Stokes’ Theorem and the Divergence Theorem — Theory
1. Big picture
Stokes’ Theorem and the Divergence Theorem are two major three-dimensional versions of the
Fundamental Theorem of Calculus. Both convert an integral over a boundary into an integral over
the object inside.
\[
\text{boundary information}
=
\text{interior information}.
\]
Stokes’ Theorem connects a curve boundary with a surface. The Divergence Theorem connects a closed
surface boundary with a volume.
2. Stokes’ Theorem
Stokes’ Theorem says:
\[
\oint_C \mathbf F\cdot d\mathbf r
=
\iint_S
(\nabla\times\mathbf F)\cdot\mathbf n\,dS.
\]
The left side is a line integral around the boundary curve \(C\).
The right side is the flux of curl through the surface \(S\).
The curve \(C\) must be the boundary of the surface \(S\).
3. Curl in Stokes’ Theorem
If
\[
\mathbf F=\langle P,Q,R\rangle,
\]
then
\[
\nabla\times\mathbf F
=
\left\langle
R_y-Q_z,\,
P_z-R_x,\,
Q_x-P_y
\right\rangle.
\]
Curl measures local rotation of the vector field. Stokes’ Theorem says that the total curl
passing through the surface equals the circulation around its boundary.
4. Orientation in Stokes’ Theorem
Orientation is essential. Once a normal direction is chosen on the surface, the boundary direction
is determined by the right-hand rule.
\[
\text{surface normal}
\quad\Longleftrightarrow\quad
\text{compatible boundary direction}.
\]
If the normal is reversed, the compatible boundary direction is also reversed.
\[
\iint_S(\nabla\times\mathbf F)\cdot(-\mathbf n)\,dS
=
-
\iint_S(\nabla\times\mathbf F)\cdot\mathbf n\,dS.
\]
5. Parametric surfaces for Stokes’ Theorem
A parametric surface is written as:
\[
\mathbf r(u,v)
=
\langle X(u,v),Y(u,v),Z(u,v)\rangle.
\]
The tangent vectors are:
\[
\mathbf r_u
=
\frac{\partial\mathbf r}{\partial u},
\qquad
\mathbf r_v
=
\frac{\partial\mathbf r}{\partial v}.
\]
The oriented area vector is:
\[
\mathbf r_u\times\mathbf r_v\,du\,dv.
\]
Reversing the cross product reverses the surface orientation:
\[
\mathbf r_v\times\mathbf r_u
=
-(\mathbf r_u\times\mathbf r_v).
\]
6. Example of Stokes’ Theorem
Let
\[
\mathbf F
=
\left\langle
-\frac{y}{2},
\frac{x}{2},
0
\right\rangle.
\]
Then
\[
\nabla\times\mathbf F
=
\langle 0,0,1\rangle.
\]
If \(S\) is the unit disk in the \(xy\)-plane with upward normal, then
\[
\iint_S(\nabla\times\mathbf F)\cdot\mathbf n\,dS
=
\iint_S1\,dA
=
\pi.
\]
The boundary line integral around the unit circle also equals \(\pi\).
7. Divergence Theorem
The Divergence Theorem says:
\[
\iint_{\partial V}
\mathbf F\cdot\mathbf n\,dS
=
\iiint_V
\nabla\cdot\mathbf F\,dV.
\]
The left side is the flux through the closed surface \(\partial V\).
The right side is the total divergence inside the volume \(V\).
8. Divergence
If
\[
\mathbf F=\langle P,Q,R\rangle,
\]
then the divergence is:
\[
\nabla\cdot\mathbf F
=
P_x+Q_y+R_z.
\]
Divergence measures source strength. Positive divergence means the field behaves like a source.
Negative divergence means the field behaves like a sink.
9. Closed surfaces
The Divergence Theorem requires a closed surface. A closed surface completely encloses a volume.
Examples include:
- a sphere,
- a box,
- a closed cylinder,
- any closed smooth surface enclosing a solid region.
The standard orientation is outward.
\[
\mathbf n
=
\text{outward unit normal}.
\]
If inward orientation is used, the flux changes sign.
10. Example of the Divergence Theorem
Let
\[
\mathbf F=\langle x,y,z\rangle.
\]
Then
\[
\nabla\cdot\mathbf F
=
1+1+1
=
3.
\]
For the unit ball,
\[
\operatorname{Vol}(V)
=
\frac{4\pi}{3}.
\]
Therefore,
\[
\iiint_V 3\,dV
=
3\cdot\frac{4\pi}{3}
=
4\pi.
\]
Thus the outward flux through the unit sphere is \(4\pi\).
11. Relationship between the theorems
Stokes’ Theorem and the Divergence Theorem are related but not the same.
A helpful memory rule:
\[
\text{Stokes: curve } \leftrightarrow \text{ surface},
\qquad
\text{Divergence: surface } \leftrightarrow \text{ volume}.
\]
13. Common mistakes
- Using Stokes’ Theorem on a closed surface: Stokes uses a surface with a boundary curve.
- Using the Divergence Theorem on an open surface: the Divergence Theorem needs a closed surface.
- Confusing curl and divergence: Stokes uses curl, while the Divergence Theorem uses divergence.
- Wrong orientation: reversing orientation reverses the sign of the flux or circulation comparison.
- Inconsistent boundary: in Stokes’ Theorem, the boundary curve must be the boundary of the chosen surface.
- Forgetting all pieces of a closed surface: a cylinder includes the side, top, and bottom.
- Expecting exact equality from numerical integration: small differences can remain unless many panels are used.
