Transcript Document

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
Sample of syllabus
 Definition of limit of a function, One sided limit, Limits at
infinity, Curve sketching, Volumes of solids of revolution by
the washer method .
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
The symbolic statement is:
lim  n  gon   Circle
n 
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
an 
n
1, ½, 1/3, ¼, …..1/10000,…., 1/10000000000000..
As n is getting bigger, what are these terms
approaching?
1
lim  0
n  n
Graphical
Examples
1
f ( x) 
x
As x gets really, really big, what is happening to the
height, f(x)?
1
lim  0
x  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
1
lim sin
x0
x
X
2/π
Sin(1/x) 1
2/3π
2/5π
2/7π
2/9π
2/11π
X
0
-1
1
-1
1
-1
Limit does
not exist
Differential Equations and
Mathematical Modeling
Sample of syllabus
 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.
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
b
f ( x)dx

is defined informally to be the signed area of the region in the xya
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:
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
 𝐹 = 𝑀𝑎 (more difficult when 𝐹 depends on position,
or on time)
 Newton’s Law of Cooling
 Population Dynamics
Ordinary differential equations
Definition:
A differential equation is an equation containing an
unknown function and its derivatives.
Examples:
dy
1.
 2 x  3,
dx
d2y
dy
2.
3
 ay  0,
2
dx
dx
4
d y  dy 
3.

  6 y  3,
3
dx
 dx 
where y is dependent variable and x is independent
variable.
3
Physical Origin
1. Newton’s Low of Cooling
dT
  T  Ts 
dt
where dT/dt is rate of cooling of the liquid , And
T- Ts is temperature difference between the liquid
T its surrounding Ts.
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2. Growth and Decay
dy
y
dt
where y is the quantity present at any time
3. Geometric Origin
1. For the family of straight lines
y  c1 x  c2 ,
the differential equation is
d2y
 0.
2
dx
2. For the family of curves
y  c1e2 x  c2 e 3x
The differential equation is
d 2 y dy
  6 y  0.
2
dx
dx
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Introduction to
mathematical Modeling
with ODEs
The Five Stages of Modeling
1.
2.
3.
4.
5.
Ask the question.
Select the modeling approach.
Formulate the model.
Solve the model. Validate if possible.
Answer the question.
Example:
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
dN
N (t  t )  N (t )
 lim
,
dt t 0
t
it represents the change is N with respect to
time.
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
Total density
at time t+Dt
≈ Total density at time t + increase in density due to
reproduction during time interval Dt
 Rewrite: [N(t+Dt) – N(t)]/Dt ≈ KN(t)
Our Cell Division Model: Getting the ODE
• Take the limit as Dt → 0
dN dT  KN
“Exponential growth” (Malthus:1798)
• Analytic solution possible here.
N (t )  N 0 e
N 0  N( 0 )
Kt
Exponential Growth: Realistic?
June 2005
Lisette de Pillis HMC Mathematics
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.
Sample of syllabus
 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.
USES OF LINEAR
ALGEBRA
CRYPTOGRAPHY
SPACE
EXPLORATION
GAME
PROGRAMMING
ELECTRICAL NETWORKS
MATRICES
IN
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LIGHTS
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STORELIGHTS
INFORMATION
LIGHTS
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CHANGING
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