Abstract:

This lecture provides a rigorous mathematical derivation of Stokes' theorem, presenting it as a direct generalization of Green's theorem from two to three dimensions. The process begins by extending the concept of circulation from a 2D plane to a line integral along a three-dimensional boundary curve CS of a surface S. By expressing the dz component of the integral as a total differential in terms of x and y, the 3D line integral is projected into a 2D line integral. Green's theorem is then applied to convert this into a surface integral.

A detailed and careful calculation of the resulting derivatives, accounting for all dependencies via the chain rule, leads to a complex expression. This expression is then insightfully rearranged into the scalar product of two new vector quantities. The first is identified as the Rotation (or Curl) of the vector field (rot F or ∇ x F), a new differential operator. The second is the normal vector to the surface, which defines the differential surface element vector. The lecture culminates in the final form of Stokes' theorem, equating the circulation around a boundary curve to the flux of the rotation of the field through the surface. It concludes by noting that this integral is independent of the specific surface for a given boundary and is always zero for a closed surface.

Deriving Stokes' Theorem from Green's Theorem: A Step-by-Step Guide

• 0:00 The lecture's objective is to generalize Green's theorem from two dimensions to three, which results in the formulation of Stokes' theorem.
• 1:53 The first step in the generalization is to extend the components of the problem to three dimensions: the vector field is expanded from F(P,Q) to F(P,Q,R), and the area of integration becomes a surface S in space defined by z = f(x,y) with a boundary curve CS.
• 9:52 The concept of circulation Γ is formally extended to 3D as the line integral of the vector field F along the boundary curve CS of the surface.
• 13:14 A key step in the derivation is using the total differential (dz = (∂z/∂x)dx + (∂z/∂y)dy) to transform the 3D line integral into a 2D line integral in the XY-plane, making it suitable for the application of Green's theorem.
• 16:20 Green's theorem is applied to the 2D line integral, converting it into a complex 2D surface integral. The lecturer then meticulously calculates the required partial derivatives using the chain rule.
• 29:45 In a crucial creative step, the complex result from the previous calculation is recognized and rewritten as the scalar product (dot product) of two distinct vectors.
• 34:56 The first of these two vectors is defined as a new, fundamental quantity: the Rotation (or Curl) of the vector field F. This is introduced as rot F and is shown to be equivalent to the cross product of the Nabla operator and the vector field (∇ x F).
• 45:31 The second vector in the scalar product is identified as the normal vector to the surface S. Multiplied by dx dy, this forms the differential surface element vector d**F**.
• 48:52 By combining these new definitions, the final, elegant form of Stokes' Theorem is presented: the line integral of F around a boundary curve (the circulation) is equal to the surface integral of the rotation of F over the surface bounded by that curve: ∮ F ⋅ dr = ∬ (rot F) ⋅ d**F**.
• 51:39 The lecture concludes with two important implications of the theorem:
  • The value of the integral depends only on the boundary curve, not the specific surface it encloses.
  • The integral of the rotation of a vector field over any closed surface (which has no boundary) is always zero.