UNIVERSITY OF MUMBAI
Engineering Mathematics-III Course Code ; CEC 301
Bachelor of Engineering in Civil Engineering
(REV-
2019 ‘C’ Scheme) from Academic Year 2019 – 20.
Course
Objectives:
1. To familiarize with the Laplace Transform,
Inverse Laplace Transform of various functions, its applications.
2. To acquaint with the concept of Fourier Series,
its complex form and enhance the problem solving skills.
3. To familiarize with the concept of complex
variables, C-R equations with applications.
4. To study the application of the knowledge of
matrices and numerical methods in complex engineering problems.
Course Outcomes: Learner will be able to….
1. Apply the concept of Laplace transform to solve
the real integrals in engineering problems.
2. Apply the concept of inverse Laplace transform of
various functions in engineering problems.
3. Expand the periodic function by using Fourier
series for real life problems and complex engineering problems.
4. Find orthogonal trajectories and analytic
function by using basic concepts of complex variable theory
5. Apply
Matrix algebra to solve the engineering problems. 6. Solve Partial differential
equations by applying numerical solution and analytical methods for one
dimensional heat and wave equations.
Module
1: Laplace Transform
1.1 : Definition
of Laplace transform, Condition of Existence of Laplacetransform, Laplace
Transform (L) of Standard Functions like and .
1.2 Properties
of Laplace Transform: Linearity, First Shifting theorem, Second Shifting
Theorem, change of scale Property, multiplication by t, Division by t,
1.3. Laplace Transform of derivatives and integrals
(Properties without proof).
1.4 Evaluation of integrals by using Laplace
Transformation.
Self-learning
topics: Heaviside’s Unit Step
function, Laplace Transform. Of Periodic functions, Dirac Delta Function.
Module2
: Inverse Laplace Transform
2.1 Inverse
Laplace Transform, Linearity property, use of standard formulae to find inverse
Laplace Transform, finding Inverse Laplace transform using derivative
2.2 Partial fractions method & first shift
property to find inverse Laplace transform.
2.3 Inverse Laplace transform using Convolution
theorem (without proof) Self-learning Topics: Applications to solve initial and
boundary value problems involving ordinary differential equations.
Module3
: Fourier Series:
3.1 Dirichlet’s
conditions, Definition of Fourier series and Parseval’sIdentity (without proof)
3.2 Fourier
series of periodic function with period 2π and2l, Fourier series of even and
odd functions, Half range Sine and Cosine Series. Self-learning Topics: Complex
form of Fourier Series, orthogonal and orthonormal set of functions, Fourier
Transform.
Module4
: Complex Variables:
4.1 Function f(z) of complex variable, limit,
continuity and differentiability of f(z), Analytic function, necessary and
sufficient conditions for f(z) to be analytic (without proof),
4.2 Cauchy-Riemann
equations in Cartesian coordinates (without proof)
4.3 Milne-Thomson
method to determine analytic function f(z) when real part (u) or Imaginary part
(v) or its combination (u+v or u-v) is given.
4.4 Harmonic
function, Harmonic conjugate and orthogonal trajectories Self-learning Topics:
Conformal mapping, linear, bilinear mapping, cross ratio, fixed points and
standard transformations
Module5
: Matrices:
5.1 Characteristic equation, Eigen values and Eigen
vectors, Properties of Eigen values and Eigen vectors. (No theorems/proof)
5.2 Cayley-Hamilton
theorem (without proof): Application to find the inverse of the given square
matrix and to determine the given higher degree Polynomial matrix.
5.3 Functions of square matrix, Similarity of
matrices, Diagonalization of matrices Self-learning Topics: Verification of
Cayley Hamilton theorem, Minimal polynomial and Derogatory matrix &
Quadratic Forms (Congruent transformation & Orthogonal Reduction)
Module6
: Numerical methods for PDE
6.1
Introduction of Partial Differential equations, method of separation of
variables, Vibrations of string, Analytical method for one dimensional heatand
wave equations. (only problems)
6.2 Crank Nicholson method, Bender Schmidt method 06
Hrs. Self-learning Topics: Analytical methods of solving two and three
dimensional problems.
End
Semester Examination:
Weightage of each module in end semester examination
will be proportional to number of respective lecture hours mentioned in the
curriculum. 1. Question paper will comprise of total six questions, each
carrying 20 marks 2. Question 1 will be compulsory and should cover maximum
contents of the curriculum 3. Remaining questions will be mixed in nature (for
example if Q.2 has part (a) from module 3 then part (b) will be from any module
other than module 3) 4. Only Four questions need to be solved.
