Alexandr Draganov - Mathematical Tools for Real-World Applications: A Gentle Introduction for Students and Practitioners

Mathematical Tools for Real-World Applications

A Gentle Introduction for Students and Practitioners

Alexandr Draganov

The MIT Press

Cambridge, Massachusetts

London, England

2022 Massachusetts Institute of Technology

All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher.

The MIT Press would like to thank the anonymous peer reviewers who provided comments on drafts of this book. The generous work of academic experts is essential for establishing the authority and quality of our publications. We acknowledge with gratitude the contributions of these otherwise uncredited readers.

Library of Congress Cataloging-in-Publication Data

Names: Draganov, Alexandr, author.

Title: Mathematical tools for real-world applications: a gentle introduction for students and practitioners / Alexandr Draganov.

Description: Cambridge, Massachusetts: The MIT Press, [2022] | Includes bibliographical references and index.

Identifiers: LCCN 2021046200 | ISBN 9780262543965 (paperback)

Subjects: LCSH: Mathematics. | Problem solvingMathematical models.

Classification: LCC QA39.3.D73 2022 | DDC 510dc23/eng/20211119

LC record available at https://lccn.loc.gov/2021046200

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To Luda

The reader who follows my drift with sufficient attention will easily see that nothing is less in my mind than ordinary Mathematics, and that I am expounding quite another science, of which these illustrations are rather the outer husk than the constituents. Such a science should contain the primary rudiments of human reason, and its province ought to extend to the eliciting of true results in every subject.

Ren Descartes, Rules for the Direction of the Mind

Indeed, I have found that it is usually in unimportant matters that there is a field for the observation, and for the quick analysis of cause and effect which gives the charm to an investigation.

Arthur Conan Doyle, The Adventures of Sherlock Holmes

Two hikers on a trail: units

A circle and a straight line: units

Designing satellite coverage: units

Law of sines and law of cosines: units

Archimedess spiral: units

A circle and an ellipse: units

A triangle, an inscribed circle, and a circumscribed circle: units

Two hikers on a trail: limiting cases

The quadratic equation: limiting cases

A circle and a straight line: limiting cases

Parameter domains for the sum and for the difference of two radicals

A circle inscribed in a right triangle: limiting cases

Small and large angles for a right triangle with an inscribed circle

Designing satellite coverage: limiting cases

Two circles inscribed in an angle: limiting cases

A circle and a parabola: limiting cases

Detecting an error in the circle and parabola problem

Linear regression: limiting cases

A spherical cap: limiting cases

Law of cosines: limiting cases

Detecting a vessel by two radars: limiting cases

A torus: limiting cases

Archimedess spiral: limiting cases

A circle and an ellipse: limiting cases

A circle and a hyperbola: limiting cases

A triangle, an inscribed circle, and a circumscribed circle: limiting cases

Symmetry in the fall of Newtons apple

Two hikers on a trail: symmetry

Waves in a pond and on a beach

Changing a variable to impose symmetry

A circle and a straight line: symmetry

A circle inscribed in a right triangle: symmetry

Designing satellite coverage: symmetry

Two circles inscribed in an angle: symmetry

Linear regression: symmetry

A spherical cap: symmetry

A circle inscribed in a right triangle: symmetry

Herons formula: symmetry

A triangle, an inscribed circle, and a circumscribed circle: symmetry

A circle and a hyperbola: symmetry

Two hikers on a trail: symmetry

Archimedess spiral: symmetry

A circle and an ellipse: symmetry

Detecting a vessel by two radars: symmetry

The Pythagorean theorem and a scaling argument

Scaling of light intensity with the distance from the source

The universe in Olberss paradox

Rope tension on a capstan for a small angle

Linear regression: scaling

A torus: scaling

A spherical cap: scaling

Rectangle inscribed in a right triangle: scaling

A circle and an ellipse: scaling

Archimedess spiral: scaling

Two circles inscribed in an angle: scaling

Monthly mortgage payments as a function of the interest rate

A spherical cap: order of magnitude estimates

A torus: order of magnitude estimates

Archimedess spiral: order of magnitude estimates

Achilles and the tortoise: an algebraic solution

Achilles and the tortoise: an iterative solution

Convergence criterion for the MSA

The product of two linear expressions: successive approximations

The quadratic equation: successive approximations

Archimedess spiral: successive approximations

Designing satellite coverage: successive approximations

A circle and a parabola: successive approximations

Criteria for h for the circle and parabola problem

A small quadratic term in the circle and parabola problem

Approximations for detecting a vessel by two radars

CCPDF for the roulette game

Airplane landing error

Airplane landing: shift invariance

CCPDF for the GPD and the normal distribution

The step function

Step functions as building blocks for a general input

A PLL response to a step function input

PLL responses for two different values of fN

Distances traveled by two hikers on a trail

The intersections between a circle and a straight line

The intersections between a circle and an ellipse

The intersections between a circle and a hyperbola

The intersections between a circle and a parabola

Designing satellite coverage

Detecting a vessel by two radars

Two circles inscribed in an angle

A circle inscribed in a right triangle

A rectangle inscribed in a right triangle

A spherical cap

Linear regression

Base units for physical quantities

Units for the coefficients in the car deceleration formula

Units for various terms in the cubic formula

Parallel solutions of two related problems

Symmetry in the quadratic equation

The hierarchy of scaling behaviors

Successive approximations for the product of two linear expressions

Convergence for the quadratic equation

Convergence for the turn angle of Archimedess spiral

Convergence for the satellite coverage problem

Intersection of a circle and a parabola (a small linear term)

Intersection of a circle and a parabola (a small quadratic term)

Approximations for parameter

Time required to reach the vicinity of the exact value by the PLL

Real scientific or engineering problems are often different from textbook ones and take a different set of tools, for two reasons.

First, real problems are usually messier and take longer to solve. A good sprinter is not necessarily a good marathoner; similarly, a long solution requires a different set of skills than a short one. You must plan out an approach in advance, break it up into manageable segments, and devise ways to check your work along the way. In math, this starts from understanding your problem well.