Expository articles


 

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Found total 224 results. Displaying results from 181 to 200.

 
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181
Tangents

This article gives you a gentle introduction to the notion of tangents so as to make your journey through Differential Geometry enjoyable.

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M.Sc.
MTTS Level :
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(Downloaded 1112 times last on 20 December 2024, 07:47 pm
182
Taylor's Theorem

Gives proofs of Taylor's theorem of single variable calculus with all important forms of the remainder. As an application, we establish the convergence of the binomial series of any index. A good place to learn some hard analysis.

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Final year B.Sc, M.Sc
MTTS Level :
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(Downloaded 738 times last on 19 December 2024, 09:35 pm
183
Existence of Continuous Functions

Starts with the existence of real valued continuous functions on a metric space, Urysohn's lemma and Tietze extension theorem in the context of metric spaces and then proceeds to prove them in the case of normal spaces.

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M.Sc.
MTTS Level :
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(Downloaded 726 times last on 22 December 2024, 08:39 pm
184
Generating Topologies— A Unified View ofSubspace, Product and Quotient Topologies

Unifies the principles underlying the construction of subspace, product and quotient topologies; also brings out the importance of universal mapping properties of these constructions.

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M.Sc.
MTTS Level :
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(Downloaded 842 times last on 19 December 2024, 09:35 pm
185
R m is homeomorphic to R n iff m = n

Starting with a definition of topological dimension, the article gives a self-contained elementary proof of the fact that $R^m$ is homeomorphic to R^n$ iff $m=n$.

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M.Sc./Ph.D
MTTS Level :
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(Downloaded 716 times last on 19 December 2024, 09:35 pm
186
Topological Groups — via Problems

Basic notions, examples and results about topological groups via graded exercises.

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M.Sc.
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(Downloaded 1105 times last on 22 December 2024, 04:08 pm
187
Outline of a Topology Course

Outline of a topology course given at University of Mumbai (very brief outline)

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M.Sc.
MTTS Level :
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(Downloaded 2985 times last on 23 December 2024, 06:47 pm
188

Outline of a topology course

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a shorter version of the next.
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(Downloaded 1582 times last on 23 December 2024, 07:16 am
189

Outline of a topology course

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contains many of my articles on topology as appendices. Only one of the three may be released in the Expository section.
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(Downloaded 2818 times last on 22 December 2024, 08:54 pm
190
Structure Theorems for Linear Maps

This is graded exercise version of the article ?Structure_linear_maps.pdf?; If you wish to have a quick idea and ready to work out the details, you may read this; if you want details to be spelt out, go to the other article.

Keywords :

M.Sc.
MTTS Level :
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(Downloaded 936 times last on 22 December 2024, 10:45 pm
191

Gives a complete straightforward proof of Tyknoff's theorem about the compactness of arbitrary products of compact spaces.

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M.Sc.
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(Downloaded 552 times last on 23 December 2024, 07:19 am
192
Uncountability of R

Gives 3 proofs, one uses LUB, another nested interval and the third the decimal expansion and diagonal trick. The 1st proof needs more details.

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B.Sc
MTTS Level :
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(Downloaded 998 times last on 19 December 2024, 09:35 pm
193
Extension of Uniformy Continuus Functions

Gives a excruciatingly detailed proof of the result on extensions of uniformly continuous functions to a complete metric space. This may also be found in my book on metric spaces.

Keywords :

Final year B.Sc, M.Sc
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(Downloaded 608 times last on 19 December 2024, 09:35 pm
194
Vector Product on R 3

This is part of Frenet.pdf, but is of independent interest. Gives the geometric meaning of the vector product and the identities involved with this concept.

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B.Sc
MTTS Level :
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(Downloaded 491 times last on 19 December 2024, 09:35 pm
195
Vector Analysis

A very brief summary with sketches of ideas underlying the classical theorems of vector analysis. Can at best be a pointer to the theme.

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Final year B.Sc, M.Sc
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(Downloaded 653 times last on 20 December 2024, 09:29 pm
196
Vector Fields on Spheres

Vector fields on $S^2$ and ends up with Milnor's proof of hairy ball theorem

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M.Sc./Ph.D
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(Downloaded 1046 times last on 23 December 2024, 06:15 am
197
Volterra’s Proof of Nonexistence of a Function

Volterra's elementary proof of the non-existence of a function which is continuous at all rationals and discontinuous at irrationals. This is usually proved using the topological nature of the set of discontinuities of a function and baire category theorem.

Keywords :

Final year B.Sc, M.Sc
MTTS Level :
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(Downloaded 637 times last on 21 December 2024, 12:27 pm
198
Every Real Sequence has a Monotone Subsequence

Gives a (nemonic/geometric) proof of the fact that any real sequence has a monotone subsequence.

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B.Sc
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(Downloaded 924 times last on 19 December 2024, 09:35 pm
199
An Elementary Proof of Weierstrass Approximation Theorem

Elementary proof of Weierstrass approximation theorem. May be compared with elem_wat.pdf

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Final year B.Sc, M.Sc
MTTS Level :
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(Downloaded 713 times last on 19 December 2024, 09:35 pm
200
Weyl’s Integral and Character Formulas for U (n)

Weyl integral and character formula for $U(n)$. Includes Braching law for $U(n)$. This was a handout to a 700 level course at Indiana University, Bloomington during the year 1983-84. There is complete set of handouts for the 2 semester course. I plan to type all of them as and when I find time.

Keywords :

Ph.D.
MTTS Level :
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(Downloaded 567 times last on 23 December 2024, 06:06 pm