# MECHANICAL ENGINEERING LIBRARY

## MECHANICAL ENGINEERING

• SUBJECT : ENGINEERING MATHEMATICS - THEORY WITH WORKED OUT EXAMPLES SELF AND CLASSROOM PRACTICE QUESTIONS
• PUBLISHERS : ACE ENGINEERING PUBLISHERS
• IDEAL FOR : GATE AND PSU's - MECHANICAL ENGINEERING STREAM

CONTENTS :
1. PROBABILITY AND STATISTICS
2. DIFFERENTIAL EQUATIONS
3. LAPLACE TRANSFORM
4. COMPLEX VARIABLES
5. NUMERICAL METHODS

BRIEF INTRODUCTION :

PROBABILITY BASIC TERMINOLOGY :

Exhaustive events : A set of events is said to be exhaustive,  if it includes all the possible events.  For in tossing a coin there are two exhaustive cases either head or tail and there is no third possibility.

Mutually exclusive events : If the occurrence of one of the events procludes the occurrence of all other then such a set of events is said to be mutually exclusive.  Just as tossing a coin,  either head comes up or the tail and both can't happen at the same time,  i.e.,  these are two mutually exclusive cases.

Equally likely events : If one of the events cannot be expected to happen in preference to another then such events are said to be equally likely.  For instance,  in tossing a coin,  the coming of the head or the tail is equally likely.

PROBABILITY AND SET NOTATIONS

Random experiment : Experiments which are performed essentially under the same conditions and whose results cannot be predicted are known as random experiments.  e g.,  Tossing a coin or rolling a die are random experiments.

Sample space : The set of all possible outcomes of a random experiments called sample space for that experiment and is denoted by S.  The elements of the sample space S are called the sample points e.g.,  On tossing coin,  the possible outcomes are the head(H) and the tail(T)

Thus S={H, T}  a of a sample space S

Event : The outcome of a random experiment is called an event.  Thus every subset is an event.  The null set is also an event and is called an possible event.  Probability of an impossible event is zero

Differential Equations :

Mathematicians have tried in vain to this day to discover some order in the sequence of prime A numbers and we have reason to believe that it is a mystery into which the human mind will never penetrate.

Introduction:  Differential equations arise from many problems in mechanical bending of beams,  conduction of heat,  velocity of t. The chemical reactions etc. Differential equations are used to model the physical systems.  The systems which change con time are called dynamic Systems.  For example, models which describe heat transfer chemical engineering which depend on time,  electronic circuits with time dependent current and voltages are dynamic systems.  The phase A differential equation is an equation which involves differential coefficients or differentials.
A partial differential equation is that in which there are two or more independent variables and partial differential coefficients with respect to any of them.

The degree of a differential equation is the degree of the highest derivative occuring in it, after the equation has been expressed in a form free from radicals and fractions as far as the derivatives are concerned.

Practical approach to differential equations :
• The study of a differential equation consists of three phases.
• Formulation of differential equation from the given physical situation called modelling
• Solution of this differential equation and evaluating the arbitrary constants from the given conditions
• Physical interpretation of the solutions.

Formation of Differential Equation : A differential equation is formed in an attempt to eliminate certain arbitrary constants from a relation in the variables and constants

Solution of a differential equation : A solution if a differential equation is a relation between the variables which satisfies the given differential equation.

The general solution of a differential equation is that in which the number of arbitrary constants is equal to the order of the differential equation.

A particular solution is that which can be obtained from the general solution by giving particular values to the arbitrary constants.

A singular solution is that which cannot be obtained from the general solution by giving particular values to the arbitrary constants.

Laplace Transform

Introduction : Laplace transform is used for solving differential and integral equations. In physics and engineering, it is used for analysis of linear time-invariant systems such as electrical​ circuits, harmonic oscillators,  optical devices and mechanical​ systems.

The Fourier transform named after Joseph finite Fourier,  is a mathematical transformation employed to transform signals between time(or spatial)  domain and frequency domain,  which has many applications in physics and engineering. It is reversible,  being able to transform from either domain to the other.  The term itself refers to both the transform operation and to the function it produces.

In mathematics and signal processing,  the Z transform converts a time domain signal,  which is a sequence of real or complex numbers,  into a complex frequency domain representation. It can be considered as a discrete-time equivalent of the Laplace transform.  This similarity is explored in the theory of time scale calculus.

## MECHANICAL ENGINEERING

• SUBJECT : ENGINEERING MATHEMATICS - THEORY WITH WORKED OUT EXAMPLES SELF AND CLASSROOM PRACTICE QUESTIONS
• PUBLISHERS : ACE ENGINEERING PUBLISHERS
• IDEAL FOR : GATE AND PSU's - MECHANICAL ENGINEERING

CONTENTS :
1. LINEAR ALGEBRA
2. CALCULUS
3. VECTOR CALCULUS

BRIEF INTRODUCTION TO LINEAR ALGEBRA :

INTRODUCTION : Linear algebra comprises of the theory and applications of linear system of equation,  linear transformations and eigen value problems.  In linear algebra,  we make a systematic use of matrices and to a lesser extent determinants and their properties.  Determinants were first introduced for solving linear systems and have important engineering applications in systems of differential equations,  electrical networks,  eigen value problems and so on.  Many complicated expressions occurring in electrical and mechanical systems can be elegantly simplified by expressing them in the form of determinants Cayley matrices in the year 1860.  But it was not the twentieth century was well-  discovered advanced that engineers heard of them.  These days,  however,  matrices have been found to be of great utility in many branches of applied mathematics such as algebraic​ differential equations,  mechanics theory of electrical circuits,  nuclear physics,  aerodynamics and astronomy.  With the advent of computers,  the usage of matrix methods has been facilitated.

RANK OF A MATRIX : If we select any rows and r columns from any matrix A,deleting all the other rows and columns,  then the determinant formed by these r x r elements is called the minor of A of order r.  Clearly,  there will be a number of different minors of the same order,  got by deleting different rows and columns from the same matrix.

Definition : A matrix is said to be of rank r when

(i) it has at least one non-zero minor of order r,  and
(ii) every minor of order higher than r vanishes.  Briefly,  the rank of a matrix is the largest order of any non-vanishing minor of the matrix.
If a matrix has a non-zero minor of order r,  its rank is greater than or equal to r.
If all minors of a matrix of order r + 1 are zero,  its rank is less than or equal to r
The rank of a matrix A shall be denoted by p(A).

Elementary transformation of a matrix :

The following operations,  three of which refer to rows and three to columns are known transformations L The interchange of any two rows(columns).  IT.  The multiplication of any row(column by a non-zero number.  the corresponding elements of III.  The addition of a constant multiple of the elements of any row(column to any other row(column)  Notation.  The elementary row transformations will be denoted by the following symbols:
(i) Rij for the interchange of the ith and jth rows.
(ii)  kRi for multiplication of the ith row by k.
(iii) Ri + pRj for addition to the i'th row,  p times the j'th row.  writing C in place of R

The corresponding column transformation will be denoted by matrix.  While the value of the minors may get. Elementary transformations do not change he order or rank of a changed by the transformation I and II,  their zero or non-zero character remains if one can be obtained from the(3)  Equivalent matrix.  Two matrices A and B are said to be equivalent the same other by a sequence of elementary transformations.  Two equivalent matrices have the same order and rank.  The symbol is used for equivalence