Integral Theorems

QUIZ

Vector calculus connects line, surface, and volume integrals through three powerful theorems:
Gauss’s Divergence Theorem, Green’s Theorem, and Stokes’ Theorem.
These theorems convert integrals from one form to another and are central in electromagnetism, fluid flow, and mathematical physics.

1. Green’s Theorem

Green’s theorem connects a line integral around a closed curve with a double integral over the enclosed area.

Statement

For a vector field
$\mathbf{A} = P(x,y)\,\hat{i} + Q(x,y)\,\hat{j}$,

\[\oint_C (P\,dy + Q\,dx) = \iint_R \left( \frac{\partial Q}{\partial y} - \frac{\partial P}{\partial x} \right) dA\]
  • $C$ = closed curve bounding region $R$
  • Positive orientation = counterclockwise

Interpretation

Green’s theorem is the 2D version of Stokes’ theorem.

It converts:

  • A line integral (circulation) → into A surface integral of scalar curl

Applications

  • Computing area:
    Using $P=-y/2, Q=x/2$

    \[A = \frac{1}{2}\oint_C (x\,dy - y\,dx)\]

  • Used heavily in:

    • Fluid flow (vorticity)
    • Complex analysis
    • Planar physics problems

2. Gauss’s Divergence Theorem

Statement

Gauss’s theorem relates the flux of a vector field through a closed surface to the divergence of the field inside the volume.

\[\iiint_V (\nabla\cdot \mathbf{A})\, dV = \iint_{\partial V} \mathbf{A}\cdot d\mathbf{S}\]

Left Side → Divergence integrated over the volume
Right Side → Net flux through the surface bounding the volume

  • $V$ = volume
  • $\partial V$ = closed surface bounding $V$
  • $d\mathbf{S} = \hat{n}\, dS$ (outward normal)

Physical Meaning

  • Divergence measures net “outflow” of the vector field from a point.
  • Gauss’s theorem says:
    Total outflow from a volume = sum of divergence at each point inside.

This is exactly how Gauss’s law in electrostatics is written.

Example (Electrostatics)

Electric flux through a closed surface:

\[\iint_S \mathbf{E} \cdot d\mathbf{S} = \frac{Q_{\text{enc}}}{\varepsilon_0}\]

Taking divergence:

\[\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}\]

Using Gauss theorem:

\[\iiint_V \frac{\rho}{\varepsilon_0}\, dV = \iint_S \mathbf{E}\cdot d\mathbf{S}\]

3. Stokes’ Theorem

Stokes’ theorem converts a line integral around a closed loop into a surface integral of curl over the surface bounded by that loop.

Statement

\[\oint_{\partial S} \mathbf{A} \cdot d\mathbf{l} = \iint_S (\nabla \times \mathbf{A}) \cdot d\mathbf{S}\]
  • $\partial S$ = boundary curve of the surface
  • $d\mathbf{S} = \hat{n}\, dS$
  • Orientation follows right-hand rule

Physical Meaning

  • Curl measures local rotation/vorticity.
  • Stokes theorem says:
    Total circulation around boundary = sum of rotations inside the surface.

Example (Faraday’s Law in Electromagnetism)

Faraday’s law states:

\[\oint_C \mathbf{E}\cdot d\mathbf{l} = -\frac{d\Phi_B}{dt}\]

Using Stokes theorem:

\[\oint_C \mathbf{E}\cdot d\mathbf{l} = \iint_S (\nabla\times \mathbf{E})\cdot d\mathbf{S}\]

Thus,

\[\nabla\times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}\]

Summary

Theorem Converts Boundary
Gauss Surface integral $\leftrightarrow$ volume integral of divergence Closed surface
Green Line integral $\leftrightarrow$ area (2D curl) Closed curve in plane
Stokes Line integral $\leftrightarrow$ surface integral of curl Curve bounding surface

Green is a special case of Stokes.
Gauss is the divergence counterpart.