87 Result(s)
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Chapter
Claes Johnson
Analysis
Math. Applications in Chemistry
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Chapter
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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Fourier Analysis Tool Bag
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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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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...
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Complex Numbers
In this chapter, we introduce the set of complex numbers ℂ. A complex number, typically denoted by z, is an ordered pair z = (x, y) of real numbers x and y, where x represents the real part of z and y the imagina...
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Rational Numbers
We learn in school that a rational number r is a number of the form \(r = \frac{p}{q} = p/q\) , where p and q are integers with q≠ 0. Suc...
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Analytic Functions Tool Bag
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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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Differentiation Rules
We now state and prove some rules for computing derivatives of combinations of functions in terms of the derivatives of the functions in the combination. These rules of differentiation form a part of Calculus ...
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What is a Function?
The concept of a function is fundamental in mathematics. We already met this concept in the context of the Dinner Soup model, where the total cost was 15x (dollars) if the amount of beef was x (pounds). For every...
