The document discusses applications of calculus and matrices in engineering, describing how maxima and minima are used to optimize problems like material usage and piping design, and how matrices and determinants are applied in areas like graph theory, nuclear reactor modeling, and cryptography. It also provides examples of using maxima and minima to determine optimal illumination or boat paths, and examples of matrix applications in areas like computer graphics, statistics, and automatic thesaurus compilation.

1. CAREER POINT UNIVERSITY
KOTA, RAJASTHAN
REAL WORLD APPLICATIONS OF CALCULUS AND
RELEVANCE OF MATRICES
SUBMITTED TO:
DR. SONA RAJ
FACULTY
MATHEMATICS
SUBMITTED BY:
JAYA KAUSHIK
K12200
POOJA PAREEK
K12505
B.TECH(EE)
2ND SEMESTER/ 1 YEAR

2. CONTENTS
MAXIMA AND MINIMA
DEFINITION
 POINTS ON AGRAPH: CRITICAL AND SADDLE
FINDING FUNCTIONAL MAXIMA MINIMA
APPLICATIONS IN REAL WORLD AND ENGINEERING
MATRIX AND DETERMINANTS
DEFINITION
DETERMINANTS
EIGEN VECTORS AND EIGEN VALUES
COLLINEARITY OF POINTS
APPLICATIONS IN ENGINEERING AND REAL WORLD

3. DEFINITION
GLOBAL EXTREMA: if f( c) < f( x) for all x in
domain f, f( c) is the global maximum value of f.
If f( c) < f( x) for all x in domain f, f( c) is the
global minimum value of f.
LOCAL EXTREMUM: if f( c) > f(x) for all x in
domain f, in some open interval containing c, f(
c) is a local maximum value of f. if f( c),< f( x)
for all x in domain f in some open interval
containing c, f( c) is a local minimum value of f.

4. Finding global maxima and minima is the goal of mathematical
optimization. If a function is continuous on a closed interval,
then by the extreme value theorem global maxima and minima
exist. Furthermore, a global maximum(or minimum)either must
be a local maximum (or minimum) in the interior of the domain,
or must lie on the boundary of the domain. So a method of
finding a global maximum (or minimum) is to look at all the
local maxima (or minima) in the interior, and also look at the
maxima (or minima) of the points on the boundary, and take the
largest (or smallest) one.
imum or local minimum by using the first derivative test, second
derivative test, or higher-order derivative test, given sufficient
differentiability.
FINDING FUNCTIONAL MAXIMA AND MINIMA

6. Critical point
A critical point or stationary point of a
differentiable function of a real or complex variable
is any value in its domain where its derivative is 0
or undefined For a differentiable function of
several real variables, a critical point is a value in
its domain where all partial derivatives are zero.
SADDLE POINT
A saddle point is a point in the domain of a
function that is a stationary point but not a local
extremum.

7. Location of the third point on the parabola for largest triangle if a line and a
parabola intersects at point A and C.
For finding out the the position of the cars when they are nearest to each other
Time of collision of cars by finding their velocity of approach
To determine how fast the ship leaving from its starting point

8. Founds application in finding the best illumination on the circular walk surrounding
the area if the light is to be placed at the centre of the walk of radius a
Inscribe a circular cylinder of maximum convex surface area in a given circular cone

9. REAL WORLD APPLICATIONS OF MAXIMA MINIMA
IN EXCONOMICS BUSINESS AND ENGINEERING:
IN MANUFACTURING BUSINESS IT IS USUALLY POSSIBLE TO EXPRESS PROFIT AS
A FUNCTION OF THE NO OF UNITS SOLD BY FINDING MAXIMA MINIMA.
THE SHAPE OF A CONTAINER CAN BE DETERMINED BY MINIMIZING THE USE OF
MATERIAL.
DESIGN OF PIPING SYSTEM BASED ON MINIMIZING THE PRESSURE DROP.
IN LINEAR ALGEBRA AND GAME THEORY:
LINEAR PROGRAMMING CONSISTS OF MAXIMIZING OR MINIMIZING A
PARTICULAR QUANTITY WHILE REQUIRING CERTAIN CONSTRAINTS BE IMPOSED ON
OTHER QUANTITIES
MINIMIZING THE COST OF PRODUCTION OF AUTOMOBILE GIVES CERTAIN
KNOWN CONSTRAINTS ON THE COST OF EACH PART AND THE TIME SPENT BY
EACH LABOURER.

11. To find out the height for max attraction that a wire bent in
the form of circle of radius and exerts upon a particle in axis
of circle
To find out shortest and most economical path of
motorboat
Minimum length of the cables joining at one point
Water flowing into cylindrical tank
Rate of movement of shadow on the ground
Water flowing to the rectangular and triangular trough
Nearest distance from a given point to curve
Time rates : lengthening of shadow and movement of its tip
in 3d space

12. DETERMINANTS
Every square matrix has a determinant. The
determinant has the same elements as the matrix,
but they are enclosed between vertical bars instead
of brackets. you have learned a method for
evaluating a 2 x 2 determinant.

