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.
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(Downloaded 1461 times last on 16 March 2026, 06:56 am
(Downloaded 1461 times last on 16 March 2026, 06:56 am
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
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MTTS Level :
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(Downloaded 1224 times last on 15 March 2026, 09:04 am
(Downloaded 1224 times last on 15 March 2026, 09:04 am
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.
Keywords :
M.Sc.
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MTTS Level :
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(Downloaded 1545 times last on 14 March 2026, 12:32 pm
(Downloaded 1545 times last on 14 March 2026, 12:32 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.
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MTTS Level :
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(Downloaded 1429 times last on 15 March 2026, 08:48 pm
(Downloaded 1429 times last on 15 March 2026, 08:48 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
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MTTS Level :
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(Downloaded 1292 times last on 14 March 2026, 11:40 am
(Downloaded 1292 times last on 14 March 2026, 11:40 am
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 2153 times last on 13 March 2026, 06:55 pm
(Downloaded 2153 times last on 13 March 2026, 06:55 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.
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(Downloaded 5792 times last on 16 March 2026, 12:35 am
(Downloaded 5792 times last on 16 March 2026, 12:35 am
188
Outline of a topology course
Keywords :
a shorter version of the next.
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(Downloaded 3436 times last on 15 March 2026, 05:23 am
(Downloaded 3436 times last on 15 March 2026, 05:23 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 6207 times last on 16 March 2026, 12:15 am
(Downloaded 6207 times last on 16 March 2026, 12:15 am
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.
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MTTS Level :
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(Downloaded 1530 times last on 14 March 2026, 03:01 pm
(Downloaded 1530 times last on 14 March 2026, 03:01 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 1127 times last on 15 March 2026, 10:47 am
(Downloaded 1127 times last on 15 March 2026, 10:47 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.
Keywords :
B.Sc
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MTTS Level :
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(Downloaded 1305 times last on 15 March 2026, 11:02 am
(Downloaded 1305 times last on 15 March 2026, 11:02 am
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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MTTS Level :
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(Downloaded 1366 times last on 14 March 2026, 10:48 pm
(Downloaded 1366 times last on 14 March 2026, 10:48 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.
Keywords :
B.Sc
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MTTS Level :
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(Downloaded 649 times last on 15 March 2026, 09:58 pm
(Downloaded 649 times last on 15 March 2026, 09:58 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.
Keywords :
Final year B.Sc, M.Sc
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(Downloaded 855 times last on 14 March 2026, 04:21 pm
(Downloaded 855 times last on 14 March 2026, 04:21 pm
196
Vector Fields on Spheres
Vector fields on $S^2$ and ends up with Milnor's proof of hairy ball theorem
Keywords :
M.Sc./Ph.D
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MTTS Level :
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(Downloaded 2583 times last on 16 March 2026, 06:34 am
(Downloaded 2583 times last on 16 March 2026, 06:34 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
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MTTS Level :
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(Downloaded 887 times last on 14 March 2026, 07:11 pm
(Downloaded 887 times last on 14 March 2026, 07:11 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.
Keywords :
B.Sc
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MTTS Level :
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(Downloaded 1302 times last on 15 March 2026, 09:31 pm
(Downloaded 1302 times last on 15 March 2026, 09:31 pm
199
An Elementary Proof of Weierstrass Approximation Theorem
Elementary proof of Weierstrass approximation theorem. May be compared with elem_wat.pdf
Keywords :
Final year B.Sc, M.Sc
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(Downloaded 1052 times last on 15 March 2026, 07:41 pm
(Downloaded 1052 times last on 15 March 2026, 07:41 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.
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MTTS Level :
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(Downloaded 905 times last on 15 March 2026, 09:41 pm
(Downloaded 905 times last on 15 March 2026, 09:41 pm