Section 6.9 Stokes’ Theorem

6.9 Stokes’ Theorem

In this section, we study Stokes’ theorem, a higher-dimensional generalization of Green’s theorem. This theorem, like the Fundamental Theorem for Line Integrals and Green’s theorem, is a generalization of the Fundamental Theorem of Calculus to higher dimensions. Stokes’ theorem relates a vector surface integral over surface [latex]S[/latex] in space to a line integral around the boundary of [latex]S[/latex]. Therefore, just as the theorems before it, Stokes’ theorem can be used to reduce an integral over a geometric object [latex]S[/latex] to an integral over the boundary of [latex]S[/latex].

Theorem: Stokes’ Theorem

Let [latex]S[/latex] be a piecewise smooth oriented surface with a boundary that is a simple closed curve [latex]C[/latex] with positive orientation. If [latex]\overrightarrow{F}[/latex] is a vector field with component functions that have continuous partial derivatives on an open region containing [latex]S[/latex], then 

\[\int_{C}\overrightarrow{F}\cdot d\overrightarrow{r} =\int\int_{S}\text{curl}\overrightarrow{F}\cdot dS\]

\[=\int\int_{S}\text{curl}\overrightarrow{F}\cdot\overrightarrow{N}dS.\]

\[=\int\int_{D}\text{curl}\overrightarrow{F}\cdot(\overrightarrow{s_{u}}\times\overrightarrow{s_{v}})dA\]

Example 1: Find [latex]\int\int_{S}\text{curl}\overrightarrow{F}\cdot dS[/latex] using Stokes’ Theorem, where [latex]\overrightarrow{F}=\lt y,3x,x^{2}\gt[/latex] and [latex]S[/latex] is the surface, oriented outward, bounded by paraboloid [latex]z=4-x^{2}-y^{2}[/latex] and [latex]xy[/latex]-plane. 

Exercise 1: Find [latex]\int\int_{S}\text{curl}\overrightarrow{F}\cdot dS[/latex] using Stokes’ Theorem, where [latex]\overrightarrow{F}=\lt 2y,x,y^{2}\gt[/latex] and [latex]S[/latex] is the surface, oriented outward, bounded by paraboloid [latex]z=-9+x^{2}+y^{2}[/latex] and [latex]xy[/latex]-plane. 

Example 2: Find [latex]\int\int_{S}\text{curl}\overrightarrow{F}\cdot dS[/latex] using Stokes’ Theorem, where [latex]\overrightarrow{F}=\lt y\text{sin}(x),y\text{cos}(x),ze^{x}\gt[/latex] and [latex]S[/latex] is the hemisphere [latex]x^{2}+y^{2}+z^{2}=9[/latex] with [latex]x\geq0[/latex], oriented toward positive [latex]x[/latex]-axis.

Exercise 2: Find [latex]\int\int_{S}\text{curl}\overrightarrow{F}\cdot dS[/latex] using Stokes’ Theorem, where [latex]\overrightarrow{F}=\lt x\text{cos}(y),y\text{sin}(y),z^{2}\gt[/latex] and [latex]S[/latex] is the hemisphere [latex]x^{2}+y^{2}+z^{2}=4[/latex] with [latex]y\geq0[/latex], oriented toward positive [latex]y[/latex]-axis.

Example 3: Calculate the line integral [latex]\int_{C}\overrightarrow{F}\cdot d\overrightarrow{r}[/latex], where [latex]\overrightarrow{F}=\lt xy,x+z,yz\gt[/latex] and [latex]C[/latex] is the boundary of part of the plane [latex]x+2y+4z=8[/latex] in the first octant. 

Exercise 3: Calculate the line integral [latex]\int_{C}\overrightarrow{F}\cdot d\overrightarrow{r}[/latex], where [latex]\overrightarrow{F}=\lt x+y,yz,xz\gt[/latex] and [latex]C[/latex] is the boundary of part of the plane [latex]2x+y+3z=6[/latex] in the first octant. 

Example 4: Calculate the line integral [latex]\int_{C}\overrightarrow{F}\cdot d\overrightarrow{r}[/latex], where [latex]\overrightarrow{F}=\lt yz,y+z,x+y\gt[/latex] and [latex]C[/latex] is curve of intersection of plane [latex]z=x+3[/latex] and the cylinder [latex]x^{2}+y^{2}=4[/latex].

Exercise 4: Calculate the line integral [latex]\int_{C}\overrightarrow{F}\cdot d\overrightarrow{r}[/latex], where [latex]\overrightarrow{F}=\lt x-y,xz,y+z\gt[/latex] and [latex]C[/latex] is curve of intersection of plane [latex]y=2x+5[/latex] and the cylinder [latex]x^{2}+z^{2}=9[/latex].

Example 5: Find [latex]\int\int_{S}\text{curl}\overrightarrow{F}\cdot dS[/latex] using Stokes’ Theorem, where [latex]\overrightarrow{F}=\lt x^{2}z,e^{xz},e^{xy}\gt[/latex] and [latex]S[/latex] is the half ellipsoid [latex]x^{2}+9y^{2}+9z^{2}=9[/latex] with [latex]x\geq0[/latex], oriented toward positive [latex]x[/latex]-axis.

Group work:

1. Find [latex]\int\int_{S}\text{curl}\overrightarrow{F}\cdot dS[/latex] using Stokes’ Theorem, where [latex]\overrightarrow{F}=\lt e^{xy},e^{yz},yz^{3}\gt[/latex] and [latex]S[/latex] is the half ellipsoid [latex]4x^{2}+y^{2}+4z^{2}=4[/latex] with [latex]y\geq0[/latex], oriented toward positive [latex]y[/latex]-axis.

2. Calculate the line integral [latex]\int_{C}\overrightarrow{F}\cdot d\overrightarrow{r}[/latex], where [latex]\overrightarrow{F}=\lt x^{2}y,x^{3},yz\gt[/latex] and [latex]C[/latex] is curve of intersection of surface [latex]z=x^{2}+y^{2}[/latex] and the cylinder [latex]x^{2}+y^{2}=4[/latex] oriented positive.

3. Calculate the line integral [latex]\int_{C}\overrightarrow{F}\cdot d\overrightarrow{r}[/latex], where [latex]\overrightarrow{F}=\lt y^{2}z,yz,x^{2}y\gt[/latex] and [latex]C[/latex] is curve of intersection of surface [latex]x=9-y^{2}-z^{2}[/latex] and the cylinder [latex]z^{2}+y^{2}=5[/latex] oriented positive.