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 970 times last on 18 January 2025, 12:35 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 691 times last on 18 January 2025, 12:35 am

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 945 times last on 18 January 2025, 12:35 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 1072 times last on 18 January 2025, 09:25 pm

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 1654 times last on 22 January 2025, 11:47 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.
MTTS Level :

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(Downloaded 656 times last on 18 January 2025, 12:35 am

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 1158 times last on 18 January 2025, 12:35 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 656 times last on 18 January 2025, 07:52 pm

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 1412 times last on 18 January 2025, 12:35 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 831 times last on 22 January 2025, 10:57 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 3738 times last on 21 January 2025, 10:53 pm

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 500 times last on 18 January 2025, 12:35 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 552 times last on 20 January 2025, 05:38 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 1943 times last on 22 January 2025, 09:43 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 636 times last on 18 January 2025, 12:35 am

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 823 times last on 21 January 2025, 03:53 pm

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.

Keywords :

M.Sc./Ph.D
MTTS Level :

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(Downloaded 907 times last on 22 January 2025, 01:01 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 2650 times last on 22 January 2025, 09:19 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 1861 times last on 22 January 2025, 09:20 am

Principle of Induction

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

Keywords :

B.Sc
MTTS Level :

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(Downloaded 884 times last on 18 January 2025, 12:36 am