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AU Mathematics III Sem III
Unit 1 : Vector Calculus(D i f f e r e n t i a t i o n o f v e c t o r s , c u r v e s i n s p a c e , V e l o c i t y a n d a c c e l e r a t i o n , R e l a t i v e v e l o c i t y a n d
a c c e l e r a t i o n , S c a l a r a n d V e c t o r p o i n t f u n c t i o n s , V e c t o r o p e r a t o r ", " a p p l i e d t o s c a l a r p o i n t
f u n c t i o n s , G r a d i e n t , " a p p l i e d t o v e c t o r p o i n t f u n c t i o n s , D i v e r g e n c e a n d c u r l , P h y s i c a l
i n t e r p r e t a t i o n s o f ", F a n d " × F , " a p p l i e d t w i c e t o p o i n t f u n c t i o n s , " a p p l i e d t o p r o d u c t s o f
p o i n t f u n c t i o n s , i n t e g r a t i o n o f v e c t o r s , L i n e i n t e g r a l , C i r c u l a t i o n , W o r k , S u r f a c e i n t e g r a l - f l u x ,
G r e e n s t h e o r e m i n t h e p l a n e , S t o k e s t h e o r e m , V o l u m e i n t e g r a l , D i v e r g e n c e t h e o r e m , I r r o t a t i o n a l a n d s o l e n o i d a l f i e l d s , G r e e n s t h e o r e m , I n t r o d u c t i o n o f o r t h o g o n a l c u r v i l i n e a r c o o r d i n a t e s : C y l i n d r i c a l , S p h e r i c a l a n d p o l a r c o o r d i n a t e s ) |
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| 1. Poisson's Formula, Schwarz's Theorem, the reflection principle(Mathematics) |
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This unit describes theorems on harmonic functions such as Poisson's integral formula, Schwarz's theorem, Hermite's inequality, Reflection principle.
Title: Math 2.3 Complex Analysis - II
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| 2. Vector Spaces(Mathematics) |
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A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied ("scaled") by numbers, called scalars in this unit. The operations of vector addition and scalar multiplication have to satisfy certain requirements, called axioms.
This unit mainly deals with standard properties of Vectors, Subspaces, Linear combinations and systems of
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Title: Math 2.1 Linear Algebra
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| 3. The Residue Theorem, the Argument Principle(Mathematics) |
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This unit deals with Residue Theorem, Argument Theorem and Rouche's Theorem and related problems.
Title: Math 2.3 Complex Analysis - II
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| 4. Definition and Basic Properties of Harmonic Functions(Mathematics) |
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Harmonic functions are tremendously important for the role they play in physics and engineering where they turn out naturally in a variety of context including electronics fluid dynamics and heat transfer. This unit deals with harmonic functions and mean value property for harmonic functions.
Title: Math 2.3 Complex Analysis - II
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| 5. Elementary Matrix Operation, Rank of a Matrix, Matrix Inverse and System of Linear Equation(Mathematics) |
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In the previous two units, we have learnt about vector space and linear
transformations between two vector spaces U(F) and V(F) defined over the same field F In this unit, we shall see that each linear transformation from n-dimensional vector space U(F) to an m-dimensional vector space V(F) corresponds to an m *n matrix over a field F. Here, we see how the matrix representation of a linear transf
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Title: Math 2.1 Linear Algebra
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| 6. Linear Transformation and Matrix(Mathematics) |
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A linear map, linear mapping, linear transformation, or linear operator (in some contexts also called linear function is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or O. A. Cayley formally introduced m *n matrices in two papers in 1850 and 1858 (the term "matrix" was c
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Title: Math 2.1 Linear Algebra
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| 7. Properties of Determinant, Co-factor Expansions and Cramer's Rule(Mathematics) |
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An exposition of the theory of determinants independent of their relation to the solvability of linear equations was first given by Vandermonde in his "Memoir on elimination theory" of 1772. Laplace extended some of Vandermonde's work in his Researches on the Integral Calculus and the System of the World (1772), showing how to expand II x II determinants by co-factors. The determinant of a square
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Title: Math 2.1 Linear Algebra
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| 8. Algebra of sets - Sigma Algebra and Borel Sets(Mathematics) |
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In this unit, we introduce the concept of algebra of sets. this unit also explains the concept of sigma algebra along with some examples. This unit provides an insight into Borel sets along with some examples.
Title: Math 3.2 Measure Theory
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| 9. Metric Spaces, Completeness(Mathematics) |
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In this unit, we introduce the concept of metric spaces. We give the
properties of metric spaces and some theorems related to metric space.
Finally, we explain the Metric Completion Theorem.
