By Xin-She Yang

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This is probably the main reason why the planetary system and rings are formed. 1 Matrix 1\tiatrices are widely used in almost all engineering subjects. A matrix is a table or array of numbers or functions arranged in rows and columns. The elements or entries of a n1atrix A are often denoted as aii. 1) we say the size of A is m by n, or m x n. A is square if m = n. 3) where A is a 2 x 3 matrix, B is a 2 x 2 square matrix, and u is a 3 x 1 column matrix or column vector. 4) where (i = 1, 2, ... , m;j = 1, 2, ...

2 Determinant The determinant of a square matrix A is a number or scalar obtained by the following recursive formula or the cofactor or Laplace expansion by column or row. 11) j=l where Mij is the determinant of a minor matrix of A obtained by deleting row i and column j. 13) where A and Bare the same size (nxn). A nxn square matrix is singular if IAI = 0, and is nonsingular if and only if IAI =/= 0. 3 Inverse 1\llatrix Algebra is defined as the sum of the diagonal elements, n tr(A) = L~i = a11 + a22 + ...

Fi27r. 5 Some Special Integrals Calculus /'(0:, x) and the upper incomplete gamnla function r(a, x) so that r(x) = 'Y(O:, x) + r(a, x). 97) while the upper incomplete gamma function is defined by 1 00 r(a,x) = to:-le-tdt. Obviously, l(o:,x) ~ r(a) as X~ have 1 1 00. ) 2 erf(x) = fi'Y( ,x ). 100) From the definition, we know that the beta function is symmetric, B(x, y) = B(y, x). 101) r(x +Y). 102) which arises from heat conduction and diffusion problems as well as wave propagation problems. =O -- 1 --- 2 .....