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 $S$ in space to a line integral around the boundary of $S$. Therefore, just as the theorems before it, Stokes’ theorem can be used to reduce an integral over a geometric object $S$ to an integral over the boundary of $S$.

Theorem: Stokes’ Theorem

Let $S$ be a piecewise smooth oriented surface with a boundary that is a simple closed curve $C$ with positive orientation. If $\overrightarrow{F}$ is a vector field with component functions that have continuous partial derivatives on an open region containing $S$, 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 $\int {\int }_S\text{curl}\overrightarrow{F}\cdot dS$ using Stokes’ Theorem, where $\overrightarrow{F}=<y,3x,x^2>$ and $S$ is the surface, oriented outward, bounded by paraboloid $z=4-x^2-y^2$ and $xy$-plane. 

Exercise 1: Find $\int {\int }_S\text{curl}\overrightarrow{F}\cdot dS$ using Stokes’ Theorem, where $\overrightarrow{F}=<2y,x,y^2>$ and $S$ is the surface, oriented outward, bounded by paraboloid $z=-9+x^2+y^2$ and $xy$-plane. 

Example 2: Find $\int {\int }_S\text{curl}\overrightarrow{F}\cdot dS$ using Stokes’ Theorem, where $\overrightarrow{F}=<y\text{sin}(x),y\text{cos}(x),z{e}^x>$ and $S$ is the hemisphere $x^2+y^2+z^2=9$ with $x\ge 0$, oriented toward positive $x$-axis.

Exercise 2: Find $\int {\int }_S\text{curl}\overrightarrow{F}\cdot dS$ using Stokes’ Theorem, where $\overrightarrow{F}=<x\text{cos}(y),y\text{sin}(y),z^2>$ and $S$ is the hemisphere $x^2+y^2+z^2=4$ with $y\ge 0$, oriented toward positive $y$-axis.

Example 3: Calculate the line integral ${\int }_C\overrightarrow{F}\cdot d\overrightarrow{r}$, where $\overrightarrow{F}=<xy,x+z,yz>$ and $C$ is the boundary of part of the plane $x+2y+4z=8$ in the first octant. 

Exercise 3: Calculate the line integral ${\int }_C\overrightarrow{F}\cdot d\overrightarrow{r}$, where $\overrightarrow{F}=<x+y,yz,xz>$ and $C$ is the boundary of part of the plane $2x+y+3z=6$ in the first octant. 

Example 4: Calculate the line integral ${\int }_C\overrightarrow{F}\cdot d\overrightarrow{r}$, where $\overrightarrow{F}=<yz,y+z,x+y>$ and $C$ is curve of intersection of plane $z=x+3$ and the cylinder $x^2+y^2=4$.

Exercise 4: Calculate the line integral ${\int }_C\overrightarrow{F}\cdot d\overrightarrow{r}$, where $\overrightarrow{F}=<x-y,xz,y+z>$ and $C$ is curve of intersection of plane $y=2x+5$ and the cylinder $x^2+z^2=9$.

Example 5: Find $\int {\int }_S\text{curl}\overrightarrow{F}\cdot dS$ using Stokes’ Theorem, where $\overrightarrow{F}=<x^{2}z,e^{xz},e^{xy}>$ and $S$ is the half ellipsoid $x^2+9y^2+9z^2=9$ with $x\ge 0$, oriented toward positive $x$-axis.

Group work:

1. Find $\int {\int }_S\text{curl}\overrightarrow{F}\cdot dS$ using Stokes’ Theorem, where $\overrightarrow{F}=<e^{xy},e^{yz},y{z}^3>$ and $S$ is the half ellipsoid $4x^2+y^2+4z^2=4$ with $y\ge 0$, oriented toward positive $y$-axis.

2. Calculate the line integral ${\int }_C\overrightarrow{F}\cdot d\overrightarrow{r}$, where $\overrightarrow{F}=<x^{2}y,x^3,yz>$ and $C$ is curve of intersection of surface $z=x^2+y^2$ and the cylinder $x^2+y^2=4$ oriented positive.

3. Calculate the line integral ${\int }_C\overrightarrow{F}\cdot d\overrightarrow{r}$, where $\overrightarrow{F}=<y^{2}z,yz,x^{2}y>$ and $C$ is curve of intersection of surface $x=9-y^2-z^2$ and the cylinder $z^2+y^2=5$ oriented positive.