Bill Casselman’s Euclid: thanks to ClayMath

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.

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.

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)

Calculus : IITJEE Advanced Math tutorial problems: Part 1

Problem 1: Prove that |x| \leq \sum_{i=1}^{n} |x^{i}|

Problem 2: When does equality hold in the following theorem? |x+y| \leq |x|+|y|? Hint: Re-examine the proof of the theorem, the answer is not “when x and y are linearly dependent.”

Problem 3: Prove that |x-y| \leq |x|+|y|. When does inequality hold?

Problem 4: Prove that ||x|-|y|| \leq |x-y|?

Problem 5: The quantity |y-z| is called the distance between x and y. Prove and interpret geometrically the “triangle inequality” : |x-z| \leq |x-y|+|y-z|.

Problem 6: Let functions f and g be integrable on [a,b].

(a) Prove that |\int_{a}^{b}| \leq (\int_{a}^{b}f^{2})^{\frac{1}{2}}.(\int_{a}^{b}g^{2})^{\frac{1}{2}}. Hint: Consider separately the cases 0 = \int_{a}^{b}(f-g \lambda)^{2} for some \lambda \in \Re and 0 < \int_{a}^{b}(f-g\lambda)^{2} for all \lambda \in \Re.

(b) If equality holds, must f=g \lambda for some \lambda \in \Re? What if f and g are continuous?

(c) Show that the following theorem is a special case of (a) above: |\sum_{i=1}^{n}x^{i}y^{i}| \leq |x|.|y|, equality holds if and only if x and y are linearly dependent.

Problem 7: A linear transformation T: \Re^{n} \rightarrow \Re^{n} is norm preserving if |T(x)|=|x| amd inner product preserving if <Tx, Ty> = <x,y>

(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 1-1 and T^{-1} is of the same sort.

Problem 8:

If x, y \in \Re^{+} are non-zero, the angle between x and y, denoted \angle {(x,y)} is defined as \arccos{(\frac{<x,y>}{|x|.|y|})}, which makes sense by the following theorem :

<x,y> \equiv |\sum_{i=1}^{n}x^{i}y^{i}| \leq |x|.|y|

The linear transformation T is angle preserving if T is 1-1, and for x,y \neq 0 we have \angle {(Tx,Ty)} = \angle{(x,y)}

(a) Prove that if T is norm preserving, then T is angle preserving.

(b) If there is a basis \{ x_{1}, x_{2}, \ldots, x_{n}\} of \Re^{n} and numbers \lambda_{1}, \lambda_{2}, \ldots, \lambda_{n} such that Tx_{i}=\lambda_{i} x_{i}, prove that T is angle preserving if and only if all \lambda_{i} are equal.

(c) What are all angle preserving T: \Re^{n} \rightarrow \Re^{n}?

Problem 9: If 0 \leq \theta < \pi, let T: \Re^{2} \rightarrow \Re^{n} have the matrix \left | \begin{array}{cc} \cos{\theta} & \sin{\theta} \\ -\sin{\theta} & \cos{\theta} \end{array} \right |.

Show that T is angle preserving and if x \neq 0, then \angle{(x, Tx)}= \theta

Problem 10: If T: \Re^{m} \rightarrow \Re^{n} is a linear transformation, show that there is a number M such that |T(h)| \leq M|h| for h \in \Re^{m}. Hint: Estimate |T(h)| in terms of |h| and the entries in the matrix of T.

Problem 11: If x, y \in \Re^{n} and z, w \in \Re^{m}, show that <(x,z),(y,w)> = <x,y>+<z,w> and |(x,z)|= \sqrt{|x|^{2}+|z|^{2}}. Note that (x,z) and (y,w) denote points in \Re^{n+m}.

Problem 12: Let (\Re^{n})^{*} denote the dual space of the vector space \Re^{n}. If x \in \Re^{n}, define \phi_{x} \in (\Re^{n})^{*} by \phi_{x}(y)=<x,y>. Define T: \Re^{n} \rightarrow (\Re^{n})^{*} by T(x)=\phi_{x}. Show that T is a 1-1 linear transformation and conclude that every \phi \in (\Re^{n})^{*} is \phi_{x} for a unique x \in \Re^{n}.

Problem 13: If x, y \in \Re^{n}, then x and y are called perpendicular (or orthogonal) if <x,y>=0. If x and y are perpendicular, prove that |x+y|^{2} = |x|^{2}+|y|^{2}.


Nalin Pithwa