1. Notes

1.1. Calculus for automatons

Since from the beginning, we have been doing integrals of the form

\[\int \vec{A} \bullet \mathrm{d}\vec{B}\]

A vector (field) dotted with a differential element vector (another vector) over some range in up to three dimensions. Notice this form!

  • Identify the integral you need to do (Coulomb’s Law, Gauss’s Law, etc.),

  • figure out what each term of the generic form is,

  • and then only afterwards is it useful to think harder about the tactics of the computation.

Each part could be some unpleasant function that varies over 3D space, but that’s all it is. Take it step by step, one equation per line, make your work flow vertically, and pay no care to how many sheets of paper you are using.[1]

1.2. Divergence

Is the vector field field at this point an innie, an outie, or just passing through?

\[\vec{\nabla} \bullet \vec{A} \quad\text{or}\quad \mathop{div} \vec{A}\]

  • del dot A

  • divergence of A

The divergence is defined over a field and yields a field at each position.

The divergence is defined over a vector field and yields a scalar field at each position.

Operator called del is defined as:

\[\vec{\nabla} = \frac{\partial}{\partial x} \mathbf{\hat{a}_x} + \frac{\partial}{\partial y} \mathbf{\hat{a}_y} + \frac{\partial}{\partial z} \mathbf{\hat{a}_z}\]
NB [2]

Notice how the operator is defined as if it is a vector? That’s because the expression is a vector! This is yet another reason “del dot A” is a scalar — the dot product of two vectors is a scalar.

Interesting \(\nabla\) history

The symbol itself is called nabla and originates from the Greek word for a harp (the musical instrument). See this quote for the original suggestion for this name, which came after its definition and use in calculus.

Right-click on any equation on these pages to see how the expression was typed, then Show Math As  TeX Commands will pop up a window with the source code. Notice how the symbol ∇ is called “nabla” in the code.

2. Gauss’s Law

See the summary page — Gauss’s Law

One small (but important!) addition and this is the first of the set called “Maxwell’s equations”. So exciting!

Constant (field • surface) easy version:

\[E \cdot A = \frac{Q_{\mathrm{enclosed}}}{\epsilon_0} \quad \text{(book Equation 5.8)}\label{eq-gauss-easy1}\]
\[E = \frac{Q_{\mathrm{enclosed}}}{A \cdot \epsilon_0} \quad \text{(book Equation 5.9)}\label{eq-gauss-easy2}\]
When is it valid to use equation \(\eqref{eq-gauss-easy1}, \eqref{eq-gauss-easy2}\) when solving a problem?

The E-field across the surface of integration is:

  • zero, and/or

  • constant

Example 5.7 selects a surface made from parts that are either zero or constant.

We frequently use equation \(\eqref{eq-gauss-easy2}\) to find the E-field (magnitude) by careful choice of the surface. What do we need to ensure when choosing this surface? → When the dot product \(\vec{E}\bullet\mathrm{d}\vec{S}\) (a scalar) is constant.

\(\vec{E}\bullet\mathrm{d}\vec{S}\) corresponds to \(E \cdot A\) from equation \(\eqref{eq-gauss-easy1}\) and gives a hint on how to use this “shortcut.”

See the alternate solution for Example 5.7 which works out the mechanics of how to use this “shortcut” when the choice of surface seems odd (to me). Find this on Canvas  Chapter 5  Alternate examples 05.

3. Areas

sphere

\(4 \pi r^2\)
cylinder

\(2 \pi r^2 h\)
circle

\(\phantom{2}\pi r^2\)
1. Engineering paper is only 5¢/page, compare that to the tuition you are paying for the course.
2. nota bene in Latin, the link mentions several other Latin abbreviations, e.g. this one.