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 1131 times last on 18 January 2025, 12:36 am
(Downloaded 1131 times last on 18 January 2025, 12:36 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 756 times last on 20 January 2025, 01:32 pm
(Downloaded 756 times last on 20 January 2025, 01:32 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.
Keywords :
M.Sc.
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MTTS Level :
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(Downloaded 744 times last on 18 January 2025, 12:36 am
(Downloaded 744 times last on 18 January 2025, 12:36 am
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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(Downloaded 852 times last on 18 January 2025, 12:36 am
(Downloaded 852 times last on 18 January 2025, 12:36 am
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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(Downloaded 741 times last on 22 January 2025, 02:52 pm
(Downloaded 741 times last on 22 January 2025, 02:52 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 1123 times last on 21 January 2025, 12:20 am
(Downloaded 1123 times last on 21 January 2025, 12:20 am
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 3036 times last on 21 January 2025, 08:27 pm
(Downloaded 3036 times last on 21 January 2025, 08:27 pm
188
Outline of a topology course
Keywords :
a shorter version of the next.
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(Downloaded 1614 times last on 22 January 2025, 07:52 am
(Downloaded 1614 times last on 22 January 2025, 07:52 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 2979 times last on 22 January 2025, 06:31 pm
(Downloaded 2979 times last on 22 January 2025, 06:31 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.
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(Downloaded 967 times last on 22 January 2025, 12:31 pm
(Downloaded 967 times last on 22 January 2025, 12:31 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 590 times last on 22 January 2025, 12:14 am
(Downloaded 590 times last on 22 January 2025, 12:14 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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(Downloaded 1011 times last on 18 January 2025, 12:36 am
(Downloaded 1011 times last on 18 January 2025, 12:36 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 619 times last on 21 January 2025, 08:30 am
(Downloaded 619 times last on 21 January 2025, 08:30 am
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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(Downloaded 503 times last on 20 January 2025, 11:45 am
(Downloaded 503 times last on 20 January 2025, 11:45 am
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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MTTS Level :
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(Downloaded 666 times last on 18 January 2025, 12:36 am
(Downloaded 666 times last on 18 January 2025, 12:36 am
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 1089 times last on 20 January 2025, 08:42 pm
(Downloaded 1089 times last on 20 January 2025, 08:42 pm
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 646 times last on 18 January 2025, 12:36 am
(Downloaded 646 times last on 18 January 2025, 12:36 am
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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(Downloaded 943 times last on 21 January 2025, 03:58 pm
(Downloaded 943 times last on 21 January 2025, 03:58 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
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MTTS Level :
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(Downloaded 725 times last on 22 January 2025, 06:40 am
(Downloaded 725 times last on 22 January 2025, 06:40 am
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 586 times last on 19 January 2025, 03:31 am
(Downloaded 586 times last on 19 January 2025, 03:31 am