vector_calculus

# Vector calculus

 Stan Zurek, Vector calculus, Encyclopedia Magnetica https://E-Magnetica.pl/vector_calculus

Vector calculus - a set of mathematical operations involving derivatives and integrals of vectors which can represent functions or fields in a multidimensional space (2D, 3D, 4D, etc.) Vector calculus is an extension of the “ordinary” calculus which is performed on scalar derivatives and integrals.1)2)

Gradient $∇F$, divergence $∇·\mathbf{F}$, and curl $∇ × \mathbf{F}$ are the basic operations in vector calculus

The three main operators in vector calculus quantify changes in fields:

Vector calculus is used widely in calculations of electromagnetic phenomena. The basic operations allow extracting information about the distribution of electromagnetic fields, energy associated with the field, electromagnetic radiation, and so on. The four Maxwell's equations are typically written in the vector calculus notation.3)

The topic of vector calculus (including derivatives and integrals) is very complex and this article contains only the fundamental concepts, and is by no means exhaustive.

There are a number of excellent in-depth, but easy-to-follow books on the subject of using vector calculus in electrostatics, magnetostatics, and electromagnetic theory, for example by Purcell and Morin,4) Griffiths,1) Fleisch,5)2) to name just a few.

## Maxwell's equations

Maxwell's equations in a differential form4)
Gauss's law for electrostatics $$∇ · \mathbf{E} = \frac {\rho_{charge}}{\epsilon_0}$$
Gauss's law for magnetism $$∇ · \mathbf{B} = 0$$
Faraday's law of electromagnetic induction $$∇ × \mathbf{E} = - \frac {\partial \mathbf{B}}{\partial t}$$
Ampère's circuital law $$∇ × \mathbf{B} = \mu_0 · \mathbf{J} + \mu_0 · \epsilon_0 · \frac {\partial \mathbf{E}}{\partial t}$$

Maxwell's equations fully describe mathematically the interrelation between electric and magnetic fields. The early version of these equations were first collated by a Scottish physicist James Clerk Maxwell. Subsequently they were unified and expressed in vector notation by Oliver Heaviside, so that today four fundamental equations are used, whose physical meaning can be summarised as follows:9)

The equations can be mathematically written in many ways (e.g. differential or integral form) or different units (e.g. CGS or MKS). They can also be formulated on the basis of more fundamental theory of quantum electrodynamics.

In vacuum the equations simplify, because there are no charges, no currents and no material properties which have to be included in the constitutive equations.

## System of coordinates

Vectors and tensors can be defined or expressed in different systems of coordinates, which are used for reducing the complexity of calculations of certain problems. This is because the mathematical equations can take slightly different form in each of such systems, and therefore might be easier to solve analytically (or numerically) in a given system.

Three most important systems are: Cartesian, cylindrical, and spherical. If they are 3D then each of them requires three unique values to identify a unique vector, or a distance from some reference point, e.g. (0, 0, 0).

Example of representation of the same vector in three different systems of coordinates: Cartesian (x, y, z), cylindrical (r, φ, z), and spherical (r, φ, θ)

In a Cartesian system, there are three orthogonal directions, for example (x, y, z) and therefore each vector can be combined from three components, each expressing a linear distance from a reference point (0, 0, 0). The distances can be also represented in a relative sense defining the length and angle of the vector, but not its location as such.

In a cylindrical system, there are also three components. But a location of a given point inside a cylinder can be expressed by a radius r, angle φ, and height z of a cylindrically-shaped volume, so such set of values uniquely identifies a given point or vector. Such system is useful for performing calculations on a geometry which has one axis of rotational symmetry.

In a spherical system, the components denote a radius r and two angles φ and θ. Such system is useful for performing calculations in spherical or elliptical geometries.

The specific components differ between the systems, but they can be converted between them through strict mathematical relationships involving trigonometry.2)

Of course, the names of the component variables (for lengths and angles) can be chosen in an arbitrary way, depending on the preference in a given approach.