# Department of Mathematics

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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? Calculus comes from the Latin word calculus which were stones used for counting. Includes limits, derivatives, integrals and infinite sums. You will study the first three topics in detail next year. We will study limits in detail and derivatives a bit. Used in many science fields and economics for optimization and cost analysis. Builds on algebra, geometry and trig skills. That is why pre-calc focused so much on improving those skills. All of your previous math will come in use as you study 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. Let’s talk about the goals for the course. Calculus is the study of functions. Have you ever met a function outside a math class? Everything is a function! The temperature in this room is a function of time. Profit is a function of quantity sold, which is a function of price, which is a function of cost (among other things). Modeling is the key idea that allows us to do math and make money! This is not a proof class. But we will focus more on concepts than in your last math class. The intellectual pursuit of mathematics involves an understanding of the principles behind the machinery.

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:
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

As n is getting bigger, what are these terms approaching?
Let’s look at the sequence whose nth term is given by 1, ½, 1/3, ¼, …..1/10000,…., 1/ As n is getting bigger, what are these terms approaching?

Graphical Examples

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

Does the height, or f(x) ever get to 0?
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 X 2/π 2/3π
2/5π 2/7π 2/9π 2/11π X Sin(1/x) 1 -1 Limit does not exist

Differential Equations and Mathematical Modeling
Calculus comes from the Latin word calculus which were stones used for counting. Includes limits, derivatives, integrals and infinite sums. You will study the first three topics in detail next year. We will study limits in detail and derivatives a bit. Used in many science fields and economics for optimization and cost analysis. Builds on algebra, geometry and trig skills. That is why pre-calc focused so much on improving those skills. All of your previous math will come in use as you study calculus.

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 Let’s talk about the goals for the course. Calculus is the study of functions. Have you ever met a function outside a math class? Everything is a function! The temperature in this room is a function of time. Profit is a function of quantity sold, which is a function of price, which is a function of cost (among other things). Modeling is the key idea that allows us to do math and make money! This is not a proof class. But we will focus more on concepts than in your last math class. The intellectual pursuit of mathematics involves an understanding of the principles behind the machinery. 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 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:

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 Since functions are everywhere, so are differential equations. F=ma: if you remember the calculus you got before you got here, you know that a is the second derivative of position. You might have F a function of position, or of time, or both. This gives you a differential equation for the position function of time. NLC: the rate of change of temperature is proportional to the difference between the current temp and ambient temp Populations: Owls eat mice. You have mice which grow kind of like bacteria, and owls which die if they have no mice to eat. Interactions between owls and mice result in more owls, fewer mice. This becomes two DE; one for each species

Ordinary differential equations
Definition: A differential equation is an equation containing an unknown function and its derivatives. Examples: where y is dependent variable and x is independent variable.

Physical Origin Newton’s Low of Cooling
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 where y is the quantity present at any time

3. Geometric Origin the differential equation is
1. For the family of straight lines the differential equation is 2. For the family of curves The differential equation is

Introduction to mathematical Modeling with ODEs

The Five Stages of Modeling
Ask the question. Select the modeling approach. Formulate the model. Solve the model. Validate if possible. Answer the question. Questions: What does it mean to “solve” and to “validate”?

it represents the change is N with respect to time.
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 it represents the change is N with respect to time. From Taubes p.3: In biology, it is typical to study time-dependent phenomena. A differential equation for a function or a variable is simply an algebraic equation that involves both the function and its derivatives.

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) From Keshet: Suppose we are abler to observe that over a period of one time unit, a single bacterial cell divides, its daughters divide, and so forth, leading to a totla of K new bacterial cells. We then define the reproductive rate of the bacteria by the constant K > 0. Suppose densities are observed at two closely spaced times t and del t. Neglecting death, we then expect to find the exponential relationship. Note: in general, these models look like (stuff changing in time) = (stuff going in) + (stuff going out) 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. Exponential growth is solved by separation of variables: dN/N = Kdt. Integrate both sides: int(from 0 to t ) of dN/N = int(from 0 to t) of K ds Gives: ln(N) (from 0 to t) = Kt So: ln(N(t) – ln(N(0)) = Kt So ln(N(t)) = Kt + c (c is a constant, c = ln(N(0))) Exponentiate both sides, and we get: N(t) = N_0 e^{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? Calculus comes from the Latin word calculus which were stones used for counting. Includes limits, derivatives, integrals and infinite sums. You will study the first three topics in detail next year. We will study limits in detail and derivatives a bit. Used in many science fields and economics for optimization and cost analysis. Builds on algebra, geometry and trig skills. That is why pre-calc focused so much on improving those skills. All of your previous math will come in use as you study calculus.

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. Hhhh

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. Have you ever met a function outside a math class? Everything is a function! The temperature in this room is a function of time. Profit is a function of quantity sold, which is a function of price, which is a function of cost (among other things). Modeling is the key idea that allows us to do math and make money! This is not a proof class. But we will focus more on concepts than in your last math class. The intellectual pursuit of mathematics involves an understanding of the principles behind the machinery. 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.

CRYPTOGRAPHY SPACE EXPLORATION GAME PROGRAMMING ELECTRICAL NETWORKS
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 VECTORS ADDING

3D OBJECTS VECTOR CROSS PRODUCT

Thanks