MATHEMATICS

MADE EASY GATE HANDWRITTEN CLASSROOM NOTES

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Maths Made Easy GATE Handwritten Notes Free Download PDF

Topics Covered in this book :

Linear Algebra
Differential Calculus
Vector Calculus
Differential Equations
Numerical Methods
Complex Variables
Probability
Laplace Transforms

Partial fractions

When the degree of the numerator is equal to or higher than the degree of the denominator, the numerator must be divided by the denominator until the remainder is of less degree than the denominator . There are basically three types of partial fraction and the form of partial fraction used, where f (x) is assumed to be of less degree than the relevant denominator and A, B and C are constants to be determined. (In the latter type , ax^2+bx+c is a quadratic expression which does not factorize without containing surds or imaginary terms.) Resolving an algebraic expression into partial fractions is used as a preliminary to integrating certain functions and in determining inverse Laplace transforms

Introduction to logarithms

With the use of calculators firmly established, logarithmic tables are nowrarely used for calculation.However, the theory of logarithms is important, for there are several scientific and engineering laws that involve the rules of logarithms.

From the laws of indices: 16 = 2^4

The number 4 is called the power or the exponent or the index. In the expression 2^4, the number 2 is called the base.
In another example: 64 = 8^2. In this example, 2 is the power, or exponent, or index. The number 8 is the base.

What is a logarithm?

Consider the expression 16 = 2^4. An alternative, yet equivalent, way of writing this expression is stated as ‘log to the base 2 of 16 equals 4’.
We see that the logarithm is the same as the power or index in the original expression. It is the base in the original expression which becomes the base of the logarithm. The two statements: 16=24 and log2 16=4 are equivalent.

If we write either of them, we are automatically implying the other. In general, if a number y can be written in the form ax , then the index ‘x’ is called the ‘logarithm of y to the base of a’, i.e. if y = ax then x = loga y In another example, if we write down that 64 = 82 then the equivalent statement using logarithms is:
log8 64 = 2 In another example, ifwe write downthat: log3 81=4 then the equivalent statement using powers is: 3^4 = 81 So the two sets of statements, one involving powers and one involving logarithms, are equivalent.