real world and CS branch related application of algebra and calculus

1. 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

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

3. 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 original
differential equation into an elementary
algebraic expression. This latter can then
simply 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.

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

5. 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.

6. 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”);
}
}

7. OUTPUT OF PROGRAM
*****
***
**
*

8.  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.
E
W
N
S
Jaipur (0,0)
New Delhi (100,120)
Agra (140,20)
. .
.

9. SOLUTION
The 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:
Area  
1
2
100 120 1
140 20 1
0 0 1
Area  
1
2
[(2000  0  0)  (0  0 16800)]
Area  7400
Hence, area of the Golden Triangle is about 7400 square miles.

10.  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.

11. CRAMER”S RULE FOR A 33
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 c
k e f
l h i
det A
, y 
a j c
d k f
g l i
det A
, z 
a b j
d e k
g h l
det A

12. 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

13. 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

14. 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) = 0
where 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

15. 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.

16. 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.

17. 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.

18. 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.

19. 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.

20. 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 GRAPHSVISUALS ARE 3D. THEY ARE USED OFTEN
FOR VIDEO GAMES,ESPECIALLY PHYSIC ENGINES,PHYSIC ENGINES
DEFINE THE GAME SUCH AS GRAVITY,FICTION ETC.

21. 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.

22. REFRENCES
www.quora.com
www.functionspace.com
www.teach-nology.com
www.m.reddil.com
www.academia.edu
• Books- Das H.K
• N.P. Bali
• introduction to laplace transformation
• introduction to matirx