All things are difficult before they are easy, Thomas Fuller.

When you have eliminated the impossible, whatever remains, however improbable, must be the truth, Sherlock Holmes.

A vector field is an assignment of a vector $\vec{F}$ to each point (x, y) in a space, i.e., $\vec{F} = M\vec{i}+N\vec{j}$ where M and N are functions of x and y.

A vector field on a plane can be visualized as a collection of arrows, each attached to a point on the plane. These arrows represent vectors with specific magnitudes and directions.

Work is defined as the energy transferred when a force acts on an object and displaces it along a path. In the context of vector fields, we calculate the work done by a force field along a curve or trajectory C using a line integral. The work done by a force field $\vec{F}$ along a curve C is: W = $\int_{C} \vec{F}·d\vec{r} = \int_{C} Mdx + Ndy = \int_{C} \vec{F}·\hat{\mathbf{T}}ds$, where $\hat{\mathbf{T}}$ is the unit tangent vector.

A vector field is conservative if there exist a scalar function such that $\vec{F}$ = ∇f (the vector field is its gradient). This scalar function is known or referred to as the potential function associated with the vector field.

Theorem. Fundamental theorem of calculus for line integral. If $\vec{F}$ is a conservative vector field in a simply connected region of space (i.e., a region with no holes), and if f is a scalar potential function for $\vec{F}$ in that region, then $\int_{C} \vec{F}·d\vec{r} = \int_{C} ∇f·d\vec{r} = f(P_1)-f(P_0)$ where P_{0} and P_{1} are the initial and final points of the curve C, respectively.

The line integral of the vector field $\vec{F}$ along the curve C is defined as: $\int_{C} \vec{F}d\vec{r}$ where $d\vec{r}$ is an infinitesimal vector tangent to the curve, given by: $d\vec{r} = ⟨dx, dy, dz⟩$.

$\int_{C} \vec{F}d\vec{r} = \int_{a}^{b} (P\frac{dx}{dt} + Q\frac{dy}{dt} + R\frac{dz}{dt})dt = \int_{a}^{b} (P(x(t), y(t), z(t))\frac{dx}{dt} + Q(x(t), y(t), z(t))\frac{dy}{dt} + R(x(t), y(t), z(t))\frac{dz}{dt})dt$

Let our vector field $\vec{F} = P\hat{\mathbf{i}}+ Q\hat{\mathbf{j}}+R\hat{\mathbf{k}}$, we define the curl of $\vec{F}$ as $curl(\vec{F})=(R_y-Q_z)\hat{\mathbf{i}} + (P_z-R_x)\hat{\mathbf{j}}+(Q_x-P_y)\hat{\mathbf{k}}$

$curl(\vec{F}) = ∇ x \vec{F} = |\begin{smallmatrix}\hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}}\\ \frac{∂}{∂x} & \frac{∂}{∂y} & \frac{∂}{∂z}\\ P & Q & R\end{smallmatrix}| = (\frac{∂R}{∂y}-\frac{∂Q}{∂z})\hat{\mathbf{i}}-(\frac{∂R}{∂x}-\frac{∂P}{∂z})\hat{\mathbf{j}} + (\frac{∂Q}{∂x}-\frac{∂P}{∂i})\hat{\mathbf{k}}$

Theorem. The curl of a conservative field is $\vec{0}$. Conversely, if $curl(\vec{F})=\vec{0}$, the components **P, Q, and R have continuous first-order partial derivatives**, and its domain is open and simply-connected, the vector field is conservative.

Given a vector field $\vec{F}$ and a surface S with boundary curve C, Stokes’ Theorem states: $\oint_C \vec{F} \cdot d\vec{r} = \int \int_{S} (∇x\vec{F})·\hat{\mathbf{n}}dS$

Maxwell’s equations are a set of four fundamental laws that describe how electric and magnetic fields behave and interact with each other. These equations form the foundation of classical electromagnetism, optics, and electric circuits.

An electric field ($\vec{E}$) is an invisible force field generated by electric charge. It describes how a charged particle would be pushed (attracted) or pulled (repelled) by other charges around it. The strength and direction of the electric force, per unit charge, an electrically charged particle would feel at any point in space, are given by the electric field. Mathematically, this is expressed as: $\vec{F} = q\vec{E}$ where $\vec{F}$ is the force experienced by a charge q in the electric field $\vec{E}$.

The unit of the electric field is newtons per coulomb (N/C), which measures the force per unit charge. A positive charge creates an electric field that radiates outward, while a negative charge creates an electric field that points inward toward the charge.

A magnetic field $\vec{B}$ is another type of invisible physical field that is generated by moving electric charges (such as current-carrying wires) or by magnetic materials (as in permanent magnets). The force a charged particle q experiences in a magnetic field $\vec{B}$ depends on its velocity $\vec{v}$ and is given by the equation: $\vec{F} = q\vec{v}x\vec{B}$. It describes the force experienced by a charged particle q moving with velocity $\vec{v}$ in a magnetic field $\vec{B}$. This force is perpendicular to both the velocity of the charged particle $\vec{v}$ and the magnetic field $\vec{B}$. The magnetic field is measured in teslas (T).

