Importance of Mathematics in Engineering, Application areas of mathematics, mathematical thinking,

1. IMPORTANCE OF
MATHEMATICS IN
ENGINEERING
Dr. Chetan B. Bhatt
Principal,
Government MCA College
Maninagar, Ahmedabad

2. यथा शिखाां मयूराणाां, नागानाां मणयो यथा ।
तद्वद वेदाङ्ग िाश्त्राणाां, गशणतां मुशधिनां वतिते ॥
Vedang Jyotish (1000 BC)
"As the crown on the head of a peacock, and as the
gem on the hood of a snake,
so stands Mathematics crowned above all disciplines
of knowledge."
DR CHETAN B BHATT

3. PRESENTATION OUTLINE
Engineering
Applied Mathematics
 Application Domains
 Current Status
Mathematical Thinking and Engineering Education
Mathematical Tools
DR CHETAN B BHATT

4. ENGINEERING
DR CHETAN B BHATT

5. WHAT IS ENGINEERING?
The main task of engineering is to find and
deliver optimal solutions to real life problems,
within the given material, technological,
economic, social, and environmental
constraints, through, the application of
scientific, mathematical, technological, and
engineering knowledge.
Main activities performed by engineers are –
 Conceive, Design, Implement, Operate DR CHETAN B BHATT

6. ENGINEERING DESIGN
APPROACH
Prepare mathematical model
Simulate
Analyse
Implement
Mathematics is used at every stage and mathematical
thinking is very important in engineering
DR CHETAN B BHATT

7. REAL LIFE PROBLEM,
DESIGN, AND MATHS
Case – 1: A man weighing 60Kg sits on a three-legged
stool. What force should each leg be designed for?
Case – 2: A man weighing 60Kg sits on a four-legged
stool. What force should each leg be designed for?
DR CHETAN B BHATT

8. REAL LIFE PROBLEM,
DESIGN, AND MATHS
Case – 1: A man weighing 60Kg sits on a three-legged
stool. What force should each leg be designed for?
Mathematical problem formulation:
Three unknown and system is determinate
A
B
C
G
60
Kg
R
B
R
C
R
A
Z
X
Y
0

 Z
F 0

 X
M 0

 Y
M
DR CHETAN B BHATT

9. REAL LIFE PROBLEM,
DESIGN, AND MATHS
Case – 2: A man weighing 60Kg sits on a four-legged
stool. What force should each leg be designed for?
Mathematical problem formulation:
Four unknown and system is indeterminate
A
B
D
60
Kg
R
B
R
D
R
A
Z
X
Y
0

 Z
F 0

 X
M 0

 Y
M C
R
C
In case – 1 leg should be designed for 20
Kg.
In case – 2 leg must be designed for 30 Kg
?????
(Hambly’s Paradox)
G
DR CHETAN B BHATT

10. REAL LIFE PROBLEM,
DESIGN, AND MATHS
Case – 2: A man weighing 60Kg sits on a four-legged
stool. What force should each leg be designed for?
Mathematical problem formulation:
Four unknown and system is indeterminate
A
B
D
G
60
Kg
R
B
R
D
R
A
Z
X
Y
0

 Z
F 0

 X
M 0

 Y
M C
R
C
G
DR CHETAN B BHATT

11. EXAMPLE
If A > 0, B > 0 and A x B = 1;
What can be inferred?
A + B ≥ 2
DR CHETAN B BHATT

12. COMMENT
In engineering formulating problems in mathematical
forms with consideration of practical situations are
very important in designing.
Engineering design requires mathematical thinking as
well as thinking beyond the science and maths.
DR CHETAN B BHATT

13. APPLIED
MATHEMATICS
DR CHETAN B BHATT

14. AREAS OF MATHEMATICS
As per the Mathematical Subject Classification
(MSC) areas are –
 General / Foundation (Study if foundation of mathematics and
logic)
 Discrete mathematics and algebra (Study of structure of
mathematical abstraction)
 Analysis (Study of change and quantity)
 Geometry and topology (study of space)
 Applied mathematics and other (study of mathematical
abstraction)
DR CHETAN B BHATT

15. APPLIED MATHEMATICS
Mathematics used to solve real life problems
Many topics are developed as pure mathematics and found
unexpected application later
Historically mathematical knowledge developed for applications
 Nyay Shastra, Chand Shastra, Aryabhatiyam, Shalabha Sutra, Newtonian
mathematics, Fluid Mechanics
Many universities having department of applied mathematics and
having courses such as –
 Classical mechanics, Fluid mechanics
Then invention of computation devices led to courses like numerical
analysis, computational mathematics
DR CHETAN B BHATT