References:
1. Engineering Mathematics, Dr. B. S. Grewal,
KhannaPublication
2. Advanced Engineering Mathematics, Erwin Kreyszig,
Wiley EasternLimited,
3. Advanced Engineering Mathematics, R. K. Jain and
S.R.K. Iyengar, Narosapublication
4. Advanced Engineering Mathematics, H.K. Das, S.
Chand Publication
5. Higher Engineering Mathematics B.V. Ramana,
McGraw HillEducation
6. Complex Variables and Applications, Brown and
Churchill, McGraw-Hilleducation,
7. Text book
of Matrices, Shanti Narayan and P K Mittal, S. ChandPublication
8. Laplace
transforms, Murray R. Spiegel, Schaum’s OutlineSeries
UNIVERSITY OF MUMBAI
Engineering Mathematics-III
Course Code ; EEC 301
Bachelor of Engineering in Electronics &
Telecommunication
(REV-
2019 ‘C’ Scheme) from Academic Year 2019 – 20.
Course
Objectives:
The course is aimed 1. To learn the Laplace
Transform, Inverse Laplace Transform of various functions and its applications.
2. To understand the concept of Fourier Series, its
complex form and enhance the problem solving skill.
3. To understand the concept of complex variables,
C-R equations, harmonic functions and its conjugate and mapping in complex
plane.
4. To
understand the basics of Linear Algebra.
5. To use concepts of vector calculus to analyze and
model engineering problems.
Course
Outcomes: After successful completion of course student will
be able to:
1. Understand the concept of Laplace transform and
its application to solve the real integrals in engineering problems.
2. Understand the concept of inverse Laplace
transform of various functions and its applications in engineering problems.
3. Expand the periodic function by using Fourier
series for real life problems and complex engineering problems.
4. Understand complex variable theory, application
of harmonic conjugate to get orthogonal trajectories and analytic function.
5. Use matrix
algebra to solve the engineering problems.
6. Apply the concepts of vector calculus in real
life problems.
Module1
: Laplace Transform
1.1 Definition of Laplace transform, Condition of
Existence of Laplace transform.
1.2 Laplace Transform (L) of Standard Functions like
𝑒 𝑎𝑡
, 𝑠𝑖𝑛(𝑎𝑡),
𝑐𝑜𝑠(𝑎𝑡),
𝑠𝑖𝑛ℎ(𝑎𝑡),
𝑐𝑜𝑠ℎ(𝑎𝑡)
and 𝑡 𝑛 , 𝑛 ≥ 0.
1.3
Properties of Laplace Transform: Linearity, First Shifting theorem, Second
Shifting Theorem, change of scale Property, multiplication by t, Division by t,
Laplace Transform of derivatives and integrals (Properties without proof).
1.4 Evaluation of integrals by using Laplace
Transformation.
Self-learning
Topics:
Heaviside’s Unit Step function, Laplace Transform of
Periodic functions, Dirac Delta Function.
Module2
: Inverse Laplace Transform
2.1 Inverse Laplace Transform, Linearity property,
use of standard formulae to find inverse Laplace Transform, finding Inverse
Laplace transform using derivatives.
2.2 Partial fractions method to find inverse Laplace
transform.
2.3 Inverse Laplace transform using Convolution
theorem (without proof). Self-learning Topics: Applications to solve initial
and boundary value problems involving ordinary differential equations.
Module3
: Fourier Series:
3.1 Dirichlet’s conditions, Definition of Fourier
series and Parseval’s Identity (without proof). 3.2 Fourier series of periodic
function with period 2𝜋
and 2l.
3.3 Fourier
series of even and odd functions.
3.4 Half range Sine and Cosine Series. Self-learning
Topics: Complex form of Fourier Series, Orthogonal and orthonormal set of
functions. Fourier Transform.
Module4
: Complex Variables:
4.1 Function f(z) of complex variable, limit,
continuity and differentiability of f(z)Analytic function, necessary and
sufficient conditions for f(z) to be analytic (without proof).
4.2 Cauchy-Riemann equations in cartesian
coordinates (without proof).
4.3 Milne-Thomson method to determine analytic
function f(z)when real part (u) or Imaginary part (v) or its combination (u+v
or u-v) is given.
4.4 Harmonic function, Harmonic conjugate and
orthogonal trajectories Self-learning Topics: Conformal mapping, linear,
bilinear mapping, cross ratio, fixed points and standard transformations.
