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Math Notes

University Mathematics

ConceptComplete

Applications of the Fundamental Theorems

The integral theorems of vector calculus have far-reaching applications in physics, engineering, and pure mathematics, from Maxwell's equations to fluid dynamics and beyond.

Maxwell's Equations

Definition

Maxwell's equations in differential and integral form illustrate the integral theorems:

| Differential | Integral (via theorem) | |-------------|----------------------| | ∇⋅E=ρ/ϵ0\nabla \cdot \mathbf{E} = \rho/\epsilon_0 | ∯E⋅dS=Q/ϵ0\oiint \mathbf{E} \cdot d\mathbf{S} = Q/\epsilon_0 (divergence thm) | | ∇⋅B=0\nabla \cdot \mathbf{B} = 0 | ∯B⋅dS=0\oiint \mathbf{B} \cdot d\mathbf{S} = 0 (divergence thm) | | ∇×E=−∂B∂t\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} | ∮E⋅dr=−dΦBdt\oint \mathbf{E} \cdot d\mathbf{r} = -\frac{d\Phi_B}{dt} (Stokes' thm) | | ∇×B=μ0J+μ0ϵ0∂E∂t\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t} | ∮B⋅dr=μ0I+μ0ϵ0dΦEdt\oint \mathbf{B} \cdot d\mathbf{r} = \mu_0 I + \mu_0\epsilon_0\frac{d\Phi_E}{dt} (Stokes' thm) |

Fluid Dynamics

ExampleConservation laws

For a fluid with density ρ\rho and velocity v\mathbf{v}, conservation of mass gives: ∂ρ∂t+∇⋅(ρv)=0\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0 Integrating over a region EE and applying the divergence theorem: ddt∭Eρ dV=−∯∂Eρv⋅dS\frac{d}{dt}\iiint_E \rho\,dV = -\oiint_{\partial E} \rho\mathbf{v} \cdot d\mathbf{S} The rate of change of mass inside EE equals the net inward flux of mass through the boundary. This is the continuity equation.

Heat Equation and Laplace's Equation

ExampleDeriving the heat equation

The heat flux is q=−k∇T\mathbf{q} = -k\nabla T (Fourier's law). Conservation of energy gives: ρc∂T∂t=−∇⋅q=k∇2T\rho c \frac{\partial T}{\partial t} = -\nabla \cdot \mathbf{q} = k\nabla^2 T This is the heat equation ∂T∂t=α∇2T\frac{\partial T}{\partial t} = \alpha \nabla^2 T where α=k/(ρc)\alpha = k/(\rho c). In steady state, TT satisfies Laplace's equation ∇2T=0\nabla^2 T = 0, and solutions are harmonic functions.

RemarkGreen's identities

By combining the divergence theorem with product rules, we obtain Green's identities:

  1. ∬Sf∇g⋅dS=∭E(f∇2g+∇f⋅∇g) dV\iint_S f\nabla g \cdot d\mathbf{S} = \iiint_E (f\nabla^2 g + \nabla f \cdot \nabla g)\,dV
  2. ∬S(f∇g−g∇f)⋅dS=∭E(f∇2g−g∇2f) dV\iint_S (f\nabla g - g\nabla f) \cdot d\mathbf{S} = \iiint_E (f\nabla^2 g - g\nabla^2 f)\,dV

These identities are the starting point for potential theory, Green's functions, and the theory of elliptic PDEs. They encode the self-adjointness of the Laplacian and lead to uniqueness theorems for solutions of Laplace's and Poisson's equations.