CAREER POINT UNIVERSITYKOTA, RAJASTHAN

REAL WORLD APPLICATIONS OF CALCULUS AND RELEVANCE OF MATRICES

SUBMITTED TO:DR. SONA RAJFACULTYMATHEMATICS

SUBMITTED BY:JAYA KAUSHIKK12200POOJA PAREEKK12505B.TECH(EE)2ND SEMESTER/ 1 YEAR

CONTENTSMAXIMA AND MINIMADEFINITION POINTS ON AGRAPH: CRITICAL AND SADDLEFINDING FUNCTIONAL MAXIMA MINIMAAPPLICATIONS IN REAL WORLD AND ENGINEERING

MATRIX AND DETERMINANTSDEFINITION DETERMINANTSEIGEN VECTORS AND EIGEN VALUESCOLLINEARITY OF POINTSAPPLICATIONS IN ENGINEERING AND REAL WORLD

DEFINITIONGLOBAL 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.

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 su cient di erentiability.ffi ff

FINDING FUNCTIONAL MAXIMA AND MINIMA

Critical pointA 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.

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 approachTo determine how fast the ship leaving from its starting point

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

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 QUANTITIESMINIMIZING THE COST OF PRODUCTION OF AUTOMOBILE GIVES CERTAIN KNOWN CONSTRAINTS ON THE COST OF EACH PART AND THE TIME SPENT BY EACH LABOURER.

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 circleTo find out shortest and most economical path of motorboatMinimum length of the cables joining at one pointWater flowing into cylindrical tankRate of movement of shadow on the groundWater flowing to the rectangular and triangular troughNearest distance from a given point to curveTime rates : lengthening of shadow and movement of its tip in 3d space

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.

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

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

EigenvaluesLet 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.

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 11Δ= x y 12 x y 1

1 2 3 2 3 1 3 1 21= [x (y - y ) + x (y - y ) + x (y - y )]2

APPLICATIONS 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

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

•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, whichis good. With variables we can change them at will with very minimal hassalto observe the eects. W

• we have to take into account each materialstrength, each resonance for the building, outside stresses such as wind andweather, 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 canassume we have less errors because we have less chances to make errors. Ourchances for errors in solving this with substitution, grows increasingly likelywith every substitution made.8

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.

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