Module5
: Linear Algebra: Matrix Theory
5.1 Characteristic equation, Eigen values and Eigen
vectors, Example based on properties of Eigen values and Eigen vectors.(Without
Proof).
5.2 Cayley-Hamilton theorem (Without proof),
Examples based on verification of Cayley- Hamilton theorem and compute inverse
of Matrix.
5.3 Similarity of matrices, Diagonalization of
matrices. Functions of square matrix Self-learning Topics: Application of
Matrix Theory in machine learning and google page rank algorithms, derogatory
and non-derogatory matrices.
Module6
: Vector Differentiation and Integral
6.1 Vector differentiation: Basics of Gradient,
Divergence and Curl (Without Proof).
6.2
Properties of vector field: Solenoidal and irrotational (conservative) vector
fields.
6.3 Vector
integral: Line Integral, Green’s theorem in a plane (Without Proof), Stokes’
theorem (Without Proof) only evaluation. Self-learning Topics: Gauss’
divergence Theorem and applications of Vector calculus.
End
Semester Theory Examination (80-Marks):
Weightage to each of the modules in end-semester
examination will be proportional to number of respective lecture hours
mentioned in the curriculum.
1. Question paper will comprise of total 06
questions, each carrying 20 marks.
2. Question No: 01 will be compulsory and based on
entire syllabus wherein 4 to 5 sub- questions will be asked.
3. Remaining questions will be mixed in nature and
randomly selected from all the modules.
4. Weightage of each module will be proportional to
number of respective lecture hours as mentioned in the syllabus.
5. Total 04 questions need to be solved.
UNIVERSITY OF MUMBAI
Engineering Mathematics-III
Course Code ; CEC 301
Bachelor of Engineering in Computer Engineering
(REV-
2019 ‘C’ Scheme) from Academic Year 2019 – 20.
Course
Objectives: The course aims:
1 To learn
the Laplace Transform, Inverse Laplace Transform of various functions, its
applications.
2 To understand the concept of Fourier Series, its
complex form and enhance the problemsolving skills. 3 To understand the concept
of complex variables, C-R equations with applications.
4 To
understand the basic techniques of statistics like correlation, regression, and
curve fitting for data analysis, Machine learning, and AI.
5 To understand some advanced topics of probability,
random variables with their distribution
sand expectations.
Course
Outcomes:
On successful completion, of course, learner/student
will be able to:
1 Understand
the concept of Laplace transform and its application to solve the real
integrals in engineering problems.
2 Understand
the concept of inverse Laplace transform of various functions and its
applicationsin engineering problems.
3 Expand the
periodic function by using the Fourier series for real-life problems and
complex engineering problems.
4 Understand complex variable theory, application of
harmonic conjugate to get orthogonal trajectories and analytic functions.
5 Apply the concept of Correlation and Regression to
the engineering problems in data science, machine learning, and AI.
6 Understand
the concepts of probability and expectation for getting the spread of the data
and distribution of probabilities.
Module
1 Laplace Transform:
6 1.1
Definition of Laplace transform, Condition of Existence of Laplace transform.
1.2 Laplace Transform (L) of standard functions like
𝑒 𝑎𝑡
, 𝑠𝑖𝑛(𝑎𝑡),
𝑐𝑜𝑠(𝑎𝑡),
𝑠𝑖𝑛ℎ(𝑎𝑡),
𝑐𝑜𝑠ℎ(𝑎𝑡)and𝑡 𝑛 , 𝑛 ≥ 0.
1.3 Properties of Laplace Transform: Linearity,
First Shifting Theorem, Second Shifting Theorem, Change of Scale,
Multiplication byt, Division by t, Laplace Transform of derivatives and
integrals (Properties without proof).
1.4 Evaluation of real improperintegrals by using
Laplace Transformation.
1.5 Self-learning Topics:Laplace Transform: Periodic
functions, Heaviside’s Unit Step function, Dirac Delta Function, Special
functions (Error and Bessel)
Module
2 Inverse Laplace Transform :
2.1 Definition of Inverse Laplace Transform,
Linearity property, Inverse Laplace Transform of standard functions, Inverse
Laplace transform using derivatives.
2.2 Partial fractions method to find Inverse Laplace
transform.
2.3 Inverse Laplace transform using Convolution
theorem (without proof)
2.4
Self-learning Topics: Applications to solve initial and boundary value problems
involving ordinary differential equations.
Module
3 Fourier Series:
3.1
Dirichlet’s conditions, Definition of Fourier series and Parseval’s
Identity(withoutproof).
