105 Result(s)
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Chapter
Claes Johnson
Analysis
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Applications Tool Bag
In this section we collect the basic models of engineering and science expressed as differential equations. For specification of boundary and initial values we refer to the text.
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Mountain Climbing
Every new body of discovery is mathematical in form, because there is no other guidance we can have. (Darwin)
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The Right to Not Know
To my knowledge the President had no prior knowledge of the bugging plan, and such knowledge by Nixon, would be difficult to believe. (Nixon’s campaign Deputy Director Jeb Stuart Magruder, testifying under oat...
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Fourier Analysis Tool Bag
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Do We live in “The Best of Worlds”?
The Best World has the greatest variety of phenomena regulated by the simplest Laws. (Leibniz)
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Jazz/Pop/Folk vs. Classical Music
Music is the pleasure the human soul gets from counting, without knowing it. (Leibniz)
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An Agenda
... there is no study in the world which brings into more harmonious action all the faculties of the mind than [mathematics], ... or, like this, seems to raise them, by successive steps of initiation, to highe...
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Numerical Quadrature
In some cases, we can compute a primitive function (or antiderivative or integral) of a given function analytically, that is we can give give a formula for the primitive function in terms of known functions. F...
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The Square Root of Two
We met the equation x 2 = 2 in the context of the Muddy Yard model, trying to determine the length of the diagonal of a square with side length 1.
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Introduction to Modeling
We start by giving two basic examples of the use of mathematics for describing practical situations. The first example is a problem in household economy and the second is a problem in surveying, both of which ...
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The Solar System
There is talk of a new astrologer who wants to prove that the earth moves and goes around instead of the sky, the sun, the moon, just as if somebody were moving in a carriage or ship might hold that he was sit...
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Trigonometric Functions
In this chapter, we shall study the following initial value problem for a second order differential equation: Find a function u(x) defined for x ≥ 0 satisfying
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The Bisection Algorithm for f(x) = 0
We now generalize the Bisection algorithm used above to compute the positive root of the equation x2 − 2 = 0, to compute roots of the equation
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Level Curves/Surfaces and the Gradient
It would make no sense to overload the student with all kinds of little things that might be of occasional use. Instead, it is important that students become familiar with ways to think mathematically, recogni...
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Natural Numbers and Integers
In this chapter, we recall how natural numbers and integers may be constructively defined, and how to prove the basic rules of computation we learn in school. The purpose is to give a quick example of developi...
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Optimal Control
We’re making the right decisions to bring the solution to an end. (George W. Bush)
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Fourier Series
Yesterday was my 21st birthday, at that age Newton and Pascal had already acquired many claims to immortality. (Fourier 1789, age 21)
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Techniques of Integration
It is not generally possible to find an explicit formula for a primitive function of a given arbitrary function in terms of known elementary functions, by which we mean the polynomials, rational functions, root f...
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The Function y = x r
We showed above that we can solve the equation x 2=a for any positive rational number a using the Bisection algorithm.
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Adaptive Solvers for IVPs
On two occasions I have been asked (by members of Parliament), “Pray, Mr Babbage, if you put into the machine wrong figures, will the right answer come out?”. I am not able rightly to apprehend the kind of con...
