§16.7 Stokes’ Theorem
Lecture notes
Key equations
Curl: for a vector field $\F = M\i + N\j + P\k,$
\[
\begin{aligned}
\curl\F = \nabla\times\F &=
\begin{vmatrix}
\i & \j & \k \\[.5em]
\pd {}x & \pd{} y & \pd{} z\\[.5em]
M & N & P
\end{vmatrix}\\
&= \left(\pd Py - \pd Nz\right)\i +
\left(\pd Mz - \pd Px\right)\j +
\left(\pd Nx - \pd My\right)\k
\end{aligned}
\]
Stokes' theorem: for an oriented smooth surface $(S,\n)$ with boundary curve $C=\partial S,$
\[ \iint_S (\nabla\times\F)\cdot\n\,d\sigma = \oint_C \F\cdot d\r \]
Curl
Stokes' theorem
Interpretation
Examples
-
Verify Stokes' theorem for the vector field
\[ \F = 2yz\i - y^2\j + xy\k \]
over the cone
\[ S\colon\: z^2 = x^2 + y^2,\quad 0\le z\le 2 \]
and the disk
\[ D\colon\: x^2 + y^2 = 4,\quad z=2. \]
-
Verify Stokes' theorem for
\[ \F = \frac1{x^2+y^2}(-y\i + x\j) \]
over the annulus
\[ A\colon\: 1\le x^2+y^2 \le 4 \]