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)
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CAREER POINT UNIVERSITYKOTA,RAJASTHAN
ASSIGNMENT
Subject:- Mahematics Topic:- Study of Vector calculus, Maxima-Minima and Convergent and
Divergent series
Submitted to:-Dr. Sona Raj ma’amProfessorMathematics Department
Submitted by:-Vasim khan Gori (K12519)Mohammeed Aquib(K12678)Kapil Dev Rawal(K13010)
INTRODUCTIONVector 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 .
INTRODUCTION
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 .
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 .
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 = 2rh + 2r2 (given)
V = r2h
(We have to maximise volume. So first
reduce variables r and h in either r or h )
V= r2 ((s – 2r2)/2r) = r(s- 2r2)/2
Problem
h
r
V= rs - 2r3
dv/dr = d/dr(rs - 2r3)
dv/dr = s- 6r2
For maxima and minima dv/dr =0
So s- 6r2 = 0 or s = 6r2
Or 2rh + 2r2= 6r2 or h= 2r
Now for knowing whether for h=2r volume of cylinder is
maximum or minimum, we calculate d2v/dr2
dv/dr = s- 6r2
d2v/dr2 = -12r = -ve
So volume will maximum at h= 2r
h=2r
r
Cylinder of maximum volume for given surface area
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.
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.
Application of vector multiplication
zzyyxx BABABAABBA cos
sin , ABBABBBAAAkji
BA
zyx
zyx
BA
BA
a) WorkrdFdW
dFFdW
"
cos
b) Torque Fr
rv
v
sinrr
c) Angular velocity
1) Dot product
2) Cross product
- Example
Motion of a particle in a circle at constant speed:-
.
.,2
2
constvvv
constrrr
Differentiating the above equations,
0or 02
,0or 02
avdtvdv
vrdtrdr
“two vectors are perpendicular”
rva
avvrvar
var
vvar
vr
2
2
2
,0 & 0
0 this,atingDifferenti
,0
Divergence and divergence theorem
zV
yV
xV
VVVzyx
VV
zyx
zyx
),,(),,(div
flow of a gas, heat, electricity, or particles
vV
nV
coscoscos
))()((
VvAvtAvt
Avt
: flow of water
amount of water crossing A’ for t
1) Physical meaning of divergence
),,( 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 dydzdxxVdydzVV x
xx
)]1()2([
axis-z along ,
axis-y along ,
dxdydzzV
dzdxdyyV
z
y
In this way,
dxdydzdxdydzdxdydzzV
yV
xV zyx VV
div
“Divergence is the net rate of outflow per unit volume at a point.”
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.
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.
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.
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 .
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.
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.