CAREER POINT UNIVERSITY

SUBMITTED TO : Dr. SONA RAJ

SUBMITTED BY : ABDULQADIR EZZY (K12430) YASH MALHOTRA HIMANK MAHESHWARI B.TECH/2ND SEM COMPUTER SCIENCE

SECTION : A

REAL WORLD APPLICATION OF ALGEBRA AND CALCULUS

CONTENTS

INTRODUCTION OF LAPLACE TRANSFORMATION INTRODUCTION OF MATRIX AND DETERMINANTS INTRODUCTION OF CALCULUS EXAMPLE REAL WORLD APPLICATIONS APPLICATION RELATED TO BRANCH (CS) CONCLUSION REFERENCES

INTRODUCTION OF LAPLACE Transformation in mathematics deals with the conversion of one function to

another function that may not be in the same domain.

Laplace transform is a powerful transformation tool, which literally transforms the originaldifferential equation into an elementaryalgebraic expression. This latter can thensimply be transformed once again, into the solution of the original problem.

This transform is named after the mathematician and renowned astronomer Pierre Simon Laplace who lived in France.

INTRODUCTION OF MATRIX

In mathematics , a matrix is a rectangular array of numbers , symbols or expressions arranged in rows and columns.

INTRODUCTION OF CALCULUS

As is well known, the mathematical formalism of calculus is widely and successfully used in natural sciences . However, this does not mean that the problem of validity of calculus is now completely solved, or that the foundations of calculus are not in need of formal-logical analysis. In my view, standard calculus cannot be considered as absolute truth if there is no formal-logical substantiation of this calculus.

EXAMPLE:

MATRIX USE IN C-LANGUAGE

#include<stdio.h>main()int r , c; clrscr(); for(r=1;r<=4;r++){for(c=r;c<=4;c++)}printf(“*”);}printf(“\n”);}}

OUTPUT OF PROGRAM

***********

USING MATRICES IN REAL LIFE

The Golden Triangle is a large triangular region in the India.The Taj Mahal is one of the many wonders that lie within the boundaries of this triangle. The triangle is formed by the imaginary lines that connect the cities of New Delhi, Jaipur, and Agra. Use a determinant to estimate the area of the Golden Triangle. The coordinates given are measured in miles.

EW

N

S

Jaipur (0,0)

New Delhi (100,120)

Agra (140,20)

. ..

SOLUTIONThe approximate coordinates of the Golden Triangle’s three vertices are: (100,120), (140,20), and (0,0). So the area of the region is as follows:

Area12

100 120 1140 20 1

0 0 1

Area12

[(200000) (0016800)]

Area7400

Hence, area of the Golden Triangle is about 7400 square miles.

GRAPHIC USES OF MATRIX MATHEMATICS

Graphic software uses matrix mathematics to process linear transformations to render images. A square matrix, one with exactly as many rows as columns, can represent a linear transformation of a geometric object. For example, in the Cartesian X-Y plane, the matrix       reflects an object in the vertical Y axis. In a video game, this would render the upside down mirror image of a castle reflected in a lake.If the video game has curved reflecting surfaces, such as a shiny silver goblet, the linear transformation matrix would be more complicated, to stretch or shrink the reflection.

CRAMER”S RULE FOR A 33 SYSTEM

Let A be the co-efficient matrix of the linear system: ax+by+cz= j, dx+ey+fz= k, and gx+hy+iz=l.

IF det A ≠0, then the system has exactly one solution. The solution is:

x

j b ck e fl h i

detA, y

a j cd k fg l i

detA, z

a b jd e kg h l

detA

EXAMPLE

The atomic weights of three compounds are shown. Use a linear system and Cramer’s rule to find the atomic weights of carbon(C ), hydrogen(H), and oxygen(O).

Compound Formula Atomic weight

Methane CH4 16

Glycerol C3H8O3 92

Water H2O 18

1) Solving Ordinary Differential Equation

Problem:Y" + aY' + bY = G(t) subject to the initial conditions Y(0) = A, Y' (0) = B where a, b, A, B are constants.

