4dspace : Expository articles

Expository articles

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Found total 224 results. Displaying results from 61 to 80.

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Outline of a Course in Field Theory (Expanded Version)

This is an expanded version of filed_theory_04082005.pdf

Keywords :

M.Sc.
MTTS Level :

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(Downloaded 1018 times last on 03 April 2025, 09:54 am

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.
MTTS Level :

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(Downloaded 731 times last on 03 April 2025, 06:41 pm

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 954 times last on 03 April 2025, 09:54 am

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.
MTTS Level :

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(Downloaded 1133 times last on 04 April 2025, 02:38 am

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.
MTTS Level :

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(Downloaded 1905 times last on 04 April 2025, 04:30 am

Cauchy’s Theorem implies Fundamental Theorem of Algebra

Incomplete

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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 684 times last on 03 April 2025, 09:40 pm

Fundamental Theorem of Algebra via Linear Algebra

Modified version of Derksen's proof of FTA via linear algebra. A must read for budding algebraists!

Keywords :

B.Sc./M.Sc
MTTS Level :

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(Downloaded 1181 times last on 03 April 2025, 09:54 am

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.

Keywords :

M.Sc./Ph.D
MTTS Level :

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(Downloaded 672 times last on 03 April 2025, 09:55 am

The Fundamental Theorem of Algebra

Offers many simple proofs of the fundamental theorem of algebra

Keywords :

B.Sc./M.Sc
MTTS Level :

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(Downloaded 1465 times last on 03 April 2025, 10:49 am

Fubini Theorem on R n

Fubini theorem in $R^n$. (The source file contains some more material but it is commented out.)

Keywords :

M.Sc.
MTTS Level :

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(Downloaded 886 times last on 03 April 2025, 09:54 am

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.

Keywords :

M.Sc.
MTTS Level :

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(Downloaded 3968 times last on 04 April 2025, 04:52 am

Illustration of Gauss Theorem in Theory of Surfaces

Looks at two examples to bring out the meaning of the theorem.

Keywords :

M.Sc.
MTTS Level :

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(Downloaded 510 times last on 03 April 2025, 09:55 am

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.
MTTS Level :

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(Downloaded 588 times last on 03 April 2025, 09:54 pm

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
MTTS Level :

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(Downloaded 1978 times last on 03 April 2025, 09:55 am

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 656 times last on 03 April 2025, 09:40 pm

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
MTTS Level :

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(Downloaded 877 times last on 04 April 2025, 02:02 am

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
MTTS Level :

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(Downloaded 985 times last on 03 April 2025, 09:54 am

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
MTTS Level :

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(Downloaded 2731 times last on 04 April 2025, 08:07 am

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.
MTTS Level :

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(Downloaded 1977 times last on 04 April 2025, 04:33 am

Principle of Induction

Statement of principles of induction and well-ordering and exercises.

Keywords :

B.Sc
MTTS Level :

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(Downloaded 900 times last on 03 April 2025, 09:54 am