Kategorier
Algebra matematik 5

Matrices

A matrix is a square or retangular array of numbers or function of numbers that obeys certain laws. The numbers are distinguished by two subscripts ij. The first indicating the row and the second indicating the column (vertical). in which the number appears. a13 is ther number in the first row snd third column.
It is worth mentioning that it is not a number as the determinant but an array of numbers.

Aij =Bij if and only if all the corresponding elements are equal in the two matrices.

Addition is performed by adding the respective numbers in the corresponding places in each matrix wuth each other.
A+B = B´+A so matrix addition it is commutative.

Division is performed by multipying yjr elements in row i by the elemetns in column j.
Therfore matrixmultiplication is anti-commutatative and AB not equal to BA.
A diagonal matrix is a matrix with zeroes in every position except in the diagonal. The trace of such a matrix is the sum of the diagonal elements.

A diagonal matrix with ones in all position of the diagonal is called an identity matrix. 

The identity matrix is defined by the operation AI = A.
The inverse of a matrix is defined by the relation A A-1 =I.

A tensor can be thought of as a three dimensional matrix.

Kategorier
Calculus

Spherical harmonics

The spherical harmonics are functions describing the angular dependence of many physical problems e.g solutions to the Schrödinger equation for the hydrogen atom.

If the latitiude is denoted by v and x= cosv then  the equation is

(d/dx){(1 – x2)dPndx} + n(n+1)Pn = 0.

Pn is the spherical harmonics or the Legendre polynomials of degree n.

a more detailde description is found below:

Spherical harmonics