Filling the gaps in the “Electromagnetics”-textbook: Surface Integral of Electric vector field (part 1: Taylor Expansion)

I am studying Electromagnetics with the textbook “Electromagnetism (by Shigenobu Sunagawa) “(SBN-10‏: ‎ 4000077449). This textbook assumes a strong foundation in mathematics, so many intermediate steps in the calculations are skipped. For that reason, there were a lot of parts that were hard to understand for me, so I filled in the intermediate steps by consulting an AI each time. Even worse, when I continued reading later, I found that I couldn’t recall the logic, even when looking back at my own notes. I realize that I need to keep proper records to refer and remember any time.

In the explanation part of “div”, there is a figure like that:

It is necessary to calculate the integration E・n through surface A for derivation of div. The textbook describes it as:

This is a result on the premise that vector field E(x) has same vector through anywhere of surface A. (I have used ‘x’ to represent the position vector) But vector field E(x) can have different vector through the surface A. E(x+Δx,y,z) and E(x,y,z) are vector only at bottom-left corner, and E(x) is different at the other point of each surface. So, the figure above must draw like below:

Because of that reason, integral of E⋅n through the surface is not simple. We have to figure out the formula of E in the surface first, using Taylor expansion.

Taylor expansion

Taylor expansion allows us to approximate a function’s value at a nearby point using its derivatives at a known point(a) and a small displacement (h).

Taylor expansion for a single variable

This series has a structure (derivatives at a known point)×(displacement). For multiple variables, we can extend simply with applying the same rules. In the single-variable case, the first-order term is simply the derivative times the displacement:

In the multivariable case, we do the same thing for each variable independently and sum them up:

We got the formula below through that process:

Taylor expansion for multivariable functions

We can simplify with using dot product of operator ∇ = (∂/∂x, ∂/∂y, ∂/∂z) and h = (Δx , Δy , Δz).

Taylor expansion for multivariable functions (another display with using ∇ operator)

Applying to vector E on the surface A

Since the unit normal vector on face A is n=(1,0,0), the dot product E⋅n reduces to the x-component of E only.

And displacement of x is 0 on the surface A. The x-component of E on face A is described using the theorem above as:

This is the full expression for x-component of E​ at surface A without any omissions. This concludes Part 1. In the next part, we will perform the surface integral over face A. To be continued.