Department of Mathematics

Mata Sundri College(University of Delhi)

Introduction of Discipline

Course II

Prerequisite: For the Discipline Course II, The student must have studied Mathematics Upto 10+2 Level.

The course structure of Discipline-II in Mathematics is

a blend of pure and applied papers. The study of this

course would be beneficial to students belonging to

variety of disciplines such as Economics, Physics,

Engineering, Management Sciences, Computer

Sciences, Operational research and Natural sciences.

The course has been designed to help one pursue a

masters degree in Mathematics and also helps in

various competitive examinations. The first two

courses on Calculus and Linear Algebra are central to

both pure and applied mathematics. The next two

courses differential equations & Mathematical

modeling and Numerical methods with practical

components are of applied nature.

The course on Differential Equations and Mathematical

Modeling deals with modeling of much Physical,

technical, or biological process in the form of

differential equations and their solution procedures.

The course on Numerical Methods involves the design

and analysis of techniques to give approximate but

accurate solutions of hard problems using iterative

methods. The last two courses on Real Analysis and

Abstract Algebra provides an introduction to the two

branches of Pure Mathematics in a rigorous and

definite form.

What is Calculus?

From Latin, calculus, a small stone used for

counting

A branch of Mathematics including limits,

derivatives, integrals, and infinite sums

Used in science, economics, and engineering

Builds on algebra, geometry, and

trigonometry with two major branches

differential calculus and integral calculus

Definition of limit of a function, One sided limit, Limits at infinity, Curve sketching, Volumes of solids of revolution by the washer method .

Sample of syllabus

Vector valued functions: Limit, Continuity, Derivatives, integrals, Arc length, Unit tangent vector Chain Rule, Directional derivatives, Gradient,

Tangent plane and normal line, Extreme values, Saddle points and so on.

Introduction to Limits

What is a limit?

A Geometric Example

Look at a polygon inscribed in a circle

As the number of sides of the polygon increases, the polygon is getting closer to becoming a circle.

If we refer to the polygon as an n-gon, where n is the number of sides we can make some

Mathematical statements:

As n gets larger, the n-gon gets closer to being a circle

As n approaches infinity, the n-gon approaches the circle

The limit of the n-gon, as n goes to infinity is the circle

limn

n go Ci cn r le

The symbolic statement is:

The n-gon never really gets to be the

circle, but it gets close - really, really

close, and for all practical purposes, it

may as well be the circle. That is what

limits are all about!

Numerical Examples

Let’s look at the sequence whose nth term is given by

1, ½, 1/3, ¼, …..1/10000,…., 1/10000000000000..

As n is getting bigger, what are these terms approaching?

1na n

01

limn n

Graphical

Examples

1( )f x

x

As x gets really, really big, what is happening to the height, f(x)?

01

limx x

As x gets really, really small, what is happening to the height, f(x)?

Does the height, or f(x) ever get to 0?

Nonexistence

Examples

Oscillating Behavior

Discuss the existence of the limit

0

1limsinx x

X 2/π 2/3π 2/5π 2/7π 2/9π 2/11π X 0

Sin(1/x)

1 -1 1 -1 1 -1 Limit does not exist

Differential Equations and Mathematical Modeling

First order ordinary differential equations: Basic concepts and ideas, Modeling: Exponential growth and decay, Direction field, Separable equations, Modeling: Radiocarbon dating, Mixing problem

Orthogonal trajectories of curves, Existence and uniqueness of solutions, Second order differential equations: Homogenous linear equations of second order Partial differential equations: Basic Concepts and

definitions, Mathematical problems, First order equations: Classification, Construction, Geometrical interpretation, Method of characteristics and so on.

Sample of syllabus

The Derivative of a function of a real variable measures the sensitivity to change of a quantity (a function or dependent variable) which is determined by another quantity (the independent variable). It is a fundamental tool of calculus

Example:

Velocity is the rate of change of the position of an object, equivalent to a specification of its speed and direction of motion, e.g. 60 km/h to the north. Velocity is an important concept in kinematics, the branch of classical mechanics which describes the motion of bodies.

As a change of direction occurs while the cars turn on the curved track, their velocity is not constant.

Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus. Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral

is defined informally to be the signed area of the region in the xy-plane bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b, such that area above the x-axis adds to the total, and that below the x-axis subtracts from the total.