16. APPLICATION DOMAINS
Physical Science and Engineering:
Classical Mechanics:
Calculus, trigonometry, vector space, tensor, matrix algebra,
differential equation, integral equation, integral transform, infinite
series, and complex variables etc.
techniques of abstraction, generalization and logical deduction are
used.
Time – optimization, energy – optimization i.e. optimization
techniques.
DR CHETAN B BHATT

17. APPLICATION DOMAINS
Fluid Dynamics:
It uses an area of fluid mechanics (which is considered as one of the area of
mathematics)
This is one of the widely applied area of mathematics used for
understanding volcanic eruption, flight, ocean currents etc.
Civil and mechanical engineers still base there model on this work. One of
the most widespread numerical analysis techniques for working with such
models are Finite Element Methods (FEM).
Also very important in atmospheric sciences, in dynamic metrology, and
weather forecasting.
Computational fluid dynamics: used to solve Navier – Stoke equations for
specified initial and boundary conditions for subsonic, transonic, and
hypersonic flows. DR CHETAN B BHATT

18. APPLICATION DOMAINS
Computer Science:
Number theory, algorithms, numerical analysis, logic, cryptography, set theory,
etc.
AI and Soft-computing is one of the areas extensively based on mathematics.
Signals and Systems:
Various transforms such as Laplace, Fourier Transform, wavelet transform etc.,
Lyapunov function, matrix algebra, statistics etc.
Systems Biology:
Using mathematics to prepare model for biological phenomenon and study
various aspects
VLSI Design:
Optimization, Automata theory etc.
DR CHETAN B BHATT

19. APPLICATION DOMAIN
Biological Sciences
Mathematical genetics, mathematical ecology, mathematical neuro-
physiology
Mathematical theory of epidemic, population, biomechanics
Social Sciences
Sociology, Economics, Finance, psychology, insurance
Mathematics in Art
Music, painting, sculpture, linguistics
and also mathematics in ∞
DR CHETAN B BHATT

20. MATHEMATICAL
SCIENCE IN 21ST
CENTURY
DR CHETAN B BHATT

21. LAST TWO DECADES
In the last two decades, two separate revolutions have brought digital
media out of the pre – internet age.
 Both are grounded in mathematical science
 One is matured
 Another has just began but is already redefining the limits of feasibility in some
areas of biological imaging, communication, remote sensing, and other field of
science.
DR CHETAN B BHATT

22. WAVELET REVOLUTION
 The central idea of wavelet is that for most real world images, we
don’t need all the details (bytes) in order to learn something useful.
For example, in 10 megapixel image of a face, vast majority of the pixels do not
give us useful information
 Wavelets were discover and rediscover dozens of time in 20th centure
Physicists were trying to localize waves in time and frequency
Geologist trying to interpret earth movements from seismograms
In 1984, it was discovered that all these disparate, ad hoc techniques for
decomposing a signal into its most informative pieces were really the same.
 This is typically the role of the mathematical science in engineering
and science. Because they are independent of a particular scientific
context, the mathematical science can bridge discipline
DR CHETAN B BHATT

23. COMPRESSED SENSING
 In 2004, the central premise of the wavelet revolution was turned on
its head with some simple questions:
Why do we even bother to acquiring 10 millions pixels of information, if we are
going to discard 90 percent or 99 percent of it with compressed algorithm.
Why don’t we acquire only 1 percent of the information to start with?
 This realization helped to start a second revolution, called
“Compressed Sensing”, also known as “Sparse Sampling” .
How can we know which 1 percent of the information is most relevant?
The answer is by mathematical method called “L1 minimization”
DR CHETAN B BHATT

24. APPLICATION OF
COMPRESSED SENSING
 Magnetic Resonance Imaging
Small kids as patient unable to hold still
Kaleidoscope camera to take randomly weighted average of many randomly
selected pixels.
 Facial recognition
 Short ware IR cameras
 Astronomy
 Transmission Electron Microscope
 Photography
and may more ....
DR CHETAN B BHATT

25. SUBDIVISION SURFACES
APPLICATION
DR CHETAN B BHATT
an animated short film called “Geri’s Game,”
released by Pixar
Animation Studios in 1997, which received
an Academy Award in 1998. It was
the first animated film to use subdivision
surfaces, a mathematical technique
based on wavelet compression. Wavelets
allow computers to compress an image
into a smaller data file. Subdivision surfaces
do the reverse: They allow the
computer to create a small data file that can
be manipulated and then uncompressed
to create lifelike images of something that
never existed—in this case, an
old man playing chess in the park. The top
image shows the subdivision surface.
The image below shows an actual frame
from the movie.

