VectorPotential - crowlogic/arb4j GitHub Wiki

In physics, the vector potential is an important concept used in the formulation of electromagnetism, especially within the context of quantum mechanics and field theory. It is a vector field, denoted by A, used in the definition of the magnetic field B.

The magnetic field is described by the curl of the vector potential:

$$\mathbf{B} = \nabla \times \mathbf{A}$$

where:

  • B is the magnetic field vector.
  • $\nabla \times$ is the curl operator.
  • A is the vector potential.

In the quantum mechanics context, the Hamiltonian of a charged particle in an electromagnetic field includes the vector potential as:

$$H = \frac{(\mathbf{p} - q\mathbf{A})^2}{2m} + q\phi$$

where:

  • H is the Hamiltonian.
  • m is the particle's mass.
  • p is the momentum operator.
  • q is the charge of the particle.
  • A is the vector potential.
  • $\phi$ is the electric scalar potential.

The vector potential A at a position r due to a current distribution J can be calculated using:

$$\mathbf{A}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int \frac{\mathbf{J}(\mathbf{r'})}{|\mathbf{r} - \mathbf{r'}|} d^3\mathbf{r'}$$

Here:

  • A(r) is the vector potential at the position r.
  • $\mu_0$ is the vacuum permeability.
  • J(r') is the current density at the position r'.
  • $|\mathbf{r} - \mathbf{r'}|$ is the distance between the position r and the position r'.
  • The integral is taken over all space (where r' ranges over the entire space).

In the case of Newtonian gravity, we often talk about a scalar gravitational potential $\phi$, which is related to the gravitational field g by

$$\mathbf{g} = -\nabla\phi$$

The scalar gravitational potential $\phi$ at a position r due to a mass distribution $\rho$ can be calculated using:

$$\phi(\mathbf{r}) = G \int \frac{\rho(\mathbf{r'})}{|\mathbf{r} - \mathbf{r'}|} d^3\mathbf{r'}$$

where:

  • $\phi(\mathbf{r})$ is the gravitational potential at the position r,
  • G is the gravitational constant,
  • $\rho(\mathbf{r'})$ is the mass density at the position r',
  • $|\mathbf{r} - \mathbf{r'}|$ is the distance between the position r and the position r', and
  • the integral is taken over all space.

On a 2-dimensional plane, you might use something called the Riemann surface for your analysis. In terms of metric tensors, for complex plane (which is also a flat Riemann surface), the simplest metric tensor would be the Euclidean metric tensor. In local coordinates (z, z-bar) where z = x + iy and z-bar = x - iy, the line element of the Euclidean metric can be expressed as:

$$ds^2 = dx^2 + dy^2 = dz d\bar{z}$$

If you're studying the geometrical properties of a Riemann surface (which is a complex one-dimensional manifold), the metric tensor plays a vital role. In terms of the local coordinate z, such a metric can be written as:

$$ds^2 = \rho(z, \bar{z}) dz d\bar{z}$$

where $\rho(z, \bar{z})$ is a function that gives the "scale factor" of the metric.

A metric tensor on a surface can be expressed in terms of what are known as the first and second fundamental forms. These are central concepts in differential geometry and describe the local properties of a surface embedded in a higher-dimensional space.

  1. First Fundamental Form: The first fundamental form, often denoted I or (E, F, G), is essentially the metric tensor of the surface. It can be written as:

    $$I = Edu^2 + 2Fdudv + Gdv^2$$

  2. Second Fundamental Form: The second fundamental form, often denoted II or (L, M, N), measures how the surface bends within the ambient space. It can be expressed in terms of the local coordinates (u, v) as:

    $$II = Ldu^2 + 2Mdudv + Ndv^2$$

The Gaussian curvature K and the mean curvature H of the surface can then be expressed in terms of the coefficients of the first and second fundamental forms:

  • Gaussian Curvature: $$K = \frac{LN - M^2}{EG - F^2}$$
  • Mean Curvature: $$H = \frac{EN + GL - 2FM}{2(EG - F^2)}$$

These are the fundamental quantities describing the intrinsic and extrinsic curvature of the surface, respectively.