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Department of Mathematics Mata Sundri College (University of Delhi)

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Introduction of Discipline Course II

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

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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.

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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.

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What is Calculus?

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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

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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.

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Introduction to Limits What is a limit?

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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.

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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

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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!

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Numerical Examples

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Let’s look at the sequence whose n th term is given by 1, ½, 1/3, ¼, …..1/10000,…., 1/10000000000000.. As n is getting bigger, what are these terms approaching ?

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Graphical Examples

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

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As x gets really, really small, what is happening to the height, f(x)? Does the height, or f(x) ever get to 0?

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Nonexistence Examples

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Oscillating Behavior Discuss the existence of the limit X2/π2/3π2/5π2/7π2/9π2/11πX 0 Sin(1/x)11 1 Limit does not exist

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Differential Equations and Mathematical Modeling

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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

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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.

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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:

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

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Differential Equations

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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. Ordinary differential equations

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Physical Origin 30 1. Newton’s Low of Cooling where dT/dt is rate of cooling of the liquid, And T- T s is temperature difference between the liquid T its surrounding T s.

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2. Growth and Decay where y is the quantity present at any time

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1. For the family of straight lines the differential equation is 2. For the family of curves 32 3. Geometric Origin The differential equation is

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Introduction to mathematical Modeling with ODEs

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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.

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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. Example:

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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 )]/ D t ≈ KN ( t ) Total density at time t+Dt Total density at time t + increase in density due to reproduction during time interval Dt ≈

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Our Cell Division Model: Getting the ODE Take the limit as Dt → 0 “ Exponential growth ” (Malthus:1798) Analytic solution possible here.

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Exponential Growth: Realistic? June 2005Lisette de Pillis HMC Mathematics

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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.

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What is Linear Algebra?

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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.

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Fundamental operation with vectors in Euclidean space R n, 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

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USES OF LINEAR ALGEBRA CRYPTOGRAPHY SPACE EXPLORATION GAME PROGRAMMING ELECTRICAL NETWORKS

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MATRICES IN GAMES

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LIGHTS OUT GAME TURNTHETURNTHE LIGHTSLIGHTS OFFOFF

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TURNTHETURNTHE LIGHTSLIGHTS OFFOFF

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TURNTHETURNTHE LIGHTSLIGHTS OFFOFF

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VECTORS IN GAMES

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LIGHTS OUT GAME STORE INFORMATION

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LIGHTS OUT GAME CHANGING POSITION ADDINGADDING VECTORSVECTORS

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3D OBJECTS VECTORVECTOR CROSSCROSS PRODUCTPRODUCT

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