26. EIGENVECTORS
 In 1997, Sergey Brin and Larry Page wrote a paper on experimental
search engine that they called Google (10^100).
 The idea was to give a PageRank.
 The PageRank algorithm is like Chiken-and-Egg paradox. To compute PageRank of
a Page one need to know the PageRank of other Pages.
 The problem was identified as well known math problem, known as Eigenvector
problem.
 A vector (in this case) is just a list of numbers, such as the list of the PageRank
 If we apply PageRank algorithm to a collection of vectors, most will be changed,
but the true PageRank vector persists: It is not changed by the algorithm.
 This kind of “persistent” vector is known in mathematics as Eigenvector.
DR CHETAN B BHATT

27. EIGENVECTORS
 Century back, in 1926, the motion of electrons described by the
Erwin Schrodinger. The electrons’ orbit form complicated three
dimensional shapes that are determined by Eigenvector of
Schrodinger Equation.
 At present in genomics to find out the characteristic features of
different gene more than one Eigenvectors are used. Each specifying
different characteristics. Singular Value Decomposition (SVD) is a
purely mathematical technique to pick out characteristics features in
a giant array of data by finding Eigenvectors.
DR CHETAN B BHATT
From Eigenvector to IPO 
Google went public in 2004, its
initial stock offering raised $27
billion

28. DR CHETAN B BHATT

29. MATHEMATICAL MODEL
Mathematical model is a description of a system using mathematical
concepts and language.
The process of developing mathematical model is termed
mathematical modelling.
DR CHETAN B BHATT

30. CLASSIFICATIONS OF
MATHEMATICAL MODELS
• Linear vs. Non-linear
• Static vs. Dynamic
• Explicit vs. Implicit
• Discrete vs. Continuous
• Deterministic vs. Probabilistic (Stochastic)
• Deductive, Inductive, and Floating
DR CHETAN B BHATT

31. MATHEMATICAL TOOLS
Mathematical tools available are –
 Mathematica
 MATLAB
 Maple
 SciLAB
 Sage
DR CHETAN B BHATT

32. MATHEMATICAL
THINKING AND
ENGINEERING
EDUCATION
DR CHETAN B BHATT

33. CATEGORIES OF
KNOWLEDGE
Factual
Conceptual
Procedural
Metacognitive
The above categories are applicable to all domains.
DR CHETAN B BHATT

34. MATHEMATICAL THINKING
Mathematical thinking is an approach to solve problems.
Mathematical thinking is not solving only mathematical problem, but
to apply mathematical knowledge to formulate and solve problems.
Mathematical Thinking process involves
 Problem solving
 Reasoning and Proof
 Connection
 Representation
 Communication
DR CHETAN B BHATT

35. MATHEMATICAL THINKING
FOR ENGINEERS
Problem Solving
• Build new mathematical knowledge through problem solving
• Solve problem that arise in mathematics or in other contexts
• Apply and adapt a variety of appropriate strategies to solve problems
• Monitor and reflect on the process of mathematical problem solving
Reasoning and Proof
• Recognize reasoning and proof as fundamental aspect of mathematics
• Make and investigate mathematical conjecture
• Develop and evaluate mathematical argument and proof
• Select and use various types of reasoning and methods of proof
DR CHETAN B BHATT

36. MATHEMATICAL THINKING
FOR ENGINEERS
Connection
• Recognize and use connections among mathematical ideas
• Understand how mathematical ideas interconnect and build on one another to produce
coherent whole
• Recognize and apply mathematics in contexts outside of mathematics
Representation
• Create and use representation to organize, record, and communicate mathematical ideas
• Select, apply, and translate among mathematical representation to solve problems
• Use representations to model and interpret physical, social, and mathematical phenomena
DR CHETAN B BHATT

37. MATHEMATICAL THINKING
FOR ENGINEERS
Communication
•Organize and consolidate and communicate
mathematical process coherently and clearly to others
•Analyse and evaluate mathematical process and
strategies of others
•Use language of mathematics to express mathematical
ideas precisely
DR CHETAN B BHATT

38. RAMANUJAN NUMBER
Hardy – Ramanujan Number
1729
1729 = 13 + 123 = 93 + 103
DR CHETAN B BHATT

39. MUSIC AND MATHEMATICS
Scale C D E F G A B C
Indian Sa Re Ga Ma Pa Dha Ni Sa
Western Do Re Mi Fa So La Ti Do
1 9/8 5/4 4/3 3/2 5/3 15/8 2
• Sound
• Noise/Non-music
• Music
DR CHETAN B BHATT

40. HARMONICS
DR CHETAN B BHATT

41. MATHEMATICAL WORLD
Numbers
Symbolic relationships
Shapes
Uncertainty
Reasoning
DR CHETAN B BHATT

42. ॐ पूणिमदः पूणिशमदां पूणाित् पूणि मुदच्यते |
पूणिस्य पूणिमादाय पूणिमेवावशिष्यते ||
Thank You
(धन्यवाद)
DR CHETAN B BHATT