A surface is said to be an open surface if it is bounded by an edge. Typical examples of open surfaces are disks, sheets, or hollow cylinders (without endcaps).

It is common to define a vector, called the normal vector and denoted by $\hat{n}$, that is perpendicular to the surface at any point and has a magnitude $\left|\hat{n}\right|=1$ (unit length). In the case of an open surface, two normal vectors can be defined at any point of an open surface and either of them can be used in practice. Indeed, unlike closed surfaces which we discuss below, there are no requirements on an open surface when it comes to choosing a normal vector.

Definition: closed surface and normal vector

A surface is said to be a closed surface if it has a boundary. A closed surface therefore has no edges and one can unambiguously define its inside and its outside. Typical examples of closed surfaces are hollow spheres or cylinders (with endcaps).

By convention, a closed surface is always oriented by its outward normal. Thus, at any point of a closed surface, there is only one valid normal vector out of the possible two normal vectors $\hat{n}$ that exist.

Definition: the vector $d\overrightarrow{A}$

For any given surface, we let $dA$ denote the infinitesimal area at a given point. This term represents the smallest nonzero area that can exist at that point.

The vector $d\overrightarrow{A}$ is a clever and useful construct that consists in creating a vector that points along the normal vector $\hat{n}$ and has a magnitude equal to the infinitesimal area $dA$. It can therefore be written as

d\overrightarrow{A}=dA\ \hat{n}

which explicitly shows that the vector has magnitude $dA$ in the direction of $\hat{n}$.

While this vector is typically confusing in and off itself, the context in which it is used often makes it more meaningful and understandable. In particular, it will be especially useful to have a vector which encapsulates both the infinitesimal area $dA$ and the direction normal to the surface when defining the electric flux.

Dot Product of two vectors:

Consider two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that form an angle $\theta $ as shown below. Let $a$ denote the magnitude of $\overrightarrow{a}$ and $b$ denote the magnitude of $\overrightarrow{b}$.

The dot product $\overrightarrow{a}\cdot \overrightarrow{b}$ is a scalar (real number) that measures how much of $\overrightarrow{b}$ lies along $\overrightarrow{a}$ and is equal to

\boxed{\overrightarrow{a}\cdot \overrightarrow{b}=ab{\mathrm{cos} \left(\theta \right)\ }}

If the components of these two vectors are $\overrightarrow{a}=\left\langle a_1,a_2,a_3\right\rangle $ and $\overrightarrow{b}=\left\langle b_1,b_2,b_3\right\rangle $, their dot product can also be written

\boxed{\overrightarrow{a}\cdot \overrightarrow{b}=a_1b_1+a_2b_2+a_3b_3}

Note: the dot product $\overrightarrow{a}\cdot \overrightarrow{b}$ is also a measure of how much of $\overrightarrow{a}$ lies along $\overrightarrow{b}$ since $\overrightarrow{a}\cdot \overrightarrow{b}=\overrightarrow{b}\cdot \overrightarrow{a}$.

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