Vector Calculus Theorems Overview

The Vector Calculus Theorems

  • The fundamental theorem of calculus relates an integral to an antiderivative at the boundaries.
  • There are three major vector calculus theorems that simplify integrals using boundary conditions:
    • Stokes' Theorem
    • Green's Theorem
    • Divergence Theorem
  • These theorems are crucial for fields such as fluid dynamics and electromagnetics, enabling the transformation of complex integrals into simpler forms for computations.

Stokes' Theorem

  • Stokes' theorem establishes a relationship between a surface integral and a line integral:
    ∮<em>CF⃗⋅ds⃗=∬</em>S(∇×F⃗)⋅dS⃗\oint<em>C \vec{F} \cdot d\vec{s} = \iint</em>S (\nabla \times \vec{F}) \cdot d\vec{S}
  • In this context:
    • The curve C is the boundary of the surface S.
    • The orientation of curve C is determined by walking around C with the surface to the left.
  • Example:
    1. Let F⃗=⟨−y,2x,x+z⟩\vec{F} = \langle -y, 2x, x + z \rangle
    • Find ∬S(∇×F⃗)⋅dS⃗\iint_S (\nabla \times \vec{F}) \cdot d\vec{S} for the upper hemisphere of radius 1 by converting it to a simpler line integral around a circle of radius 1.
    1. For F⃗=⟨sin⁡(x2),ey2+x,z4+2x2⟩\vec{F} = \langle \sin(x^2), e^{y^2 + x}, z^4 + 2x^2 \rangle,
    • Compute ∮CF⃗⋅ds⃗\oint_C \vec{F} \cdot d\vec{s} where C is the boundary of the plane 2x+3y+6z=62x + 3y + 6z = 6 in the first octant. This setup consists of three line integrals around three linear segments.
    • The curl is ∇×F⃗=⟨0,4x,2x⟩\nabla \times \vec{F} = \langle 0, 4x, 2x \rangle, allowing the conversion to a flux integral involving the curl.

Green's Theorem

  • Green's theorem is a 2D version of Stokes' theorem:
    ∮<em>CF⃗⋅ds⃗=∬</em>R(∂F<em>2∂x−∂F</em>1∂y)dA\oint<em>C \vec{F} \cdot d\vec{s} = \iint</em>R \left( \frac{\partial F<em>2}{\partial x} - \frac{\partial F</em>1}{\partial y} \right) dA
  • Description:
    • The line integral is taken over a closed curve C in the xy-plane.
    • The surface integral relates to a double integral over the region R enclosed by C.
  • The combination of derivatives represents the 2D curl (z-component of the curl).
  • Green's theorem helps simplify difficult line integrals into simpler double integrals and vice versa.

Divergence Theorem

  • The divergence theorem connects the divergence of a vector field in 3D with the flux across the surface of a solid:
    ∬<em>∂DF⃗⋅dS⃗=∭</em>D∇⋅F⃗ dV\iint<em>{\partial D} \vec{F} \cdot d\vec{S} = \iiint</em>D \nabla \cdot \vec{F} \, dV
  • The left side indicates the flux integral across the surface of a region D, while the right side is a triple integral over the volume D.
  • This theorem allows you to compute flux in cases that would require multiple surface integrals through simply evaluating one triple integral, streamlining the computation process.