Question

How do you prove Stoke's theorem?

Answer 1

Stokes' theorem is a fundamental result in vector calculus that relates the flux of a vector field across a surface to the circulation of the field around the boundary curve of the surface. It states that the surface integral of the curl of a vector field over a closed surface is equal to the line integral of the vector field around the boundary curve of the surface. Mathematically, it can be written as:

∮_S (curl F) · dS = ∫_C F · dr

where:

• ∮_S denotes the surface integral over a closed surface S,
• (curl F) is the curl of the vector field F,
• dS represents an infinitesimal area vector on the surface S,
• ∫_C denotes the line integral around the boundary curve C of the surface S,
• F is the vector field, and
• dr represents an infinitesimal displacement vector along the boundary curve C.

To prove Stokes' theorem, we typically use the divergence theorem and Green's theorem as intermediate steps. Here's a step-by-step outline of the proof:

1. Start with the divergence theorem, which relates the flux of a vector field across a closed surface to the divergence of the field over the volume enclosed by the surface.

   ∮_S F · dS = ∫_V (div F) dV

   where V is the volume enclosed by the closed surface S.

2. Apply the vector identity for the divergence of a curl:

   div(curl F) = 0

   This identity holds for sufficiently smooth vector fields.

3. Rewrite the left-hand side of the divergence theorem by substituting (curl F) for F:

   ∮_S (curl F) · dS = ∫_V (div(curl F)) dV = ∫_V 0 dV = 0

4. Next, apply Green's theorem to the right-hand side of the divergence theorem. Green's theorem relates the line integral of a vector field around a closed curve to the surface integral of the curl of the field over the region bounded by the curve.

   ∫_C F · dr = ∬_R (curl F) · dS

   where R is the region bounded by the closed curve C.

5. Since Stokes' theorem relates the surface integral of (curl F) over a closed surface to the line integral of F around the boundary curve, we can rewrite the right-hand side of Green's theorem as:

   ∫_C F · dr = ∬_R (curl F) · dS = ∮_S (curl F) · dS

6. Comparing the results from steps 3 and 5, we see that:

   ∮_S (curl F) · dS = ∫_C F · dr

   which is Stokes' theorem.

Thus, by applying the divergence theorem and Green's theorem, we can derive Stokes' theorem. This proof demonstrates the underlying relationship between the curl and divergence of a vector field and how they are connected to flux and circulation.