advance engineering mathematics

Polytechnic Sem 1 – Engineering Mathematics Polytechnic Sem 1 –  Notes for Polytechnic SEM – 1 is Designed by the ” Basics in Maths” team. Here we can learn Concepts in Basic Engineering mathematics at Polytechnic Sem – I.  This Material is very Useful for Basic Engineering Mathematics Polytechnic Sem – I Students. By learning These Notes, Basic Engineering Mathematics Polytechnic Sem – I Students can Write their Exam successfully and fearlessly.  LOGARITHMS Logarithm: For ant two positive real numbers a, b, and a ≠ 1. If the real number x such then ax = b, then x is called logarithm of b to the base a. it is denoted by Standard formulae of logarithms: Logarithmic Function: Let a be a positive real number and a ≠ 1. The function f: (o, ∞) → R Defined by f(x) = PARTIAL FRACTIONS Fractions: If f(x) and g(x) are two polynomials, g(x) ≠ 0, then   is called rational fraction. Ex:    etc.  are rational fractions. Proper Fraction: A rational fraction is said to be a Proper fraction if the degree of g(x) is greater than the degree of f(x). Ex:   etc. are the proper fractions. Improper Fraction: A rational fraction is said to be an Improper fraction if the degree of g(x) is less than the degree of f(x). Ex:  etc. are the Improper fractions. Partial Fractions: Expressing rational fractions as the sum of two or more simpler fractions is called resolving a given fraction into a partial fraction. ∎ If R(x) =  is proper fraction, then Case(i): – For every factor of g(x) of the form (ax + b) n, there will be a sum of n partial fractions of the form: Case(ii): – For every factor of g(x) of the form (ax2 + bx + c) n, there will be a sum of n partial fractions of the form: ∎ If R(x) = is improper fraction, then Case (i): – If degree f(x) = degree of g(x),   where k is the quotient of the highest degree term of f(x) and g(x). Case (ii): – If f(x) > g(x) R(x) =      MATRICES AND DETERMINANTS   Matrix: A set of numbers arranged in the form of a rectangular array having rows and columns is called Matrix. •Matrices are generally enclosed by brackets like •Matrices are denoted by capital letters A, B, C, and so on •Elements in a matrix are real or complex numbers; real or complex real-valued functions. Oder of Matrix: A matrix having ‘m’ rows and ‘n’ columns is said to be of order m x n read as m by n. Ex: Types Of Matrices Rectangular Matrix: A matrix in which the no. of rows is not equal to the no. of columns is called a rectangular matrix.                   Square Matrix: A matrix in which the no. of rows is equal to no. of columns is called a square matrix. Principal diagonal (diagonal) Matrix: If A = [a ij] is a square matrix of order ‘n’ the elements a11, a22, a33, ………. An n is said to constitute its principal diagonal. Trace Matrix: The sum of the elements of the principal diagonal of a square matrix A is called the trace of the matrix. It is denoted by Tr (A). Diagonal Matrix: If each non-diagonal element of a square matrix is ‘zero’ then the matrix is called a diagonal matrix. Scalar Matrix: If each non-diagonal element of a square matrix is ‘zero’ and all diagonal elements are equal to each other, then it is called a scalar matrix. Identity Matrix or Unit Matrix: If each of the non-diagonal elements of a square matrix is ‘zero’ and all diagonal elements are equal to ‘1’, then that matrix is called unit matrix Null Matrix or Zero Matrix: If each element of a matrix is zero, then it is called a null matrix. Row matrix & column Matrix: A matrix with only one row s called a row matrix and a matrix with only one column is called a column matrix. Triangular matrices: A square matrix A = [aij] is said to be upper triangular if aij = 0   ∀ i > j A square matrix A = [aij] is said to be lower triangular matrix aij = 0  ∀ i < j   Equality of matrices: matrices A and B are said to be equal if A and B are of the same order and the corresponding elements of A and B are equal. Addition of matrices: If A and B are two matrices of the same order, then the matrix obtained by adding the corresponding elements of A and B is called the sum of A and B. It is denoted by A + B. Subtraction of matrices: If A and B are two matrices of the same order, then the matrix obtained by subtracting the corresponding elements of A and B is called the difference from A to B. Product of Matrices: Let A = [aik]mxn and B = [bkj]nxp be two matrices, then the matrix C = [cij]mxp   where Note: Matrix multiplication of two matrices is possible when no. of columns of the first matrix is equal to no. of rows of the second matrix. A m x n  . Bp x q = AB mx q; n = p Transpose of Matrix: If A = [aij] is an m x n matrix, then the matrix obtained by interchanging the rows and columns is called the transpose of A. It is denoted by AI or AT. Note: (i) (AI)I = A     (ii) (k AI) = k . AI    (iii)  (A + B )T = AT + BT  (iv)  (AB)T = BTAT Symmetric Matrix: A square matrix A is said to be symmetric if AT =A If A is a symmetric matrix, then A + AT is symmetric. Skew-Symmetric Matrix: A square matrix A is said to be skew-symmetric if AT = -A If A is a skew-symmetric matrix,