Title: Math 3.3 Functional Analysis
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| 10. Vector Calculus(Mathematics) |
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We come across two types of quantities in physics, mechanics and other branches of applied mathematics,
one characterized by both magnitude and direction such as displacement, velocity, force, acceleration etc
and the other characterized by only magnitude and no direction such as mass, speed and density etc. The
former is known as vector while the later is known as scalar. In this chapter we di
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Title: Bridge Course II
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| 11. Outer measure and measurable sets(Mathematics) |
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This unit explains Lebesgue outer measure and its properties. It also explains Lebesgue measurable sets and its properties (Theorems).
Title: Math 3.2 Measure Theory
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| 12. Setting up of First Order Differential Equations and their Solutions(Mathematics) |
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Differential equations occur quite frequently in our daily life. The motion of an object can always be associated with a differential equation. The change in prices of commodities, the flow of fluids, the concentration of chemicals etc., often lead to differential equations. Such equations may depend on one or more independent variables. Further, it may include the derivatives of the first or high
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Title: Math 3.4 Mathematical Modeling
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| 13. The Metric Topology(Mathematics) |
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Our main objective is to study the definition and some properties of metric topology in this unit.
Title: Math 3.1 Topology
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| 14. Mathematical Modeling of Some Fundamental Problems(Mathematics) |
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Mathematical Modeling in terms of differential equations arises when the situation modeled involves some continuous variables varying with respect to other continuous variables and we have reasonable hypothesis about the rate of change of dependent variables with respect to independent variables. Mathematical models in terms of ordinary differential equations will be studied in this unit and the u
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Title: Math 3.4 Mathematical Modeling
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| 15. Continuous Functions(Mathematics) |
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We define continuous function and homeomorphism in this unit.
Title: Math 3.1 Topology
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| 16. Measurable functions, Littlewood's three principles(Mathematics) |
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This unit defines measurable functions along with examples. It also explains the Littlewood's three principles and states some theorems on measurable functions.
Title: Math 3.2 Measure Theory
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| 17. Banach Contraction Mapping Theorem and Applications(Mathematics) |
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In this unit, we introduce the concept of Banach Contraction Mapping
Theorem and its Applications.
Title: Math 3.3 Functional Analysis
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| 18. Closed Sets, limit points, Hausdorff spaces(Mathematics) |
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This unit deals with the definition of closed sets, limit points and neighborhood in a topological space. The Hausdorff spaces will we defined in the unit which deals with separation axioms.
Title: Math 3.1 Topology
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| 19. Programming Languages and Operating Systems(Mathematics) |
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System software is a program or a group of programs written for computer system management. The system software's are developed by the manufactures. They are supervisory programs that help executing the user's program effectively. The operating systems, language processors (such as interpreters and compliers), and utility programs (such as loaders and linkers) are examples of system software are s
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Title: Math 3.5 Computer Programming
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| 20. Building Blocks of Computer Programs(Mathematics) |
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The specification of the sequence of computational steps in a particular programming language is termed as a program. The task of developing programs is called programming. The person engaged in programming activity is called programmer. In this unit, we study on program analysis, Algorithm development, Flow chart, Decision tables and pseudo code.
Title: Math 3.5 Computer Programming
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| 21. Parallel Computers and Network(Mathematics) |
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Parallel computing is a form of computation in which many instructions are carried out simultaneously operating on the principle that large problems can often be divided into smaller ones, which are then solved concurrently (in parallel). A network computing environment is one in which an organization has linked together personal computers that have been cOlU1ected into a network.
Title: Math 3.5 Computer Programming
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| 22. Predicate Calculus(Computer Science) |
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In this unit we describe the process of derivation by which one demonstrates that a particular formula in a valid consequence of given set of premises. The method of derivation involving predicated statement calculus and also certain additional rules which are required to deal with the formulae involving quantifiers. The rules P and T, regarding the introduction of a premise at any stage of deriva
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Title: MSCS-501 Discrete Mathematics
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| 23. Lebesgue measure, nonmeasurable sets(Mathematics) |
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This unit defines Lebesgue measure and proves certain theorems given in this unit. It also helps in distinguishing measurable and non measurable sets.
Title: Math 3.2 Measure Theory
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| 24. Fundamental Aspect of Computers(Mathematics) |
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This help file presumes that you have little or no experience with the device commonly known as the PC (personal computer). Computers have become the basic necessity of any organization with serious objectives, They have made great inroads into everyone's day to day life and thinking. They are used for all sorts of problem solving ranging from simple addition to highly complex calculations in the
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Title: Math 3.5 Computer Programming
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| 25. Vector Spaces(Mathematics) |
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Abstract algebra has three basic components: groups, rings and fields. Fields play a central role in algebra. Results in field theory find important applications in algebraic number theory. Field theory happens to be the language in which a number of classical problems such as the Greek's problem of ruler and compass constructions were rephrased and solved.
Title: Math 1.1 Algebra
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