Expository articles
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Found total 224 results. Displaying results from 61 to 80.
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61
Outline of a Course in Field Theory (Expanded Version)
This is an expanded version of filed_theory_04082005.pdf
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M.Sc.
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(Downloaded 1017 times last on 31 March 2025, 01:43 pm
(Downloaded 1017 times last on 31 March 2025, 01:43 pm
62
Finite Subgroups of F ? are Cyclic
Gives two proofs of the fact that any finite subgroup of a multiplicative group of afield is cyclic. Avoids structure theorem for finite abelian groups.
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M.Sc.
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(Downloaded 729 times last on 31 March 2025, 05:20 am
(Downloaded 729 times last on 31 March 2025, 05:20 am
63
Finite Sets
A rigorous definition of finite sets and the number of elements in a finite set. A good training ground for initiating students to rigour.
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O
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(Downloaded 953 times last on 26 March 2025, 07:05 pm
(Downloaded 953 times last on 26 March 2025, 07:05 pm
64
Plancherel Theorem and Fourier Inversion Theorem
An experiment which succeeded. It is traditional to prove the Fourier inversion first and then derive Planceral theorem. Here we reverse the order and thereby learn a few tricks of analysis.
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M.Sc.
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(Downloaded 1123 times last on 01 April 2025, 04:06 pm
(Downloaded 1123 times last on 01 April 2025, 04:06 pm
65
Frenet-Serre Formulas for Space Curves
This article to introduce the readers to a modern outlook of Serre-Frenet Formulas via the Maurer-Cartan forms. While the treatment is hetrogeneous, a diligent reader can cull an elementary introduction as well as a more advanced perspective of these formulas.
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M.Sc.
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(Downloaded 1895 times last on 01 April 2025, 08:26 pm
(Downloaded 1895 times last on 01 April 2025, 08:26 pm
66
Cauchy’s Theorem implies Fundamental Theorem of Algebra
Incomplete
Keywords :
claims to have 3 proofs which use only Cauchy's theorem. I do not even remember what the third is! This serves as an antidote to the commonly held belief that you need Liouville's theorem to prove FTA. This may also be a part of FTA.pdf.
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(Downloaded 682 times last on 27 March 2025, 03:23 pm
(Downloaded 682 times last on 27 March 2025, 03:23 pm
67
Fundamental Theorem of Algebra via Linear Algebra
Modified version of Derksen's proof of FTA via linear algebra. A must read for budding algebraists!
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B.Sc./M.Sc
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(Downloaded 1180 times last on 26 March 2025, 07:08 pm
(Downloaded 1180 times last on 26 March 2025, 07:08 pm
68
The Fundamental Theorem of Algebra: Proofs via Differential Topology
Two proofs of FTA using concepts from differential topology. These will bring out the decisive role of polynomial mapping as proper maps.
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M.Sc./Ph.D
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(Downloaded 671 times last on 26 March 2025, 07:09 pm
(Downloaded 671 times last on 26 March 2025, 07:09 pm
69
The Fundamental Theorem of Algebra
Offers many simple proofs of the fundamental theorem of algebra
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B.Sc./M.Sc
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(Downloaded 1461 times last on 30 March 2025, 09:57 am
(Downloaded 1461 times last on 30 March 2025, 09:57 am
70
Fubini Theorem on R n
Fubini theorem in $R^n$. (The source file contains some more material but it is commented out.)
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M.Sc.
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(Downloaded 885 times last on 02 April 2025, 09:35 am
(Downloaded 885 times last on 02 April 2025, 09:35 am
71
Functional Analysis (2011) – Summary of Lectures
An outline of a course in functional analysis given at University of Hyderabad. Contains a lot of concrete examples, spaces of functions and good exercises. Prepares the students for future use of functional analysis in other parts of mathematics. Goes up to the spectral theom for compact self-adjoint operators, Riesz-Schauder theory of compact operators, their applications to integral equations.
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M.Sc.
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(Downloaded 3962 times last on 02 April 2025, 02:49 am
(Downloaded 3962 times last on 02 April 2025, 02:49 am
72
Illustration of Gauss Theorem in Theory of Surfaces
Looks at two examples to bring out the meaning of the theorem.
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M.Sc.
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(Downloaded 508 times last on 26 March 2025, 11:18 pm
(Downloaded 508 times last on 26 March 2025, 11:18 pm
73
Geodesics on a Sphere
Gives two proofs to show that the great circles with unit speed are the geodesics on a sphere. You may also see how the known facts are employed to prove them.
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M.Sc.
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(Downloaded 583 times last on 29 March 2025, 06:17 am
(Downloaded 583 times last on 29 March 2025, 06:17 am
74
Group Actions
Gives the definition, some interesting examples and exercises. Good way to master group actions, as students learn only the left/right and conjugacy actions in a typical algebra course.
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B.Sc./M.Sc
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(Downloaded 1977 times last on 29 March 2025, 06:36 am
(Downloaded 1977 times last on 29 March 2025, 06:36 am
75
Fundamental Theorem of Calculus in HK-Integral
The aim of this article is to bring to the attention of
Undergraduate students and college teachers a generalized Riemann integral called
Henstock-Kurzweil integral. This has attracted a growing interest
in recent years. New introductory textbooks on real analysis have
a chapter on this topic.
Keywords :
Final year B.Sc, M.Sc
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(Downloaded 653 times last on 31 March 2025, 01:12 pm
(Downloaded 653 times last on 31 March 2025, 01:12 pm
76
Any compact hypersurface in R n is a level surface (1) Orientability of Hypersurfaces in R n (2) Jordan-Brouwer Separation Theorem
In this note, we prove Jordan-Brouwer Separation theorem and the orientability of
hypersurface in $R^n$. In fact, both are easy consequences of
the main result which says that any compact hypersurface is a
level set of a smooth function defined on all of $R^n$.
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M.Sc./Ph.D
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(Downloaded 869 times last on 01 April 2025, 11:21 am
(Downloaded 869 times last on 01 April 2025, 11:21 am
77
Implicit Function Theorem for Banach Spaces
Direct proof of Implicit function theorem in the context of Banach spaces. This was in answer to questions from students as well as teachers of analysis. The proof of IMT is modified to prove this directly. Both use modified Newton's algorithm.
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M.Sc./Ph.D
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(Downloaded 983 times last on 02 April 2025, 02:04 am
(Downloaded 983 times last on 02 April 2025, 02:04 am
78
Implicit Function Theorem
The aim of this article is bring out the geometric content of the implicit function theorem of two variables. To make it self-contained, I have included a proof of the theorem. The exposition is aimed at students of mathematics from final year B.Sc. or first year M.Sc. It also brings out the geometric meaning of the Lagrange multiplier method.
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B.Sc./M.Sc
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(Downloaded 2725 times last on 02 April 2025, 09:25 am
(Downloaded 2725 times last on 02 April 2025, 09:25 am
79
Inverse Mapping Theorem
Gives a proof of Inverse Function theorem (in the context of any complete normed linear space, in particular $R^n) and its geometric meaning. It avoids contraction principle and gives a direct proof.
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M.Sc.
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(Downloaded 1975 times last on 31 March 2025, 08:24 pm
(Downloaded 1975 times last on 31 March 2025, 08:24 pm
80
Principle of Induction
Statement of principles of induction and well-ordering and exercises.
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B.Sc
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(Downloaded 899 times last on 28 March 2025, 11:25 am
(Downloaded 899 times last on 28 March 2025, 11:25 am