Fractions

When 2 is divided by 3, it may be written as 2 3 or 2/3 or 2/3. 2 3 is called a fraction. The number above the line, i.e. 2, is called the numerator and the number below the line, i.e. 3, is called the denominator.

When the value of the numerator is less than the value of the denominator, the fraction is called a proper fraction; thus 2 3 is a proper fraction. When the value of the numerator is greater than the denominator, the fraction is called an improper fraction. Thus 7 3 is an improper fraction and can also be expressed as a mixed number, that is, an integer and a proper frac- tion. Thus the improper fraction 7 3 is equal to the mixed number 21 3 .

When a fraction is simplified by dividing the numer- ator and denominator by the same number, the pro- cess is called cancelling. Cancelling by 0 is not permissible.

Square root

The square root of a number is when the index is 1 2 , and the square root of 2 is written as 21/2 or √2. The value of a square root is the value of the base which when multiplied by itself gives the number. Since 3 × 3 = 9, then √9 = 3. However, (−3) × (−3) = 9, so √9 = −3. There are always two answers when finding the square root of a number and this is shown by putting both a + and a − sign in front of the answer to a square root problem. Thus √9 = ±3 and 41/2 = √4 = ±2, and so on.

The lowest factors of 2000 are 2 × 2 × 2 × 2 × 5 × 5 × 5. These factors are written as 24 × 53, where 2 and 5 are called bases and the numbers 4 and 5 are called indices. When an index is an integer it is called a power. Thus, 24 is called ‘two to the power of four’, and has a base of 2 and an index of 4. Similarly, 53 is called ‘five to the power of 3’ and has a base of 5 and an index of 3. Special names may be used when the indices are 2 and 3, these being called ‘squared’and ‘cubed’, respectively. Thus 72 is called ‘seven squared’ and 93 is called ‘nine cubed’. When no index is shown, the power is 1, i.e. 2 means 21.

Matrix Operations

As matrix notation simplifies the calculations in solving systems of linear equations, we shall discuss different kind of matrices and operations on them. Recall that a matrix A of size m × n over a field F (here we take F as the real or complex field) is denoted by A = (aij)m × n, i = 1, 2, 3, . . . , m, and j = 1, 2 , . . . , n, and aij are from F. If m = n then A is called a square matrix. In this case the entries a11, . . . , ann are called the main diagonal or principal diagonal and other entries are called off-diagonal entries. If aij = 0 for all i and j, then A is called the null matrix or the zero matrix, and is denoted by 0. An identity matrix, denoted by I, is a square matrix whose all diagonal entries are equal to 1 and off diagonal entries are equal to zero. A square matrix A is called a diagonal matrix if all the off-diagonal entries are zero. A square matrix A = (aij)n × n is called lower (respectively upper) triangular matrix if aij = 0 whenever i > j (respectively i < j), that is, all entries above (respectively below) the main diagonal are zero. Two matrices of the same size A = (aij)m × n and B = (bij)m × n are said to be equal if aij = bij for all i, j.

If A = (aij)m × n is a matrix over F and α ε F then the scalar multiplication of A by α is the matrix αA =(α aij)m × n i.e. each entry of A is multiplied by α. If A = (aij)m × n and b = (bij)m × n are matrices of the same size over F then addition of A and B denoted by, A + B, is the matrix C = (cij)m × n , where cij = aij + bij. Scalar multiplication and addition of matrices satisfy some properties as given below. For matrices A, B and C of the same size over F and α, β ε F: (1) A + B = B + A (commutative) (2) (A + B) + C = A + (B + C) (associative) (3) A + 0 = 0+A =A, where 0 is the zero matrix of the same size as A. (4) A + (− A) = (− A)+A= 0, where − A = (− 1)A i.e. if A = (aij)m × n then − A = (− aij)m × n. (5) (α + β) A = αA + βA. (6) α (A + B) = αA + αB. (7) α (βA) = α β A.

Some Special Matrices

Here we shall discuss about some of the special type of matrices which will be used in the subsequent lectures. We consider a square matrix A = (aij)n × n. If A is a real matrix and satisfies A = AT then A is called symmetric. In this case aij = aji for all i, j. If A satisfies AT = − A then A is called skew-symmetric. In this case aij = − aji for all i, j, and therefore all diagonal entries are equal to zero. Here we take a complex square matrix A = (aij)n × n. The conjugate of A is the matrix = ( ij) n × n, where ε ij is the complex conjugate of aij. Matrix A is said to be Hermitian if (εA)T = A. In this case aij = ji and in particular aii = ii. Thus for Hermitian matrices diagonal entries are real numbers. Matrix A is said to be skew- Hermitian if (εA)T = − A. By the similar argument aij = - ji and so diagonal entries are either 0 or pure imaginary for skew-Hermitian matrices. One sees that symmetric and Hermitian matrices agree for real matrices. Similarly, skew- symmetric and skew-Hermitian matrices also agree for real matrices. A complex square matrix A = (aij)n × n is called unitary if A(εA)T =(εA)T A=I, where I is the identity matrix of the same size as A. In case of real matrices unitary matrices are called orthogonal, that is, a real matrix A is orthogonal if AAT = AT A = I.

Suppose that f(x) is a function defined on (0, l]. Suppose we want to express f(x) in the cosine or sine series. This can be done by extending f(x) to be an even or an odd function on [−l, l]. Note that there exists an infinite number of ways to express the function in the interval [−l, 0]. Among all possible extension of f there are two, even and odd extensions, that lead to simple and useful series:

Integral Surface through a given Curve - The Cauchy Problem

For the quasi linear p.d.e. (, ,) (, ,) (, ,) Pxyzz Qxyzz Rxyz x y + = , with its general integral F(, ) 0 φ ψ = where 1 φ(, ,) xyz C= and 2 ψ (, ,) xyz C= are two functionally independent solutions of the characteristic system dx dy dz PQR = = , can we find a particular integral containing the given curve C whose parametric equations are given by 0 0 x xsy ys = = ( ), ( ) and 0 z zs = ( ) where sis the parameter. This is similar to finding the arbitrary constants in the general solution of an ordinary differential equation using the initial conditions. Thus fixing the arbitrary function in the general solution of the given p.d.e by making it to pass through the given initial data is called the Cauchy Problem. Suppose z zxy = (, )is the integral surface passing through the initial data curve C then we require that the equations φ ( xsyszs C 0 00 1 ( ), ( ), ( )) = and ψ ( xsyszs C 0 00 2 ( ), ( ), ( )) = be satisfied. Now eliminating s from these two equations we obtain FCC ( 1 2 , 0 ) = orC GC 1 2 = ( ). This fixes the arbitrary function F (or G ) and produces the required surface. Let us illustrate this by considering few examples: Example 1: For the p.d.e 2 2 () () x y zx yz zx yz x y + + − =+ Find the integral surface that satisfies the Cauchy data z = 0 on the curve y x = 2 .

The complementary function of irreducible equations

FDD z f xy ( , ) (, ) ′ = Irreducible factors are treated as follows: Case 1: The particular integral 1 (, ) (, ) z f xy FDD = ′ is obtained by Expanding the operator 1 F− by the binomial theorem and then interpret the operator 1 1 D D, − −′ as integration.

Aurther

John Bird

Downlaod Book