot (dxi+dyj+dzk)\,

or \oint _{C}(e^{x}dx+2ydy-dz)=\oint _{C}F\cdot dr\, --(1)

where F=e^{x}i+2yj-k\, --(2)

Now,by Stokes'theorm,we have \oint _{C}F\cdot dr=\iint _{S}{\mathrm  {curl}}F\cdot ndS\, --(3)

But {\mathrm  {curl}}F={\begin{vmatrix}i&j&k\\{\frac  {\partial }{\partial x}}&{\frac  {\partial }{\partial y}}&{\frac  {\partial }{\partial z}}\\e^{x}&2y&-1\end{vmatrix}}\,

=[{\frac  {\partial }{\partial x}}(-1)-{\frac  {\partial }{\partial z}}(2y)]i-[{\frac  {\partial }{\partial x}}(-1)-{\frac  {\partial }{\partial z}}(e^{x})]j+[{\frac  {\partial }{\partial x}}(2y)-{\frac  {\partial }{\partial y}}(e^{x})]k=0\,

Therefore,\iint _{S}{\mathrm  {curl}}F\cdot ndS=\iint _{S}0\cdot ndS=0\, --(4)

From (1),(3) and (4),\oint _{C}(e^{x}dx+2ydy-dz)=0\,

Main Page