Ciro Santilli @cirosantilli 36

 Incoming links: Divergence

Divergence in Einstein notation  Updated 2025-04-24  +Created 1970-01-01

First we write a vector field as:

F (x_{0}, x_{1}, x_{2}) = (F^{0} (x_{0}, x_{1}, x_{2}), F^{1} (x_{0}, x_{1}, x_{2}), F^{2} (x_{0}, x_{1}, x_{2})) : R^{3} \to R^{3}

Note how we are denoting each component of

F

F^{i}

with a raised index.

Then, the divergence can be written in Einstein notation as:

\nabla \cdot F = \frac{\partial F ^{0} ( x _{0} , x _{1} , x _{2} )}{\partial x _{0}} + \frac{\partial F ^{1} ( x _{0} , x _{1} , x _{2} )}{\partial x _{1}} + \frac{\partial F ^{2} ( x _{0} , x _{1} , x _{2} )}{\partial x _{2}} = \partial_{i} F^{i} (x_{0}, x_{1}, x_{2}) = \frac{\partial F ^{i} ( x _{0} , x _{1} , x _{2} )}{\partial x ^{i}}

It is common to just omit the variables of the function, so we tend to just say:

\nabla \cdot F = \partial_{i} F^{i}

or equivalently when referring just to the operator:

\nabla \cdot = \partial_{i}

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Einstein notation for partial derivatives  Updated 2025-04-24  +Created 1970-01-01

The Einstein summation convention works will with partial derivatives and it is widely used in particle physics.

In particular, the divergence and the Laplacian can be succinctly expressed in this notation:

In order to express partial derivatives, we must use what Ciro Santilli calls the "partial index partial derivative notation", which refers to variables with indices such as

x_{0}

,

x_{1}

,

x_{2}

,

\partial_{0}

,

\partial_{1}

and

\partial_{2}

instead of the usual letters

x

,

y

and

z

.

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Maxwell's equations  Updated 2025-04-24  +Created 1970-01-01

Unified all previous electro-magnetism theories into one equation.

Explains the propagation of light as a wave, and matches the previously known relationship between the speed of light and electromagnetic constants.

The equations are a limit case of the more complete quantum electrodynamics, and unlike that more general theory account for the quantization of photon.

The equations are a system of partial differential equation.

The system consists of 6 unknown functions that map 4 variables: time t and the x, y and z positions in space, to a real number:

$E_{x} (t, x, y, z)$ , $E_{y} (t, x, y, z)$ , $E_{z} (t, x, y, z)$ : directions of the electric field $E : R^{4} \to R^{3}$
$B_{x} (t, x, y, z)$ , $B_{y} (t, x, y, z)$ , $B_{z} (t, x, y, z)$ : directions of the magnetic field $B : R^{4} \to R^{3}$

and two known input functions:

$ρ : R^{3} t o R$ : density of charges in space
$J : R^{3} \to R^{3}$ : current vector in space. This represents the strength of moving charges in space.

Due to the conservation of charge however, those input functions have the following restriction:

\frac{\partial ρ}{\partial t} + \nabla \cdot J = 0

Charge conservation

.

Also consider the following cases:

if a spherical charge is moving, then this of course means that $ρ$ is changing with time, and at the same time that a current exists
in an ideal infinite cylindrical wire however, we can have constant $ρ$ in the wire, but there can still be a current because those charges are moving
Such infinite cylindrical wire is of course an ideal case, but one which is a good approximation to the huge number of electrons that travel in a actual wire.

The goal of finding

E

and

B

is that those fields allow us to determine the force that gets applied to a charge via the Equation "Lorentz force", and then to find the force we just need to integrate over the entire body.

Finally, now that we have defined all terms involved in the Maxwell equations, let's see the equations:

d i v E = \frac{ρ}{ε _{0}}

Gauss' law

.

d i v B = 0

Gauss's law for magnetism

.

\nabla \times E = - \frac{\partial B}{\partial t}

Faraday's law

.

\nabla \times B = μ_{0} (J + ε_{0} \frac{\partial E}{\partial t})

Ampere's circuital law

.

You should also review the intuitive interpretation of divergence and curl.

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