3.2 Fourier series of periodic function with period
2π and 2l.
3.3 Fourier series of even and odd functions.
3.4 Half range Sine and Cosine Series.
3.5
Self-learning Topics: Orthogonal and orthonormal set of functions, Complex form
of Fourier Series,Fourier Transforms.
Module
4 Complex Variables:
4.1 Function f(z)of complex variable, Limit,
Continuity andDifferentiability off(z), Analytic function: Necessary and
sufficient conditions for f(z) to beanalytic (without proof).
4.2 Cauchy-Riemann equations in Cartesiancoordinates
(without proof).
4.3 Milne-Thomson method: Determine analytic
function f(z)when real part (u), imaginary part (v) or its combination (u+v /
u-v) is given.
4.4 Harmonic function, Harmonic conjugate and
Orthogonal trajectories.
4.5 Self-learning Topics: Conformal mapping, Linear
and Bilinear mappings, cross ratio, fixed points and standard transformations.
Module
5 Statistical Techniques
5.1 Karl Pearson’s coefficient of correlation (r)
5.2 Spearman’s Rank correlation coefficient (R)
(with repeated and nonrepeated ranks)
5.3 Lines of
regression
5.4 Fitting of first- and second-degree curves.
5.5 Self-learning Topics:Covariance, fitting of
exponential curve.
Module
6 Probability
6.1 Definition
and basics of probability, conditional probability.
6.2 Total
Probability theorem and Bayes’ theorem.
6.3 Discrete and continuous random variable with
probability distribution and probabilitydensity function.
6.4 Expectation, Variance, Moment generating
function, Raw and central moments up to 4th order.
6.5 Self-learning Topics: Skewness and Kurtosis of
distribution (data).
End
Semester Theory Examination:
1 The question paper will comprise a total of 6
questions, each carrying 20 marks.
2 Out of the 6 questions, 4 questions have to be
attempted.
3 Question 1, based on the entire syllabus, will
have 4sub-questions of 5 marks each and is compulsory. 4 Question 2 to Question
6 will have 3 sub-questions,each of 6, 6, and 8 marks, respectively.
5 Each
sub-question in (4) will be from different modules of the syllabus.
6 Weightage
of each module will be proportional to the number of lecture hours,as mentioned
in the syllabus.
UNIVERSITY OF MUMBAI
Engineering Mathematics-I
Course Code ; FEC 201
First Year Bachelor
of Engineering ( Common for All
Branches of Engineering )
(REV-
2019 ‘C’ Scheme) from Academic Year 2019 – 20.
Objectives
:
1. To develop the basic Mathematical skills of
engineering students that are imperative for effective understanding of
engineering subjects. The topics introduced will serve as basic tools for
specialized studies in many fields of engineering and technology.
2. To provide hands on experience using SCILAB
software to handle real life problems.
Outcomes:
Learners will be able to…
1. Illustrate
the basic concepts of Complex numbers.
2. Apply the knowledge of complex numbersto solve
problems in hyperbolic functions and logarithmic function.
3. Illustrate the basic principles of Partial
differentiation.
4. Illustrate the knowledge of Maxima, Minima and
Successive differentiation.
5. Apply principles of basic operations of matrices,
rank and echelon form of matrices to solve simultaneous equations.
6. Illustrate SCILAB programming techniques to the
solution oflinearand simultaneous algebraic equations.
Module1 Complex Numbers Pre-requisite:
Review of Complex Numbers‐Algebra of Complex
Number, Cartesian, polar and exponential form of complex number.
1.1. Statement of D‘Moivre‘s Theorem.
1.2. Expansion of sinn θ, cosnθ in terms of sines
and cosines of multiples of θ and Expansion of sinnθ, cosnθ in powers of sinθ,
cosθ
1.3. Powers and Roots of complex number.
Module 2 Hyperbolic function and Logarithm of
Complex Numbers
2.1. Circular functions of complex number and
Hyperbolic functions. Inverse Circularand Inverse Hyperbolic functions.
Separation of real and imaginary parts of all typesof Functions.
2.2 Logarithmic functions, Separation of real and
Imaginary parts of Logarithmic Functions. # Self learning topics: Applications
of complex number in Signal processing, Electrical circuits.
Module 3 Partial Differentiation
3.1 Partial Differentiation: Function of several
variables, Partial derivatives of first andhigher order. Differentiation of
composite function.
