Topic 3: Numerical calculus

Introduction

Calculus is an important mathematical tool in science, basically because it lets us break up a problem into ``smaller parts''. The power and richness of calculus rests on the concepts of differentiation and integration.

Differentiation

At its most basic, differentation is the determination of the slope of a curve. A computer can perform symbolic differentiation (e.g. in Mathematica) and get the exact answer. It can also make an approximation by numerical, such as (for example)

$$\frac{d\,y}{d\,x} = \frac{y(i+1)-y(i-1)}{x(i+1)-x(i-1)}$$

bf Integration

Basically, integration is the determination of the area under a curve. A numerical approximation might be

$$\int^{x_{max}}_{x_{min}} f(x) dx = \sum^{x_{max}}_{x_{min}} \frac{(f(i+1)+f(i))}{2} (x(i+1)-x(i))$$

Resources

There is a large body of knowledge about computer-based numerical methods for differentiation and integration. Numerical recipes is a good place to start.

Tutorial goals

Use Mathematica or Matlab to carry out a symbolic differentiation and integration

Check the result of a numerical differentation or integration

Advanced students might consider integrating a function with a singularity