Discovering Mathematics with MATLAB

This book continues the journey of discovering mathematics that started with the college-level book Discovering Mathematics with Excel (DME), in which the fundamentals of many areas of mathematics, including geometry, algebra, trigonometry, probability, logic, algorithms, number theory, vectors, calculus, and differential equations were uncovered. In this second book, these topics are explored more deeply, moving from one-dimensional functions to functions of two or more variables. Chapter 1 will review essential topics from Chapter 1 in DME, including finding slopes and areas of functions, the meaning of e, the base of natural logarithms, the fundamental theorem of calculus, and the complex plane. You will learn how to use MATLAB to produce the same results found in Chapter 1 of DME using Excel.

You will go on to use MATLAB to discover the fundamentals of functions of more than one variable. In Chapter 2, you will discover partial derivatives and the meaning of gradient, divergence and curl. While exploring multiple integrals in Chapter 3, you will discover how to use MATLAB to plot functions of more than one variable. In Chapter 4, you will discover many important topics from linear algebra, including matrices, solving linear equations, eigenvalues and eigenvectors, and solving systems of ordinary differential equations. You will explore vector fields in Chapter 5, derive Kepler’s Laws of orbital motion, and learn about electric and magnetic fields, resulting in the discovery of Maxwell’s equations. In Chapter 6, you will discover how to solve partial differential equations using MATLAB. In the final chapter on transforms, you will discover how the Fourier Transform, Fourier Series, the Discrete-Time Fourier Transform, the Discrete Fourier Transform, the Laplace Transform, and the Z-Transform are related.

Once you complete these two Discovering Mathematics books, you will be well on your way to mastering the mathematics you need to be successful in any science or engineering major. Let’s get started!

Richard E. Haskell
Darrin M. Hanna

1.  From Excel to MATLAB®
1.1  MATLAB vs. Excel
1.2  Finding Slopes Using MATLAB
1.3  Slopes and Euler’s Number e
1.4  Finding Areas Using MATLAB
1.5  The Fundamental Theorem of Calculus
1.6  Complex Numbers in MATLAB
Problems
2.  Discovering Partial Derivatives
2.1  Slopes of Surfaces
2.2  Visualizing Multidimensional Functions
2.3  Curvature of Surfaces
2.4  Partial Derivatives of Vector Fields
2.5  Partial Differential Equations
Problems
3.  Discovering Multiple Integrals
3.1  The Volume Under a Surface
3.2  The Area of a Surface
3.3  Volumes in Rectangular Coordinates
3.4  Volumes in Cylindrical Coordinates
3.5  Volumes and Areas in Spherical Coordinates
Problems
4.  Discovering Matrices
4.1  Matrix Multiplication
4.2  Solving Linear Equations
4.3  Cramer’s Rule
4.4  Gaussian Elimination
4.5  Matrix Transformations
4.6  Eigenvalues and Eigenvectors
4.7  Raising a Matrix to a Power
4.8  System of Differential Equations
Problems
5.  Discovering Vector Fields
5.1  Vector Fields and the Laws of Nature
5.2  Kepler’s Laws
5.3  The Discovery of Electricity
5.4  Faraday’s Lines of Force
5.5  Maxwell’s Electromagnetic Field
Problems
6.  Discovering Partial Differential Equation Solutions
6.1  The Heat Equation
6.2  Waves on a String
6.3  Sound Waves
6.4  Electromagnetic Waves
Problems
7.  Discovering Transforms
7.1  The Fourier Transform
7.2  Fourier Series
7.3  The Discrete-Time Fourier Transform
7.4  The Discrete Fourier Transform
7.5  The MATLAB Fast Fourier Transform Function
7.6  The Laplace Transform
7.7  The Z-Transform
Problems
Appendix A: Fourier Transform of a Gaussian Function
Appendix B: Fourier Transform of a comb Function 389
Appendix C: Fourier Transform of a Triangle Function
Index

(398 pages)