Applications of Calculus in Physics

The purpose of this book is to show students of both math and physics the deep relationship between the areas of physics and calculus. In particular:

a) How calculus can deepen our understanding of the physical concepts

b) How calculus can link together two or more seemingly unrelated physical concepts

c) How calculus can be used to help us apply physical concepts to technology

d) How calculus provides a mechanism to do physics in alternative ways

Because many students might be seeing calculus for the first time, a basic 100-page Calculus Primer has been included in the text immediately following the applications. The Calculus Primer focuses on giving the students a “crash course” in how to calculate a derivative and an integral quickly, rather than on the deep rigor taught in many calculus courses. Formal mathematical language has been intentionally avoided to help students acquire this useful tool as quickly as possible. Because this material is being included in a physics application book rather than a calculus course, the Calculus Primer includes exercises with full solutions. These step-by-step solutions are important since many students will attempt to learn this material on their own. So that the student can immediately see if he/she has executed the calculus correctly using the given rules and algorithms, the solutions have intentionally been left in an “unsimplified” form.

As the student works through the ten applications in the book, Connection Boxes are provided in the margins to map the student to the appropriate section of the Calculus Primer to learn the necessary calculus tool to complete the application. Because the Calculus Primer is designed to provide the necessary calculus tools on an “as needed” basis, both the topics covered and their order is somewhat different from that of a standard introductory calculus text. The applications are organized in terms of physical content rather than by calculus content. Consequently, physical concepts that are introduced early in a standard physics course appear earlier in this section than those introduced later in the course.

The following examples of the many applications of calculus in physics are covered in this book:
1. Displacement, Velocity, and Acceleration

2. The Work Done by a Variable Force

3. Force and Potential Energy

4. The Electrical-Mechanical Analogy

5. Impulse and Momentum

6. Kepler’s Law of Areas and Conservation of Angular Momentum

7. Shockley’s Equation and Thermodynamics

8. Simple Harmonic Oscillators

9. The Velocity of a Transverse Wave

10. Lagrangian Physics: An Alternative Way to do Physics

To help both the student and the instructor, each application also ends with a set of exercises related to the topic of the application.