15d. Gauss, Green, Stokes

[math] \newcommand{\mathds}{\mathbb}[/math]

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Let us start with a standard definition, immersing us into 3D problematics, as follows:

Definition

Given a function [math]f:\mathbb R^3\to\mathbb R[/math], its usual derivative [math]f'(u)\in\mathbb R^3[/math] can be written as [math]f'(u)=\nabla f(u)[/math], where the gradient operator [math]\nabla[/math] is given by:

[[math]] \nabla=\begin{pmatrix} \frac{d}{dx}\\ \frac{d}{dy}\\ \frac{d}{dz} \end{pmatrix} [[/math]]

By using [math]\nabla[/math], we can talk about the divergence of a function [math]\varphi:\mathbb R^3\to\mathbb R^3[/math], as being

[[math]] \lt \nabla,\varphi \gt = \left \lt \begin{pmatrix} \frac{d}{dx}\\ \frac{d}{dy}\\ \frac{d}{dz} \end{pmatrix},\begin{pmatrix} \varphi_x\\ \varphi_y\\ \varphi_z \end{pmatrix}\right \gt = \frac{d\varphi_x}{dx}+\frac{d\varphi_y}{dy}+\frac{d\varphi_z}{dz} [[/math]]

as well as about the curl of the same function [math]\varphi:\mathbb R^3\to\mathbb R^3[/math], as being

[[math]] \nabla\times\varphi= \begin{vmatrix} u_x&\frac{d}{dx}&\varphi_x\\ u_y&\frac{d}{dy}&\varphi_y\\ u_z&\frac{d}{dz}&\varphi_z \end{vmatrix} =\begin{pmatrix} \frac{d\varphi_z}{dy}-\frac{d\varphi_y}{dz}\\ \frac{d\varphi_x}{dz}-\frac{d\varphi_z}{dx}\\ \frac{d\varphi_y}{dx}-\frac{d\varphi_x}{dy} \end{pmatrix} [[/math]]

where [math]u_x,u_y,u_z[/math] are the unit vectors along the coordinate directions [math]x,y,z[/math].

All this might seem a bit abstract, but is in fact very intuitive. The gradient [math]\nabla f[/math] points in the direction of the maximal increase of [math]f[/math], with [math]|\nabla f|[/math] giving you the rate of increase of [math]f[/math], in that direction. As for the divergence and curl, these measure the divergence and curl of the vectors [math]\varphi(u+v)[/math] around a given point [math]u\in\mathbb R^3[/math], in a usual, real-life sense.

Getting back now to calculus tools, what was missing from our picture was the higher dimensional analogue of the fundamental theorem of calculus, and more generally of the partial integration formula. In 3 dimensions, we have the following result:

Theorem

The following results hold, in [math]3[/math] dimensions:

• Fundamental theorem for gradients, namely
  [[math]] \int_a^b \lt \nabla f,dx \gt =f(b)-f(a) [[/math]]
• Fundamental theorem for divergences, or Gauss or Green formula,
  [[math]] \int_B \lt \nabla,\varphi \gt =\int_S \lt \varphi(x),n(x) \gt dx [[/math]]
• Fundamental theorem for curls, or Stokes formula,
  [[math]] \int_A \lt (\nabla\times\varphi)(x),n(x) \gt dx=\int_P \lt \varphi(x),dx \gt [[/math]]

where [math]S[/math] is the boundary of the body [math]B[/math], and [math]P[/math] is the boundary of the area [math]A[/math].

Show Proof

This is a mixture of trivial and non-trivial results, as follows:

(1) This is something that we know well in 1D, namely the fundamental theorem of calculus, and the general, [math]N[/math]-dimensional formula follows from that.

(2) This is something more subtle, and we had a taste of it when dealing with the Gauss law, and its various proofs. In general, the proof is similar, by using the various ideas from the proof of the Gauss law, and this can be worked out.

(3) This is again something subtle, and again with a flavor of things that we know, from the proof of the Gauss law, and which can be again worked out.

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Getting back now to electrostatics, as a main application of the above, we have the following new point of view on the Gauss formula, which is more conceptual:

Theorem (Gauss)

Given an electric potential [math]E[/math], its divergence is given by

[[math]] \lt \nabla,E \gt =\frac{\rho}{\varepsilon_0} [[/math]]

where [math]\rho[/math] denotes as usual the charge distribution. Also, we have

[[math]] \nabla\times E=0 [[/math]]

meaning that the curl of [math]E[/math] vanishes.

Show Proof

We have several assertions here, the idea being as follows:

(1) The first formula, called Gauss law in differential form, follows from:

[[math]] \begin{eqnarray*} \int_B \lt \nabla,E \gt &=&\int_S \lt E(x),n(x) \gt dx\\ &=&\Phi_E(S)\\ &=&\frac{Q_{enc}}{\varepsilon_0}\\ &=&\int_B\frac{\rho}{\varepsilon_0} \end{eqnarray*} [[/math]]

Now since this must hold for any [math]B[/math], this gives the formula in the statement.

(2) As a side remark, the Gauss law in differential form can be established as well directly, with the computation, involving a Dirac mass, being as follows:

[[math]] \begin{eqnarray*} \lt \nabla,E \gt (x) &=&\left \lt \nabla,K\int_{\mathbb R^3}\frac{\rho(z)(x-z)}{||x-z||^3}\,dz\right \gt \\ &=&K\int_{\mathbb R^3}\left \lt \nabla,\frac{x-z}{||x-z||^3}\right \gt \,\rho(z)\,dz\\ &=&K\int_{\mathbb R^3} 4\pi\delta_x\cdot\rho(z)dz\\ &=&4\pi K\int_{\mathbb R^3}\delta_x\,\rho(z)dz\\ &=&\frac{\rho(x)}{\varepsilon_0} \end{eqnarray*} [[/math]]

And with this in hand, we have via (1) a new proof of the usual Gauss law.

(3) Regarding the curl, by discretizing and linearity we can assume that we are dealing with a single charge [math]q[/math], positioned at [math]0[/math]. We have, by using spherical coordinates [math]r,s,t[/math]:

[[math]] \begin{eqnarray*} \int_a^b \lt E(x),dx \gt &=&\int_a^b\left \lt \frac{Kqx}{||x||^3},dx\right \gt \\ &=&\int_a^b\left \lt \frac{Kq}{r^2}\cdot\frac{x}{||x||},dx\right \gt \\ &=&\int_a^b\frac{Kq}{r^2}\,dr\\ &=&\left[-\frac{Kq}{r}\right]_a^b\\ &=&Kq\left(\frac{1}{r_a}-\frac{1}{r_b}\right) \end{eqnarray*} [[/math]]

In particular the integral of [math]E[/math] over any closed loop vanishes, and by using now Stokes' theorem, we conclude that the curl of [math]E[/math] vanishes, as stated.

(4) Finally, as a side remark, both the formula of the divergence and the vanishing of the curl are somewhat clear by looking at the field lines of [math]E[/math]. However, as all the above mathematics shows, there is certainly something to be understood, in all this.

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So long for electrostatics, which provide a good motivation and illustration for our mathematics. When upgrading to electrodynamics, things become even more interesting, because our technology can be used in order to understand the Maxwell equations.

General references

Banica, Teo (2024). "Calculus and applications". arXiv:2401.00911 [math.CO].