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Math Notes

University Mathematics

ProofComplete

Vector Analysis and Tensors - Key Proof

We present a detailed proof of the Divergence Theorem (Gauss's Theorem), one of the fundamental results connecting local differential properties to global integral properties.

ProofDivergence Theorem

Theorem: For a vector field F\mathbf{F} continuously differentiable in a region VV bounded by a closed surface SS with outward normal n^\hat{\mathbf{n}}:

∭V(∇⋅F) dV=∬SF⋅n^ dS\iiint_V (\nabla \cdot \mathbf{F})\,dV = \iint_S \mathbf{F} \cdot \hat{\mathbf{n}}\,dS

Proof:

We first prove the theorem for a simple region where VV can be described as:

V={(x,y,z):(x,y)∈D, f1(x,y)≤z≤f2(x,y)}V = \{(x,y,z) : (x,y) \in D, \, f_1(x,y) \leq z \leq f_2(x,y)\}

for some domain DD in the xyxy-plane and functions f1,f2f_1, f_2.

Consider the zz-component of F=(Fx,Fy,Fz)\mathbf{F} = (F_x, F_y, F_z):

∭V∂Fz∂zdV=∬D[∫f1(x,y)f2(x,y)∂Fz∂zdz]dx dy\iiint_V \frac{\partial F_z}{\partial z}dV = \iint_D\left[\int_{f_1(x,y)}^{f_2(x,y)} \frac{\partial F_z}{\partial z}dz\right]dx\,dy

By the fundamental theorem of calculus:

=∬D[Fz(x,y,f2(x,y))−Fz(x,y,f1(x,y))]dx dy= \iint_D \left[F_z(x,y,f_2(x,y)) - F_z(x,y,f_1(x,y))\right]dx\,dy

Now consider the surface integral. The surface SS consists of three parts:

  • Top surface S2S_2: z=f2(x,y)z = f_2(x,y) with outward normal having n^⋅k^>0\hat{\mathbf{n}} \cdot \hat{\mathbf{k}} > 0
  • Bottom surface S1S_1: z=f1(x,y)z = f_1(x,y) with outward normal having n^⋅k^<0\hat{\mathbf{n}} \cdot \hat{\mathbf{k}} < 0
  • Vertical side surface where dS⋅k^=0dS \cdot \hat{\mathbf{k}} = 0

For the top surface, n^ dS=(−∂f2/∂x,−∂f2/∂y,1)dx dy\hat{\mathbf{n}}\,dS = (-\partial f_2/\partial x, -\partial f_2/\partial y, 1)dx\,dy, so:

∬S2Fz n^⋅k^ dS=∬DFz(x,y,f2(x,y))dx dy\iint_{S_2} F_z\,\hat{\mathbf{n}} \cdot \hat{\mathbf{k}}\,dS = \iint_D F_z(x,y,f_2(x,y))dx\,dy

For the bottom surface, n^ dS=(∂f1/∂x,∂f1/∂y,−1)dx dy\hat{\mathbf{n}}\,dS = (\partial f_1/\partial x, \partial f_1/\partial y, -1)dx\,dy, yielding:

∬S1Fz n^⋅k^ dS=−∬DFz(x,y,f1(x,y))dx dy\iint_{S_1} F_z\,\hat{\mathbf{n}} \cdot \hat{\mathbf{k}}\,dS = -\iint_D F_z(x,y,f_1(x,y))dx\,dy

Combining:

∬SFz n^⋅k^ dS=∭V∂Fz∂zdV\iint_S F_z\,\hat{\mathbf{n}} \cdot \hat{\mathbf{k}}\,dS = \iiint_V \frac{\partial F_z}{\partial z}dV

By similar arguments for the xx and yy components (using projections onto the yzyz and xzxz planes):

