Skip to main content
main-content
Top

About this book

Second Year Calculus: From Celestial Mechanics to Special Relativity covers multi-variable and vector calculus, emphasizing the historical physical problems which gave rise to the concepts of calculus. The book carries us from the birth of the mechanized view of the world in Isaac Newton's Mathematical Principles of Natural Philosophy in which mathematics becomes the ultimate tool for modelling physical reality, to the dawn of a radically new and often counter-intuitive age in Albert Einstein's Special Theory of Relativity in which it is the mathematical model which suggests new aspects of that reality. The development of this process is discussed from the modern viewpoint of differential forms. Using this concept, the student learns to compute orbits and rocket trajectories, model flows and force fields, and derive the laws of electricity and magnetism. These exercises and observations of mathematical symmetry enable the student to better understand the interaction of physics and mathematics.

Table of Contents

Frontmatter

1. F=ma

Abstract
Popular mathematical history attributes to Isaac Newton (1642–1727) and Gottfried Wilhelm Leibniz (1646–1716) the distinction of having invented calculus. Of course, it is not nearly so simple as that. Techniques for evaluating areas and volumes as limits of computable quantities go back to the Greeks of the classical era. The rules for differentiating polynomials and the uses of these derivatives were current before Newton or Leibniz were born. Even the fundamental theorem of calculus, relating integral and differential calculus, was known to Isaac Barrow (1630–1677), Newton’s teacher. Yet it is not inappropriate to date calculus from these two men for they were the first to grasp the power and universal applicability of the fundamental theorem of calculus. They were the first to see an inchoate collection of results as the body of a single unified theory.
David M. Bressoud

2. Vector Algebra

Abstract
While the terminology of calculus that we have at hand is certainly sufficient to prove the converse of Theorem 1.3, namely, that Newton’s law of gravity implies that planets must move in elliptical orbits with the sun at one focus, this and other arguments we are to make will be greatly simplified if we adopt a language developed in the late nineteenth century, that of vector algebra.
David M. Bressoud

3. Celestial Mechanics

Abstract
We are not quite ready to prove that Newton’s law of gravitational attraction implies Kepler’s second law. We need to take a closer look at the Calculus of vector functions in the light of the vector algebra described in the last chapter.
David M. Bressoud

4. Differential Forms

Abstract
With this chapter we begin the study of functions whose domain and range consist of several variables.
David M. Bressoud

5. Line Integrals, Multiple Integrals

Abstract
There is a curious contradiction in the standard presentation of integral calculus. It is an ancient subject rooted in the geometric investigations of Archimedes of Syracuse (287 –212 B.C.).
David M. Bressoud

6. Linear Transformations

Abstract
A linear transformation is a function,\( \vec{L}, \) from one or more real variables to one or more real variables, that satisfies the following two conditions for any values of \( \vec{a} \) and \( \vec{b} \) in the domain and any real constant c
David M. Bressoud

7. Differential Calculus

Abstract
We recall that a vector field, \( \overrightarrow F \) is differentiable at \(\vec{c}\) if and only if there exists a linear transformation, \(\vec L_c \).
David M. Bressoud

8. Integration by Pullback

Abstract
We interrupt our study of differential calculus to apply the results of Section 4 of Chapter 7 to the outstanding problem of evaluating pullbacks.
David M. Bressoud

9. Techniques of Differential Calculus

Abstract
Consider a level surface in three dimensions,f(x,y,z)=c.
David M. Bressoud

10. The Fundamental Theorem of Calculus

Abstract
For scalar functions of one variable, the fundamental theorem of calculus is a powerful tool for integration. It says that there exists an associated function, often called an antiderivative, such that the integral of the scalar function can be evaluated by looking at the antiderivative at the endpoints of the interval. Specifically, it is the following theorem.
David M. Bressoud

11. E=mc2

Abstract
Much as Newton’s contribution to calculus was less in the discovery of the techniques than in his vision of how they linked together and what could be done with them, so the equations of James Clerk Maxwell (1831–1879) were not, with one partial exception, his discovery, yet he placed his mark upon them in recognizing their basic unity and what they implied. Maxwell’s seminal paper, “A Dynamical Theory of the Electro-Magnetic Field,” read before the Royal Society of London in 1864 and published in its Philosophical Transactions in 1865, explained the nature of electromagnetic potential, revealed it to be intimately connected to the propagation of light, and set the stage for Einstein’s discovery of special relativity.
David M. Bressoud

Backmatter

Additional information