The purpose is only to share and spread the awareness of availability of this second master piece on Euclid. Thanks to Clay Math Organization for serving students world wide, and thanks to the generous Mr and Mrs Clayton. I hope my math olympiad students will enjoy this and enrich themselves mathematically.
Author: Nalin Pithwa
David Joyce’s Euclid: thanks to ClayMath
The purpose to share this here is to spread the awareness of availability of such a masterpiece by ClayMath organization, thanks of course to Mr and Mrs Clayton also.
Madhava Mathematics Competition 2021 Feb
Wisdom of V. I. Arnold, immortal Russian mathematician
Development of mathematics resembles a fast revolution of a wheel: sprinkles of water are flying in all directions. Fashion — it is the stream that leaves the main trajectory in the tangential direction. These streams of epigone works attract most attention, and they constitute the main mass, but they inevitably disappear after a while because they parted with the wheel. To remain on the wheel, one must apply the effort in the direction perpendicular to the main stream.
V I Arnold, translated from “Arnold in His Own words,” interview with the mathematician originally published in Kvant Magazine, 1990, and republished in the Notices of the American Mathematical Society, 2012.
Mathematics and Sex; Prof Clio Cresswell, Sydney, Australia on TEDx
Two semantic paradoxes
- I am lying
- The only statement on this whiteboard is false.
Whereas Russell s paradox in set theory is mathematical.
Regards
Nalin Pithwa
An elementary concept from set theory
Prove that the empty set is unique.
The Study of Mathematics : a quote
“What is best in mathematics deserves not merely to be learnt as a task, but to be assimilated as a part of daily thought, and brought again and again before the mind with ever-renewed encouragement.”
— Bertrand Russell, “The Study of Mathematics” (1902)
Ambiguity in the use of “or”
An example from daily English language usage : “In this or the next block, you will find a taxi cab” and “the child just born is just male or female”.
Calculus : IITJEE Advanced Math tutorial problems: Part 1
Problem 1: Prove that
Problem 2: When does equality hold in the following theorem? ? Hint: Re-examine the proof of the theorem, the answer is not “when x and y are linearly dependent.”
Problem 3: Prove that . When does inequality hold?
Problem 4: Prove that ?
Problem 5: The quantity is called the distance between x and y. Prove and interpret geometrically the “triangle inequality” :
.
Problem 6: Let functions f and g be integrable on .
(a) Prove that . Hint: Consider separately the cases
for some
and
for all
.
(b) If equality holds, must for some
? What if f and g are continuous?
(c) Show that the following theorem is a special case of (a) above: , equality holds if and only if x and y are linearly dependent.
Problem 7: A linear transformation is norm preserving if
amd inner product preserving if
(a) Prove that T is norm preserving if and only if T is inner product preserving.
(b) Prove that such a linear transformation T is and
is of the same sort.
Problem 8:
If are non-zero, the angle between x and y, denoted
is defined as
, which makes sense by the following theorem :
The linear transformation T is angle preserving if T is 1-1, and for we have
(a) Prove that if T is norm preserving, then T is angle preserving.
(b) If there is a basis of
and numbers
such that
, prove that T is angle preserving if and only if all
are equal.
(c) What are all angle preserving ?
Problem 9: If , let
have the matrix
.
Show that T is angle preserving and if , then
Problem 10: If is a linear transformation, show that there is a number M such that
for
. Hint: Estimate
in terms of
and the entries in the matrix of T.
Problem 11: If and
, show that
and
. Note that
and
denote points in
.
Problem 12: Let denote the dual space of the vector space
. If
, define
by
. Define
by
. Show that T is a 1-1 linear transformation and conclude that every
is
for a unique
.
Problem 13: If , then x and y are called perpendicular (or orthogonal) if
. If x and y are perpendicular, prove that
.
Regards,
Nalin Pithwa