13. The lambda modes analysis is a powerful tool for
safety analysis of nuclear reactors. It can be used
to study the steady state neutron flux
distribution inside the reactor core The lambda
modes equation is a differential eigenvalue
problem derived from the neutron diffusion
equation. The matrices associated to the
eigenvalue problem have a block structure and
the number of blocks depends on how many
levels are considered when discretizing the
energy.
The problem has been addressed with SLEPc for
solving the eigenvalue problem combined with
the linear system solvers provided by PETSc. The
code computes the eigenvectors corresponding
to a few of the largest eigenvalues. Several
benchmark reactors have been used for
validation.
NUCLEAR ENGINEERING
APPLICATION

14. CONCLUSION
MATRICES ARE UTLIZED IN NUMEROUS TECHNOLOGIES .
APPLICATION IS NOT ONLY REOUND IMPORTANT STRINED TO
GRAPH THEORY, STENOGRAPHY, LINEAR EQUATION SOLUTION.
THEY ARE IMP IMPORTANT IN CONFIDENTIAL MESSAGE
TRANSFER, COMPUTERIZED LOCKERS ALSO FINDS APPLICATION
IN BUISSNESS, CONVERSATION IN MILITARY ADMINISTRATION
SIMILARY MAXIMA MINIMA ALSO FINDS GREAT APPLICATIO IN
DESIGNING IN TYPES OF BEAMS, CACULATION OF MINIMUM
DISTANCE FOR CABLES JOINING AT THE POINT

15. Eigenvalues
Let x be an eigenvector of the matrix A. Then there must exist an
eigenvalue λ such that Ax = λx or, equivalently,
Ax - λx = 0 or
(A – λI)x = 0
If we define a new matrix B = A – λI, then
Bx = 0
If B has an inverse then x = B-10 = 0. But an eigenvector cannot be zero.
Thus, it follows that x will be an eigenvector of A if and only if B does
not have an inverse, or equivalently det(B)=0, or
det(A – λI) = 0
This is called the characteristic equation of A.

16. APPLICATIONS OF DETERMINANTS (AREA OF A
TRIANGLE)
The area of a triangle whose vertices are
is given by the expression
1 1 2 2 3 3(x , y ), (x , y ) and (x , y )
1 1
2 2
3 3
x y 1
1
Δ= x y 1
2
x y 1
1 2 3 2 3 1 3 1 2
1
= [x (y - y )+ x (y - y )+ x (y - y )]
2

17.  GRAPH DESIGN
 SOLVINGG LINEAR EQUATIONS
 COMPUTER GRAPHICS USING MATRICES FOR
PROJECTING 3D OBJECT ONTO A 2D
 DESCRIPTIVE STATISTICSUSING MATRICES FOR
DESCRIBING DATA
 IN CRYPTOGRAPHY:
FOR PRIVACY IN NETWORKS
INCLUDES ENCRYPYION AND DECRYPTION
REQUIRES SAME KEY
AUTOMATIC THESARUS COMPILITION

18. Mechanical Engineering Application: Car
Silencer Design - Uses PETSc
Simulation of Automotive Silencers. One of the design requirements of
automotive exhaust silencers is the acoustic attenuation performance
at the usual frequencies. Analytical approaches such as the plane-
wave propagation model can be considered for studying the acoustic
behavior of silencers with rectangular and circular geometries, as well
as with elliptical cross-section. In this way, the designer.
A numerical simulator based on the finite element method has been
developed [Alonson et al, 2004] solves a linear system of equations
per each excitation frequency using PETSc

19. •n this example, we are trying to
solve for the forces located in the
beams.
Since we do not have any initial
conditions, we must solve for
variables, which
is good. With variables we can
change them at will with very
minimal hassal
to observe the eects. W
• we have to take into account each material
strength, each resonance for the building, outside stresses such as wind and
weather, etc. So, while it is possible to solve for each individual variable,
it is much faster and easier to solve this way. In addition to that, we can
assume we have less errors because we have less chances to make errors. Our
chances for errors in solving this with substitution, grows increasingly likely
with every substitution made.
8

20. Diagonal Matrices
Diagonal matrix is a square matrix that has zeroes
everywhere except along the main diagonal (top left to
bottom right).
For example, here is a 3 × 3 diagonal matrix:
[70002000−1]display
style{left[begin{matrix}{7}&{0}&{0}{0}&{2}&{0}{0}&{0}
&-{1}end{matrix}right]}​⎣⎡​​​7​0​0​​​0​2​0​​​0​0​−1​​​⎦⎤
Note: The identity matrix (above) is another example
of a diagonal matrix.