3.2.Euler‘s Theorem on Homogeneous functions with
two independent variables (with proof). Deductions from Euler‘s Theorem. # Self
learning topics: Total differentials, implicit functions, Euler‘s Theorem on
Homogeneous functions with three independent variables.
Module 4 Applications of Partial Differentiation and
Successive differentiation.
4.1 Maxima
and Minima of a function of two independent variables, Lagrange‘s method of
undetermined multipliers with one constraint.
4.2
Successive differentiation: nth derivative of standard functions. Leibnitz‘s
Theorem (without proof) and problems # Self learning topics: Jacobian‘s of two
and three independent variables (simple problems)
Module 5 Matrices Pre-requisite:
Inverse of a matrix, addition, multiplication and transpose of a matrix
5.1.Types of Matrices (symmetric, skew‐
symmetric, Hermitian, Skew Hermitian, Unitary, Orthogonal Matrices and
properties of Matrices). Rank of a Matrix using Echelon forms, reduction to
normal form and PAQ form.
5.2.System of homogeneous and non –homogeneous
equations, their consistency and solutions. # Self learning topics: Application
of inverse of a matrix to coding theory.
Module 6 Numerical Solutions of Transcendental
Equations and System of Linear Equations and Expansion of Function.
6.1 Solution of Transcendental Equations: Solution
by Newton Raphson method andRegula –Falsi.
6.2 Solution of system of linear algebraic
equations, by (1) Gauss Jacobi Iteration Method, (2) Gauss Seidal Iteration
Method.
6.3 Taylor‘s Theorem (Statement only) and Taylor‘s
series, Maclaurin‘s series (Statement only).Expansion of sin(x), cos(x),
tan(x), sinh(x), cosh(x), tanh(x), log(1+x), ( ), ( ), ( ). # Self learning
topics: Indeterminate forms, L‐ Hospital Rule,
Gauss Elimination Method, Gauss Jordan Method.
End
Semester Examination In question paper weightage of each
module will be proportional to number of respective lecture hours as mention in
the syllabus.
1. Question
paper will comprise of 6 questions, each carrying 20 marks.
2. Question
number 1 will be compulsory and based on maximum contents of the syllabus
3. Remaining questions will be mixed in nature (for
example, if Q.2 has part (a) from module 3 then part (b) will be from other
than module 3)
4. Total four questions need to be solved.
Internal Assessment
Test Assessment consists of two class tests of 20 marks each. The first class
test is to be conducted when approx. 40% syllabus is completed and second class
test when additional 35% syllabus is completed. Duration of each test shall be
one hour.
References
1. Higher
Engineering Mathematics, Dr. B. S. Grewal, Khanna Publication
2. Advanced Engineering Mathematics, Erwin Kreyszig,
Wiley Eastern Limited, 9th Ed.
3. Engineering Mathematics by Srimanta Pal and
Subodh,C. Bhunia, Oxford University Press
4. Matrices,
Shanti Narayan, .S. Chand publication.
5. Applied Numerical Methods with MATLAB for
Engineers and Scientists by Steven Chapra, McGraw Hill
About the Course
The crash course will
cover all 6 units within one month. Students will be provided with most likely
questions.
Topics Covered
All six units
We are conducting such crash courses from last 2 years and it has achieved 100%
result.
Who should attend
Regular and Backlog students
for MIII.
First and Second year engineering
students.
M1 , M2, M3, M4.
Pre-requisites
Knowledge of
Differentiation and Integration
What you need to bring
Notebook
Key Takeaways
Knowledge and
confidence required to solve paper
Date and Time
Not decided yet.
For online Class :
Student needs a mobile
phone (or Laptop ) with well connected
to the internet and ZOOM app from play store installed in it.
The teacher will guide
to all students online, there will be teaching, learning, assessment, test,
online notes and learning notes and learning materials will be provided.
Sufficient number of
problems from boards point of view, moderate answers, correction, everything
will be held on this platform.
Student can ask queries
after session or while session also.
About
the Trainer
Name :
Prof. Jagdish Patil
B.Tech ( Elex )
ME. ( EXTC )
22 Years of Experience
I am teaching also since
last 22 years. Also as professor in reputed college.
Mumbai Academics Is Mumbai’s First Dedicated Professional Training Center For Training With Spoke And Hub Model With Multiple Verticles . The Strong Foundation Of Mumbai Academics Is Laid By Highly Skilled And Trained Professionals, Carrying Mission To Provide Industry Level Input To The Freshers And Highly Skilled And Trained Software Professionals/Other Professional To IT Companies. 
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