This document contains an assignment submitted by three students on the topics of vector calculus, maxima-minima, and convergent and divergent series. It includes introductions to each topic, examples of applications in engineering and real life, and conclusions. References and websites visited in preparing the assignment are listed at the end.

1. CAREER POINT UNIVERSITY
KOTA,RAJASTHAN
ASSIGNMENT
Subject:- Mahematics
Topic:- Study of Vector calculus, Maxima-Minima and Convergent and
Divergent series
Submitted to:-
Dr. Sona Raj ma’am
Professor
Mathematics Department
Submitted by:-
Vasim khan Gori (K12519)
Mohammeed Aquib(K12678)
Kapil Dev Rawal(K13010)

2. INTRODUCTION
Vector calculus is a branch of mathematics that engineering students
typically become introduced to during their first or second year at the
university. It is used extensively in physics and engineering, especially in
topics like electromagnetic fields and fluid mechanics. Vector calculus is
usually part of courses in multivariable calculus, and lays the foundation for
further studies in mathematics, for example in differential geometry and in
studies of partial differential equations. The basic objects in vector calculus
are scalar fields and vector fields, and the most basic algebraic operations
consist of scalar multiplication, vector addition, dot product, and cross
product. These basic operations are usually taught in a prior course in
linear algebra. In vector calculus, various differential operators defined on
scalar or vector fields are studied, which are typically expressed in terms of
the del operator .

3. INTRODUCTION

4. INTRODUCTION
In this topic we will study of the convergent and divergent
series which is a very important for determine the nature
of a series we have to find Sn . Since it is not possible to
find Sn for every series , we have to device tests for
convergence without involving Sn .
This topic comes under the sequence and series which is
commonly used in higher mathematics .

5. Convergent and divergent series
represents in real life
 Consider a handball court inside a cube,where every wall
including the ceiling is fair game to bounce a ball off of .bounce
a ball at an angle off one wall ,so that it bounces off numerous
diffferent walls .
 In an ideal world of no friction ,etc.,the ball will bounce foreever
and never anywhere.this is a divergent series.In the real
world,evenvually the ball will converge and come to rest
somewhere.
 There seems to be an avoidance of divergent series in real life.
so when divergence appear in real life ,that means there is a
factor that you haven’t considered .

6.  A cylinder has a fixed surface area .Establish a relation between radius and height
of a cylinder for which it’s volume is maximum.
S = 2rh + 2r2 (given)
V = r2h
(We have to maximise volume. So first
reduce variables r and h in either r or h )
V= r2 ((s – 2r2)/2r) = r(s- 2r2)/2
Problem
h
r

7. V= rs - 2r3
dv/dr = d/dr(rs - 2r3)
dv/dr = s- 6r2
For maxima and minima dv/dr =0
So s- 6r2 = 0 or s = 6r2
Or 2rh + 2r2= 6r2 or h= 2r
Now for knowing whether for h=2r volume of cylinder is
maximum or minimum, we calculate d2v/dr2

8.  dv/dr = s- 6r2
 d2v/dr2 = -12r = -ve
 So volume will maximum at h= 2r
h=2r
r
Cylinder of maximum volume
for given surface area

9. APPLICATIONS IN MECHANICAL
ENGG.
Engineering Materials:- Structure and properties of engineering materials,heat
treatment, stress-strain diagrams for engineering materials.
Metal Casting:- Design of patterns, modules and cores; solidification and cooling; riser
gating design and other design considerations.
Engineering Mechanics:- Free body diagram and equilibrium; trusses and frames;
virtual work; kinematics and dynamics of rigid bodies in plane motion, including impulse
and momentum and energy formulations.

10. Uses of maxima and minima
 For marketing purposes we require vessels of different shapes for which
fabrication cost is less but they could contain more material e.g. 1 litre container
of ghee.
 For getting more rectangular land area when total perimeter of land is given.
 In factories using resources so that the fabrication cost of commodity become
less.