∬SFx n^⋅i^ dS=∭V∂Fx∂xdV\iint_S F_x\,\hat{\mathbf{n}} \cdot \hat{\mathbf{i}}\,dS = \iiint_V \frac{\partial F_x}{\partial x}dV ∬SFy n^⋅j^ dS=∭V∂Fy∂ydV\iint_S F_y\,\hat{\mathbf{n}} \cdot \hat{\mathbf{j}}\,dS = \iiint_V \frac{\partial F_y}{\partial y}dV

Adding these three equations:

∬SF⋅n^ dS=∭V(∂Fx∂x+∂Fy∂y+∂Fz∂z)dV=∭V(∇⋅F) dV\iint_S \mathbf{F} \cdot \hat{\mathbf{n}}\,dS = \iiint_V \left(\frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}\right)dV = \iiint_V (\nabla \cdot \mathbf{F})\,dV

For arbitrary regions, decompose into a finite union of simple regions. The interior boundaries cancel (opposite normals), leaving only the outer boundary.

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RemarkPhysical Interpretation

The divergence theorem states that the total flux of F\mathbf{F} through the closed surface SS equals the total divergence within the volume VV. If F\mathbf{F} represents a flow velocity times density, then:

  • ∇⋅F\nabla \cdot \mathbf{F} is the rate of "creation" of the quantity per unit volume
  • ∬SF⋅n^ dS\iint_S \mathbf{F} \cdot \hat{\mathbf{n}}\,dS is the net outward flow through the boundary

The theorem expresses conservation: what is created inside must flow out through the boundary.

ExampleApplication to Electrostatics

For the electric field E\mathbf{E} and charge density ρ\rho, Gauss's law in differential form is:

∇⋅E=ρϵ0\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}

Applying the divergence theorem:

∬SE⋅n^ dS=1ϵ0∭Vρ dV=Qencϵ0\iint_S \mathbf{E} \cdot \hat{\mathbf{n}}\,dS = \frac{1}{\epsilon_0}\iiint_V \rho\,dV = \frac{Q_{\text{enc}}}{\epsilon_0}

This is Gauss's law in integral form: the electric flux through a closed surface equals the enclosed charge divided by ϵ0\epsilon_0. This form is often more useful for calculating fields with high symmetry (spherical, cylindrical, planar).

ProofStokes' Theorem (Sketch)

Theorem: For a vector field F\mathbf{F} on a surface SS bounded by curve CC:

∬S(∇×F)⋅n^ dS=∮CF⋅dℓ\iint_S (\nabla \times \mathbf{F}) \cdot \hat{\mathbf{n}}\,dS = \oint_C \mathbf{F} \cdot d\boldsymbol{\ell}

Proof Sketch: Parametrize SS by r(u,v)\mathbf{r}(u,v) with (u,v)∈D(u,v) \in D. Then:

∬S(∇×F)⋅n^ dS=∬D(∇×F)⋅(∂r∂u×∂r∂v)du dv\iint_S (\nabla \times \mathbf{F}) \cdot \hat{\mathbf{n}}\,dS = \iint_D (\nabla \times \mathbf{F}) \cdot \left(\frac{\partial\mathbf{r}}{\partial u} \times \frac{\partial\mathbf{r}}{\partial v}\right)du\,dv

Using the identity for curl in parametric form and applying Green's theorem in the uvuv-plane transforms this to:

∮∂DF(r(u,v))⋅∂r∂udu+F(r(u,v))⋅∂r∂vdv\oint_{\partial D} \mathbf{F}(\mathbf{r}(u,v)) \cdot \frac{\partial\mathbf{r}}{\partial u}du + \mathbf{F}(\mathbf{r}(u,v)) \cdot \frac{\partial\mathbf{r}}{\partial v}dv

which equals ∮CF⋅dℓ\oint_C \mathbf{F} \cdot d\boldsymbol{\ell} by the chain rule.

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These fundamental theorems provide the mathematical foundation for integral formulations of physical laws and are essential tools in mathematical physics.