Calculus and Differential Equations

In this chapter, we will discuss various topics related to calculus. Calculus is the branch of mathematics that concerns the processes of differentiation and integration. Geometrically, the derivative of a function represents the gradient of the curve of the function, and the integral of a function represents the area below the curve of the function. Of course, these characterizations only hold in certain circumstances, but they provide a reasonable foundation for this chapter.

We start by looking at calculus for a simple class of functions: the polynomials. In the first recipe, we create a class that represents a polynomial and define methods that differentiate and integrate the polynomial. Polynomials are convenient because the derivative or integral of a polynomial is again a polynomial. Then, we use the SymPy package to perform symbolic differentiation and integration on more general functions. After that, we see methods for solving equations using the SciPy package. Next, we turn our attention to numerical integration (quadrature) and solving differential equations. We use the SciPy package to solve ordinary differential equations and systems of ordinary differential equations, and then use a finite difference scheme to solve a simple partial differential equation. Finally, we use the fast Fourier transform to process a noisy signal and filter out the noise.

In this chapter, we will cover the following recipes:

• Working with polynomials and calculus
• Differentiating and integrating symbolically using SymPy
• Solving equations
• Integrating functions numerically using SciPy
• Solving simple differential equations numerically
• Solving systems of differential equations
• Solving partial differential equations numerically
• Using discrete Fourier transforms for signal processing