Vectors

The cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here
$E$
), and is denoted by the symbol  X.
Given two linearly independent vectors a and b, the cross product, a × b (read “a cross b”), is a vector that is perpendicular to both a and b,^[1] and thus normal to the plane containing them.

https://en.wikipedia.org/wiki/Cross_product

 

The exterior product or wedge product of vectors is an algebraic construction used in geometry to study  areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors u and  v denoted by  
  is called a bivector and lives in a space called the exterior square, a vector space that is distinct from the original
space of vectors. The magnitude^[4] of  can be interpreted as the area of the parallelogram with sides u  and v which in three dimensions can also be computed using the cross product of the two vectors.

https://en.wikipedia.org/wiki/Exterior_algebra#Inner_product

An eigenvector (/ˈaɪɡənˌvɛktər/) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by , is the factor by which the eigenvector is scaled.

In rotational motion of a rigid body, the principal axes are the eigenvectors of the inertia matrix.

real symmetric matrices have real eigenvalues.

If A is an n × n matrix, then the sum of the n eigenvalues of A is the trace of A and the product of the n eigenvalues is the determinant of A.

 

Tensors:

A tensor is an n-dimensional array satisfying a particular transformation law.

Not every Matrix is a rank 2 Tensor.

A Rank 2 tensor can be represented as a matrix of numbers — in conjunction with an associated transformation law.

A Rank 3 tensor can be represented as a 3-dimensional array of numbers — in conjunction with an associated transformation law.

 

https://rukshanpramoditha.medium.com/real-world-examples-of-0d-1d-2d-3d-4d-and-5d-tensors-100b0837ced4

Rank 3 example: Levi-Civita_symbol

https://en.wikipedia.org/wiki/Levi-Civita_symbol

Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime).

A covariant tensor, denoted with a lowered index (e.g., a_mu) is a tensor having specific transformation properties. A contravariant tensor is a tensor having specific transformation properties (cf., a covariant tensor). https://rinterested.github.io/statistics/tensors2.html https://www.mathpages.com/rr/s5-02/5-02.htm https://math.stackexchange.com/questions/8170/intuitive-way-to-understand-covariance-and-contravariance-in-tensor-algebra

Maxwell Equations

https://www.fiberoptics4sale.com/blogs/electromagnetic-optics/a-plain-explanation-of-maxwells-equations#:~:text=4.-,Ampere-Maxwell’s%20Law,field%20that%20changes%20with%20time.

https://en.wikipedia.org/wiki/Displacement_current

Liénard–Wiechert potential classical electromagnetic effect of a moving electric point charge

What is the proof of the Biot-Savart law from Maxwell equations

https://www.khanacademy.org/science/in-in-class-12th-physics-india/moving-charges-and-magnetism

 

Magnetic field of a moving charge