Introduction

In this article, we will go over some of the basic principles of vector analysis/ vector calculus that are of relevance in the study of electromagnetism and electrodynamics, both of which will be introduced in future documents. This article will focus on mathematical methods required to introduce various concepts in electrodynamics, such as flux, potential etc.

As it turns out, with every major theory in classical physics, there seems to be a primary underlying tool in mathematics that is of paramount importance to the theory. For example, the laws of Newtonian mechanics were formulated in the language of calculus. That is, the kinematics and dynamics of moving bodies are essentially statements on how different physical variables are associated with integrals and derivatives of each other. For example, the momentum is proportional to the derivative of position (the velocity), and the force is proportional to the second derivative of position (the acceleration). In fact, the idea of derivatives and integrals were originally formulated by sir Issac Newton himself, out of a need for tools to describe generalized motion in 3 dimensions. While it is debated upon as to who was the “true founding father” of calculus (Newton vs. Leibniz), the fact remains that the formulation of calculus was critical for the development of classical mechanics

When it comes to the theory of electromagnetism, previous methods that involved concepts like field lines to represent the electric force seemed to fail due to the introduction of un-testable heuristics, such as “A point charge radially emits infinite field lines” and other statements of similar nature. The final refinements to the comprehensive theory of electromagnetism were done by James Clerk Maxwell in the late 19th century. He described the entirety of all electromagnetic interactions in the universe in 4 simple equations, a feat that is recognized with merit even today. At some point in this article, we hope to introduce these 4 equations to the reader.

Vectors

While we assume familiarity with vectors and vector algebra, we shall nonetheless recap the relevant information concerning vectors to avoid ambiguity.

While not completely rigorous, a vector can be viewed as a quantity with magnitude and direction, as this definition of a vector suffices for discussions on electromagnetism. In this article, we shall use an arrow on top of the variable - $\vec{A}$ to denote a vector, and regular text - $A$ to denote scalar quantities. In most cases, if a vector and a scalar quantity have the same names, the scalar will be the magnitude of the vector with the same label. For example, $\vec{F}$ is the force vector, while $F$ is the magnitude of the force. Exceptions to this rule will be clearly stated.

Vectors obey linear combination rules. let $\vec{a}$ and $\vec{b}$ be vectors. Then, $$\vec{c} = k_1\vec{a} + k_2\vec{b}$$ is also a vector. $k_1,k_2$ are arbitrary constants. As it turns out, the validity of the above statement is actually used to formulate a rigorous definition of vectors, but this is not a topic that we will explore further.

Although there is no further theory of vectors required for the completion of a basic understanding of electrodynamics, we encourage the reader to explore ideas like linear vector spaces, dual vector spaces, as well as further topics in linear algebra, as these will be of relevance for the discussion on quantum mechanics.

Functions of Multiple Variables

Scalar Fields

Just as functions can take as input a variable $x$, it can also take as input $(x,y,z)$, or multiple variables at the same time. We call such functions multivariate functions, and we use the notation $f(x,y,z)$. An example of a multivariate function would be:$$f(x,y,z) = x^2 + 4y + 6z^3$$ While this idea might seem fairly new, it is actually just a change in notation of well known concepts, where we take various variables to be functions of one another rather than just writing equations without context. For example, if we take Newton’s law of gravitation: $$F_g = \frac{GMm}{r^2}$$ We can choose to write the gravitational force $F_g$ as a function of $M$,$m$,and $r$: $$F_g(M,m,r) = \frac{GMm}{r^2}$$ While this might seem cumbersome and lengthy,as you shall soon see, the usefulness of writing quantities as functions of other quantities proves to useful since we have an extensive set of tools (calculus) to work with functions. When we have a function that explicitly takes as input 3 space coordinates $(x,y,z)$, we call such a function a scalar field, since it essentially assigns a scalar value to every point in space. An example of a scalar field could be the temperature function: $$T(x,y,z)$$ Which tells you what the temperature is at every point in the room (since the temperature might be higher as some points, but lower at others.)

Vector Fields

While a scalar field assigns a scalar value to every point in space, a vector field, as it should be evident from the name, assigns a vector value to every point in space. Since a vector field is both a vector valued quantity and a function, we use the following notation to represent a vector field: $$\vec{F}(x,y,z)$$ With the arrow denoting that it gives vector values, and the $(x,y,z)$ denoting that it’s a function of position in space.

An example of a vector field would be the gravitational force field. It is a function of spacial position, and it has an associated vector value at each point. That is, the gravitational force (given by Newton’s law of gravitation) is stronger near a source of mass (a planet), but weaker as you move further away. This is essentially equivalent to saying that the magnitude of gravitational force varies with position/distance. And since the fact that forces are vector-valued quantities, that makes the gravitational force field a vector field. As it turns out, the electric and magnetic force fields are are also vector fields, and applying rules of calculus on these vector fields encompasses the entirety of electrodynamics.

Since vector fields are vector-valued, any operations that we do to vectors can be done with vector fields. We can take linear combinations of vector fields, take the dot and cross product of 2 vector fields. However, any operation we do to functions (taking derivatives, integrals etc.) cannot be done as it stands, due to the inclusion of multiple variable. We shall resolve this incompatibility in the next section.

Multivariate Calculus in Brief

Based on what we have discussed so far, a thorough comprehension of the above material (which can be obtained through consultation of various other resources) should result in a natural and intuitive understanding of vector calculus. As has been stated previously, this article does not aim to provide a deep understanding of vector calculus, nor does it attempt at endowing significant problem solving aptitude. Our primary objective is to state some of the important results obtained, and what concepts (at a high level) are involved in deriving them.

All of the standard rules of calculus that might have previously used remain true for multivariate calculus. For example, the product and chain rules (for computation of derivatives) remain largely unchanged. The main concept introduced while extending calculus to scalar fields (as opposed to functions of a single variable) is, surprisingly, a notational matter rather than a conceptual one (as has been the theme for a vast majority of this article).

The first and most important idea that needs to be introduced is the idea of a partial derivative, donated by:$$\frac{\partial}{\partial x}$$ The partial derivative is computed in exactly the same way, barring the fact that all the other variables that are not being differentiated with respect to are treated as constants, and the differentiation is performed keeping $x$ as the variable. The best way to understand partial derivatives is to just go through the derivatives of a few simple functions: $$\frac{\partial}{\partial x}\ x^2 + y^2 = 2x$$ $$\frac{\partial}{\partial x}\ xy = y$$ $$\frac{\partial}{\partial x}\ e^x \sin{y} \ln{z} = e^x \sin{y} \ln(z)$$ $$\frac{\partial}{\partial y}\ e^x \sin{y} \ln{z} = e^x \cos{y} \ln{x}$$ $$\frac{\partial}{\partial z}\ e^x \sin{y} \ln{z} = \frac{e^x \sin{y}}{z}$$ Similarly, the partial derivative can be taken for any multivariate function or scalar field of your choice.

Another very interesting part of multivariate calculus involves the Del operator, denoted by $\nabla$. The Del operator is defined in the following way:

$$\vec{\nabla} = \begin{pmatrix} \frac{\partial}{\partial x} \ \frac{\partial}{\partial y} \ \frac{\partial}{\partial z} \end{pmatrix}$$

In most cases, the arrow is omitted as the definition of Del is usually assumed. Similar to a derivative, the Del operator can act on scalar fields. But, since it also takes the form of a vector, it can also act on vector fields by means of the dot and cross product. A vast majority of vector calculus is concerned with the usage of Del in various different contexts, which we will explore in future articles.