Gauss-Coulomb law (Gauss’ Law) relates the behaviour of electric fields to the distribution of electric charge. It states that **the divergence of the electric field ($\vec{E}$) is proportional to the charge density ρ**, which is the amount of electric charge per unit volume. The mathematical expression is: $\vec{∇}·\vec{E} = \frac{ρ}{\epsilon_0}$ where $\epsilon_0$ is a physical constant known as the permittivity of free space. It is a constant that characterizes the strength of the electric field in a vacuum.

To understand the implications of Gauss’s Law, let’s consider an arbitrary closed surface S. We want to study the flux of the electric field through this closed surface. Flux is a measure of how much of the (electric) field passes through the surface. According to Gauss’s Law, the total flux of the electric field through the surface is proportional to the total charge enclosed within the volume D enclosed by the surface S. (Figure iii):

$\oint_S \vec{E} \cdot d\vec{S} =$[Using the Divergence Theorem, which connects the flux through a closed surface to the divergence of the field within the volume D enclosed by the surface S] $\int \int \int_{D} div \vec{E}dV = \int \int \int_{D} \vec{∇}·\vec{E}dV$= [Substituting Gauss’s Law into this equation gives:] = $\frac{1}{\epsilon_0} \int \int \int_{D} ρdV$ =[The right-hand side of this equation represents the total charge Q enclosed within the volume D] $\frac{Q}{\epsilon_0}$.

This means that the total electric flux through any closed surface is equal to the total charge enclosed within the volume D divided by ϵ_{0}.

Faraday’s Law is one of the four Maxwell’s equations, which form the foundation of electromagnetism. It describes how a changing magnetic field can create an electric field. This phenomenon is the principle behind many modern technologies such as electric generators, transformers, and induction motors.

Imagine a magnetic field $\vec{B}$ as an invisible force field that can exert forces on electrically charged particles. Now, **if this magnetic field changes over time, it will create or induce an electric field**. An electric field, in turn, is a force field that exerts forces on electric charges, causing them to move. This movement of electric charges due to this induced electric field is what we call an electromotive force (EMF) or voltage.

Faraday’s law of electromagnetic induction states that a changing magnetic field over time creates a rotating (or curling) electric field. It creates an electric field that loops around in circles. This circular electric field can create a voltage if there’s a conducting path (like a wire loop) for electric charges to move through. Mathematically, it is represented as: $∇×\vec{E} = -\frac{∂\vec{B}}{∂t}$

- $∇×\vec{E}$ is called the “curl” of the electric field $\vec{E}$. It measures how much the electric field “curls”, “rotates” or “circulates” around a point.
- $-\frac{∂\vec{B}}{∂t}$ represents how the magnetic field ($\vec{B}$) is changing over time t, i.e., the rate of change of the magnetic field over time. The negative sign reflects Lenz’s Law. It indicates the direction of the induced electric field is such that it opposes the change in the magnetic field.

According to Stokes’ Theorem, the circulation (looping) of an electric field around a closed path C can be related to the rate of change of the magnetic field ($\vec{B}$) over a surface S bounded by that path (Figure iv):

$\oint_C \vec{E} \cdot d\vec{r} = \int \int_{S}(∇ x \vec{E})·d\vec{S} = $

The line integral on the left-hand side of Stoke’s theorem represents the circulation of the electric field around the closed loop C. It is the work done per unit charge in moving a charge around the closed loop in the presence of the induced electric field. The surface integral on the right-hand side of Stoke’s theorem involves the curl of the electric field ($∇ x \vec{E}$) integrated over the surface S bounded by the loop.

By Faraday’s law, $\int \int_{S}(∇ x \vec{E})·d\vec{S} = \int \int_{S} (-\frac{∂\vec{B}}{∂t})·d\vec{S}$. It allows us to express the voltage (or EMF) around the loop ($\oint_C \vec{E} \cdot d\vec{r}$) as the negative rate of change of the magnetic field ($\frac{∂\vec{B}}{∂t}$) passing through the surface: $\oint_C \vec{E} \cdot d\vec{r} = \int \int_{S} (-\frac{∂\vec{B}}{∂t})·d\vec{S}$

It’s helpful to mention Gauss’s Law for Magnetism, which states that magnetic fields always have a north and a south pole, and why isolated “magnetic charges”, analogous to electric charges, do not exist, and therefore **the divergence of the magnetic field is always zero**, $∇ x \vec{B} = 0$. In other words, magnetic field lines always form closed loops and do not begin or end at a point.

The final piece of the puzzle is the Maxwell-Ampère Law, which describes how both electric currents and changing electric fields generate magnetic fields. It’s expressed as: $∇ x \vec{B} = μ_0\vec{J}+ε_0μ_0\frac{∂\vec{E}}{∂t}$ where:

- Curl of the Magnetic Field, $∇ x \vec{B}$ measures the tendency of the magnetic field to rotate or circulate around a point.
- $μ_0\vec{J}$ is the contribution from the current density $\vec{J}$, describing how electric currents create magnetic fields around them.
- $ε_0μ_0\frac{∂\vec{E}}{∂t}$. A changing (time-varying) electric field also generates a magnetic field, even in the absence of a current.

This content is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License and is based on MIT OpenCourseWare [18.01 Single Variable Calculus, Fall 2007].

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