The theorems of vector calculus
Green, Gauss and Stokes say the same thing in different dimensions: what happens inside a region is determined by what happens on its boundary. Maxwell’s equations come from there.
01One idea, three statements
The three theorems that close vector calculus say the same thing in different guises: what happens on the boundary is determined by what happens inside. It is the same structure as the Fundamental Theorem of Calculus, but with one more dimension each time.
| Theorem | Boundary | Interior |
|---|---|---|
| Fundamental theorem of calculus | \( f(b) - f(a) \): two points | \( \displaystyle\int_a^b f' \, dx \) |
| Green | \( \displaystyle\oint_C \vec{F} \cdot d\vec{r} \) | \( \displaystyle\iint_D (\nabla \times \vec{F})_z \, dA \) |
| Stokes | \( \displaystyle\oint_C \vec{F} \cdot d\vec{r} \) | \( \displaystyle\iint_S (\nabla \times \vec{F}) \cdot \hat{n} \, dS \) |
| Divergence (Gauss) | \( \displaystyle\oiint_S \vec{F} \cdot \hat{n} \, dS \) | \( \displaystyle\iiint_V \nabla \cdot \vec{F} \, dV \) |
When the interior is hard, you compute the boundary; when the boundary is hard, you compute the interior. All of field engineering lives on that choice. Gauss’s law solves the field of a charged sphere in two lines because it trades a volume integral for a surface integral.
02Green’s theorem
For a simple closed curve C in the plane, traversed counterclockwise, enclosing the region D:
A striking case: with P = −y/2 and Q = x/2 the integrand equals 1, so the integral on the right is the area of D. This makes it possible to measure the area of a figure by traversing only its outline, and it is the principle of the planimeter, the mechanical instrument used to measure areas on a drawing before digital calculation existed.
On the left, the circulation around the boundary; on the right, the curl summed over the interior. The theorem says they are equal.
03The divergence theorem
Applied to the electric field, it turns the integral form of Gauss’s law into the first Maxwell equation: if the flux through any closed surface is the enclosed charge over ε₀, and the charge is the integral of the density ρ, then the two volume integrands have to coincide point by point:
Applied to the current density it gives the continuity equation, ∇·J = −∂ρ/∂t: charge does not disappear, and from there comes Kirchhoff’s current law when nothing accumulates.
04Stokes’ theorem
The remarkable thing is that the left-hand side does not mention S: any surface that has C as its boundary gives the same result. Applied to the electric field, it turns the integral form of Faraday’s law into the differential form ∇ × E = −∂B/∂t, and applied to the magnetic field it gives ∇ × H = J + ∂D/∂t.
If Stokes’ theorem is applied to the loop around a capacitor, one can choose a surface that is crossed by the wire —and sees current— or another that passes between the plates —where there is no wire—. Since the result has to be the same, an extra term is needed: the ∂D/∂t that Maxwell added. That term is what makes electromagnetic waves exist, and with them all radio communication.
05The four equations, now complete
| Integral form | Theorem used | Differential form |
|---|---|---|
| \( \oiint_S \vec{E} \cdot \hat{n} \, dS = \dfrac{Q}{\varepsilon_0} \) | Divergence | \( \nabla \cdot \vec{E} = \dfrac{\rho}{\varepsilon_0} \) |
| \( \oiint_S \vec{B} \cdot \hat{n} \, dS = 0 \) | Divergence | \( \nabla \cdot \vec{B} = 0 \) |
| \( \oint_C \vec{E} \cdot d\vec{r} = -\dfrac{d\Phi_B}{dt} \) | Stokes | \( \nabla \times \vec{E} = -\dfrac{\partial \vec{B}}{\partial t} \) |
| \( \oint_C \vec{H} \cdot d\vec{r} = I + \dfrac{d\Phi_D}{dt} \) | Stokes | \( \nabla \times \vec{H} = \vec{J} + \dfrac{\partial \vec{D}}{\partial t} \) |
All of this course’s vector calculus ends here: in the four equations that govern any circuit, any transmission line and any antenna. What follows in the degree program —from radio-frequency measurements to impedance matching— is reading them in particular cases.
06In the lab
Verify Green’s theorem for F = (−y, x) over the circle of radius 2: the direct circulation gives 2πr² = 8π and the double integral of the curl gives 2·area = 2·4π = 8π. Repeat with the rectangle [0,3] × [0,2].
Digitize the outline of an irregular figure with twenty points and apply Green’s area formula in discrete form. Compare with the area measured by counting grid squares.
For a circular loop in a uniform field, compute the flux through the flat disk and through a hemisphere resting on the same boundary. Verify that they coincide, as Stokes’ theorem predicts.
07Common mistakes
- Traversing the boundary clockwise and forgetting the change of sign.
- Applying Green’s theorem to a region with a hole without subtracting the circulation around the inner boundary. This is exactly what happens with the field of a current-carrying wire.
- Using Green’s theorem with a field that is not differentiable at some interior point, such as 1/ρ at the origin.
- Confusing the normal in Stokes’ theorem: its direction is fixed by the right-hand rule relative to the direction of travel around the boundary.
- Applying the divergence theorem to an open surface: it has to be closed.
08Self-assessment
What is the circulation of F = (−y, x) around the circle of radius 3?
By Green’s theorem: curl = 2, so it equals 2·area = 2·9π = 18π ≈ 56.5.
Why does ∮(−y dx + x dy)/2 give the enclosed area?
Because ∂Q/∂x − ∂P/∂y = 1/2 + 1/2 = 1, and the double integral of 1 over D is its area.
Which Maxwell equation comes from applying the divergence theorem to Gauss’s law?
∇·E = ρ/ε₀.
What does ∇·B = 0 mean?
That isolated magnetic charges do not exist: whatever enters a closed surface also leaves it, and the lines of B always close on themselves.
Two different surfaces share the same boundary. Can the flux of the curl differ?
No: by Stokes’ theorem, both give the same circulation around the boundary.
What term had to be added for Stokes’ theorem to be consistent in a capacitor?
The displacement current ∂D/∂t, Maxwell’s contribution, which predicted electromagnetic waves.
09Further reading
- James Stewart. Multivariable Calculus. 7th ed. (Spanish edition, “Cálculo de varias variables. Trascendentes tempranas”), Cengage Learning, 2012. Sections 16.4 to 16.9: Green, Stokes and divergence with many solved exercises.
- Jerrold E. Marsden and Anthony J. Tromba. Vector Calculus. 5th ed. (Spanish edition, “Cálculo vectorial”), W. H. Freeman, 2003. Treats the three theorems as cases of a single one and explains the question of orientation well.
- Matthew N. O. Sadiku. Elements of Electromagnetics. Spanish edition, “Elementos de electromagnetismo,” 5th ed., Alfaomega, 2018. The passage from the integral to the differential form of the four Maxwell equations, step by step.
- Richard P. Feynman. The Feynman Lectures on Physics, Volume II: Mainly Electromagnetism and Matter. Spanish edition, “Física, volumen II: electromagnetismo y materia”; Addison-Wesley, 1964. Chapters 2 and 3 explain vector calculus from physical intuition, without losing rigor.