91影视

An integral of the form${鈭�}_C\mathit{F}路d\mathit{r}$.

"Let $S$be an oriented, smooth or piecewise-smooth surface bounded by a curve$C$. Suppose that $\mathit{n}$is an oriented unit normal vector of $S$and $C$has a parametrization that traverses $C$in the counterclockwise direction with respect to$\mathit{n}$.
If a vector field$\mathit{F}(x,y,z)={\mathit{F}}_{\mathbf{1}}(x,y,z)\mathit{i}+{\mathit{F}}_{\mathbf{2}}(x,y,z)\mathit{j}+{\mathit{F}}_{\mathbf{3}}(x,y,z)\mathit{k}$ is defined on$S,$then,${鈭�}_c\mathit{F}(x,y,z)路d\mathit{r}=鈭�{鈭�}_{S}curl\mathit{F}(x,y,z)路\mathit{n}dS"$.

If $S$is an oriented, smooth or piecewise-smooth surface bounded by a curve $C$, Stokes' Theorem relates a line integral of a vector field around the boundary curve$C$ to a surface integral of the curl of the vector field.
When the line integral has piecewise-continuous boundary, for example, if the boundary curve$C$ is a rectangle or triangle , it requires the evaluation of several smooth pieces.
If you use the right-hand side of Stokes' Theorem, you obtain the same result with a single area integral.