Solution: Laplace transform of Y(t) be y(s), or, more concisely, y. Then solve for y in terms of s. Take the inverse transform, we obtain the desired solution Y.

APPLICATION OF LAPLACE TRANSFORMATION

2) Solving Partial Differential Equation

Problem: Solve

with the boundary conditions U(x, 0) = 3 sin 2πx, U(0, t) = 0 and U(1, t) = 0where 0 < x < 1, t > 0.Solution: Taking Laplace transform of both sides with respect to t,

Substituting in the value of U(x, 0) and rearranging, we get

where u = u(x, s) = L[U(x, t]. The general solution of (1) is Determine the values of c1 and c2. Taking the Laplace transform of those boundary conditions that

involve t, we obtain c1 =0, c2 = 0. Thus (2) becomes

Inversion gives

3) Solving Electrical Circuits Problem

Problem: From the theory of electrical circuits we know, where C is the capacitance, i = i(t) is the electric current , and v = v(t) is the voltage. We have to find the correct expression for the complex impedance of a capacitor. Solution: Taking the Laplace transform of this equation, we obtain,

Where, and

Solving for V(s) we have

We know, So we find: which is the correct expression for the complex impedance of a capacitor.

APPLICATION OF CALCULUS

An electrical engineer uses integration to determine the exact length of power cable needed to connect two substations that are miles apart. Because the cable is hung from poles, it is constantly curving. Calculus allows a precise figure to be determined.

OTHER APPLICATION OF CALCULUS

Space flight engineers frequently use calculus when planning lengthy missions. To launch an exploratory probe, they must consider the different orbiting velocities of the Earth and the planet the probe is targeted for, as well as other gravitational influences like the sun and the moon. Calculus allows each of those variables to be accurately taken into account.

Real world applications of calculus

With calculus, we have the ability to find the effects of changing conditions on a system. By studying these, you can learn how to control a system to make it do what you want it to do. Because of the ability to model and control systems, calculus gives us extraordinary power over the material world.

Biologists use differential calculus to determine the exact rate of growth in a bacterial culture when different variables such as temperature and food source are changed. This research can help increase the rate of growth of necessary bacteria, or decrease the rate of growth for harmful and potentially threatening bacteria.

APPLICATION IN CS

ONE OF THE AREAS OF COMPUTER SCIENCE IN WHICH MATRIX MULTIPLICATION IS PARTICULARLY USEFUL IS GRAPHICS , SINCE A DIGITAL IMAGE IS BASICALLY A MATRIX TO BEGIN WITH : THE ROWS AND COLUMNS OF THE MATRIX CORRESPONS TO ROWS AND COLUMNS OF THE PIXELS , AND NUMERICAL ENTRIES CORRESPONDS TO THE PIXEL’S COLOR VALUES.

WE ARE USING INTEGRATED CIRCUITS INSODE THE CPU. LAPLACE TRANSFORMATION HELP US TO FIND OUT THE CURRENT AND SOME CRITERIA FOR THE ANALYSING THE CIRCUITS.

USES OF CALCULUS IN COMPUTER SCIENCE IS CREATING VISUALS OR GRAPHS OFTEN THE GRAPHS\VISUALS ARE 3D. THEY ARE USED OFTEN FOR VIDEO GAMES,ESPECIALLY PHYSIC ENGINES,PHYSIC ENGINES DEFINE THE GAME SUCH AS GRAVITY,FICTION ETC.

CONLUSION

OVERALL THE CONCLUSION IS ALGEBRA AND CALCULUS IS USED IN VARIOUS FIELDS LIKE IN VARIOUS ENGINEERING FIELDS , REAL WORLD APPLICATIONS , SOLVING VARIOUS EQUATIONS IN MATHS.

REFRENCES

www.quora.comwww.functionspace.comwww.teach-nology.comwww.m.reddil.comwww.academia.edu

• Books- Das H.K• N.P. Bali• introduction to laplace transformation• introduction to matirx

Thank You