11. Application of vector multiplication
zzyyxx BABABAABBA  cos

sin, ABBA
BBB
AAA
kji
BA
zyx
zyx 

BA


BA


a) Work
rdFdW
dFFdW



 cos
b) Torque Fr


rv

 
v

sinr
r




c) Angular velocity
1) Dot product
2) Cross product
- Example

12. Motion of a particle in a circle at constant speed:-
.
.,
2
2
constvvv
constrrr




Differentiating the above equations,
0or02
,0or02


av
dt
vd
v
vr
dt
rd
r






“two vectors are perpendicular”
r
v
a
avvrvar
var
vvar
vr
2
2
2
,0&0
0this,atingDifferenti
,0












13. Divergence and divergence theorem
z
V
y
V
x
V
VVV
zyx
VV
zyx
zyx
















 ),,(),,(div

flow of a gas, heat, electricity, or particles
vV 
nV 





coscos
cos
))()((
Vv
AvtAvt
Avt
: flow of water
amount of water crossing A’ for t
1) Physical meaning of divergence

14. ),,( zyx VVVV 

- Rate at which water flows across surface 1 dydzV x )1(
- Rate at which water flows across surface 2 dydzV x )2(
- Net outflow along x-axis dydzdx
x
V
dydzVV x
xx 







 )]1()2([
axis-zalong,
axis-yalong,
dxdydz
z
V
dzdxdy
y
V
z
y
















In this way,
dxdydzdxdydzdxdydz
z
V
y
V
x
V zyx
VV 













div
“Divergence is the net rate of outflow per unit volume at a point.”

15. APPLICATIONS
1. Vector Magnitude and Direction
Consider the vector shown in the diagram. The vector is drawn pointing toward the
upper right. The origin of the vector is, literally, the origin on this x-y plot.

16. Let’s say the vector is the horizontal wind. The magnitude of the wind is called the
wind speed. Now suppose each grid box corresponds to a wind speed of one meter
per second (1 m s-1 ). If we take a ruler to the page, we find that each grid box is half
an inch wide. So a vector that’s ½ inch long on this particular graph would have a
magnitude of 1 m s-1. A vector that’s an inch long would be 2 m s-1, a vector that’s 1½
inches long would be 3 m s-1.

17. Meteorologists express wind direction as the direction the wind is coming from, not
going towards. So we must add 180 degrees to get the compass heading on the
opposite side of the compass dial: this wind direction is 238 degrees.

18. Convergent and divergent series
represents in real life
 Consider a handball court inside a cube,where every wall
including the ceiling is fair game to bounce a ball off of
.bounce a ball at an angle off one wall ,so that it bounces off
numerous diffferent walls .
 In an ideal world of no friction ,etc.,the ball will bounce
foreever and never anywhere.this is a divergent series.In the
real world,evenvually the ball will converge and come to rest
somewhere.
 There seems to be an avoidance of divergent series in real
life. so when divergence appear in real life ,that means there
is a factor that you haven’t considered .

19. Conclusion
There are many uses of the given three branches of
mathematics in various engineering field as well as in other
fields of daily life as we just discussed in this presentation.
Vector calculus is widely used in physics and the maxima –
minima concept is being used to get more profit by minimum
investment. The convergent and divergent series describes the
concept of a work being done for infinite time period.

20. REFERENCE
1. Weisstein, Eric W."Perp Dot Product."FromMathWorld--A Wolfram Web
Resource.
2. Michael J. Crowe (1967).A History of Vector Analysis : The Evolution of the
Idea of a Vectorial System. Dover Publications; Reprint edition.ISBN 0-486-
67910-1.
3. Barry Spain (1965)Vector Analysis, 2nd edition, link fromInternet Archive.
4. J.E. Marsden (1976).Vector Calculus. W. H. Freeman & Company.ISBN 0-
7167-0462-5
5. Chen-To Tai (1995).A historical study of vector analysis. Technical Report RL
915, Radiation Laboratory, University of Michigan.

21. Visited Websites
1. www.mathclasses.in
2. www.seriesslides.com
3. www.mathsworld.com
4. http://math.rice.edu/~lanius/misc/calcu.html
5. http://www.Karlscalculus.org/calculus.html#toc
6. http://www.mathalino.com/reviewer/differential-calculus/21-
23-solved-problems-in-maxima-and-minima