Most of us learn to calculate areas using formulas drilled into memory: length times width for rectangles, πr² for circles. Let’s see another approach hiding in vector calculus – one that works for any shape, no matter how irregular. Enter Stokes’ theorem.

§The Trick

Stokes’ theorem relates a line integral around a closed curve to a surface integral over the region it encloses. For area calculation, we can exploit this relationship with a clever choice of vector field.

The key insight: we need a field F whose curl equals 1. Why? Because Stokes’ theorem tells us:

∮C F · dr = ∫∫R (∇ × F) · n dA

For planar regions, the curl ∇ × F = (∂F₂/∂x - ∂F₁/∂y)k points perpendicular to the plane, so this becomes:

∮C F · dr = ∫∫R (∂F₂/∂x - ∂F₁/∂y) dA

If we make (∂F₂/∂x - ∂F₁/∂y) = 1, the right side becomes ∫∫R 1 dA, which is just the area. Any F satisfying this works—common choices include F = (0, x), F = (-y, 0), or F = (-y/2, x/2). Let’s see them in action.

§Example 1: The Unit Square

Consider a square with vertices at (0,0), (1,0), (1,1), and (0,1). We’ll traverse the boundary counterclockwise.

Using F = (0, x), the line integral ∮C x dy becomes:

  • Bottom edge (y=0, dy=0): contributes 0
  • Right edge (x=1, y: 0→1): ∫₀¹ 1 dy = 1
  • Top edge (y=1, dy=0): contributes 0
  • Left edge (x=0, y: 1→0): ∫₁⁰ 0 dy = 0

Total: 1 square unit. Exactly what we expected.

§Example 2: The Unit Circle

Now for something curvier. The unit circle x² + y² = 1 can be parameterized as r(t) = (cos t, sin t) for t ∈ [0, 2π].

This time, let’s use the symmetric field: F = (-y/2, x/2). The line integral ∮C (-y/2 dx + x/2 dy) becomes:

∮C (-y/2 dx + x/2 dy) = ∫₀²π (-sin(t)/2 · (-sin t) + cos(t)/2 · cos t) dt

= ∫₀²π (sin²(t)/2 + cos²(t)/2) dt = ∫₀²π (sin²(t) + cos²(t))/2 dt

Using the identity sin²(t) + cos²(t) = 1:

∫₀²π 1/2 dt = π

Again, π square units—same answer, even cleaner calculation. The symmetric choice F = (-y/2, x/2) often simplifies circular and elliptical boundaries.

§Why This Matters

With this technique, we can calculate areas of arbitrary shape.