The term integral may also refer to the related notion of the antiderivative, a function F whose derivative is the given function f. In this case, it is called an indefinite integral and is written:

( )b

af x dx

( ) ( )F x f x dx

A definite integral of a function can be represented as the signed area of the region bounded by its graph.

Differential Equations

Describe the way quantities change with respect to other quantities (for instance, time)

The laws of science are easily expressed by DE (more difficult when depends on position, or

on time)Newton’s Law of CoolingPopulation Dynamics

Definition: A differential equation is an equation containing

an unknown function and its derivatives.

2

2

43

3

1. 2 3,

2. 3 0,

3. 6 3,

dyx

dx

d y dyay

dx dx

d y dyy

dx dx

Examples:

where y is dependent variable and x is independent variable.

Ordinary differential equations

Physical Origin

30

1. Newton’s Low of Cooling

sTTdt

dT

where dT/dt is rate of cooling of the liquid , And

T- Ts is temperature difference between the liquid

T its surrounding Ts.

2. Growth and Decay

dyy

dt

where y is the quantity present at any time

1 2 ,y c x c

2

20.

d y

dx

1. For the family of straight lines

the differential equation is

2. For the family of curves

xx ececy 32

21

2

26 0.

d y dyy

dx dx

32

3. Geometric Origin

The differential equation is

Introduction to mathematical

Modeling with ODEs

The Five Stages of Modeling

1. Ask the question.2. Select the modeling approach.3. Formulate the model.4. Solve the model. Validate if

possible.5. Answer the question.

If N (representing, eg, bacterial density, or number of tumor cells) is a continuous function of t (time), then the derivative of N with respect to t is another function, called dN/dt, whose value is defined by the limit process

it represents the change is N with respect to time.

0

( ) ( )lim ,t

dN N t t N t

dt t

Example:

Our Cell Division Model: Getting the ODE

Let N(t) = bacterial density over time Let K = the reproduction rate of the

bacteria per unit time (K > 0) Observe bacterial cell density at times t

and (t + Dt). Then N(t +Dt) ≈ N(t) + K N(t) Dt

Rewrite: [N(t+Dt) – N(t)]/Dt ≈ KN(t)

Total density at time t+Dt

Total density at time t + increase in density due to reproduction during time interval Dt

≈

Our Cell Division Model: Getting the ODE

• Take the limit as Dt → 0

“Exponential growth” (Malthus:1798)

• Analytic solution possible here.

)N(N

eNtN Kt

0

)(

0

0

KNdTdN

Exponential Growth: Realistic?

June 2005

Exponential growth models of physical

phenomena only apply within limited regions,

as unbounded growth is not physically realistic.

Although growth may initially be exponential,

the modelled phenomena will eventually enter

a region in which previously ignored negative

feedback factors become significant (leading to

a logistic growth model) or other underlying

assumptions of the exponential growth model,

such as continuity or instantaneous feedback,

break down.

What is Linear Algebra?

Linear algebra is the branch of

mathematics concerning vector

spaces and linear mappings between such

spaces. It is study of lines, planes, and

subspaces and their intersections using

algebra. Linear algebra assigns vectors as the

coordinates of points in a space, so that

operations on the vectors define operations

on the points in the space.

Fundamental operation with vectors in Euclidean space Rn, Linear combination of vectors, Dot product and their properties, Cauchy−Schwarz inequality, Triangle inequality, Projection vectors.

Linear combination of vectors, Row space, Eigenvalues, Eigenvectors, Eigenspace, Characteristic polynomials, Diagonalization of matrices.

Orthogonal and orthonormal vectors, Orthogonal and orthonormal bases, Orthogonal complement, Projection theorem (Statement only), Orthogonal projection onto a subspace, Application: Least square solutions for inconsistent systems and so on.

Sample of syllabus

USES OF LINEAR ALGEBRA

CRYPTOGRAPHY

SPACE EXPLORATION

GAME PROGRAMMING

ELECTRICAL NETWORKS

MATRICES IN

GAMES

LIGHTS OUT GAME LIGHTS OUT GAME

TURN

THE

LIGHTS

OFF

LIGHTS OUT GAME LIGHTS OUT GAME

TURN

THE

LIGHTS

OFF

LIGHTS OUT GAME LIGHTS OUT GAME

TURN

THE

LIGHTS

OFF

VECTORS IN

GAMES

LIGHTS OUT GAME STORE INFORMATION

LIGHTS OUT GAME CHANGING POSITION

ADDING

VECTORS

3D OBJECTS

VECTOR

CROSS

PRODUCT

Thanks