In "making" things, be it cool graphics, simulations, or tangible objects, mathematical concepts often play crucial roles. However, when you encounter topics like differentiation, integration, and diffusion, they might seem daunting, bringing back memories of complex equations and rigorous proofs. Seriously, when I see those formulas on the Wikipedia pages, they are just overwhelming.


The goal here is to introduce them from a different angle than ordinary math books. Especially for the purpose of creating things on a computer, you don't necessarily need to dive deep into their theoretical intricacies. More often than not, numerical methods, usually just combinations of basic additions and loops come to our rescue, providing good enough approximation and simplifying representation of these concepts.


In these articles, we will discuss summation, differentiation, integration, and diffusion, trying to demystify these concepts with demos and examples. We will prioritize using the concepts to create something, rather than on their rigor, so that if you come across them, you can translate them into simple code that works.




The summation represented by Sigma ($\sum$) is essentially a repetition of addition. It is commonly expressed using a simple loop in code. Summation is highly useful for approximating integrals, as we will discuss later, and for other concepts like Fourier series.

シグマ ($∑$) で表される総和は要は足し算の繰り返しで、コードでは大抵単純なループを使って表現できます。総和は下で取り上げる積分や、他にもフーリエ級数などを近似する時に活躍します。

Summation 総和



Differentiation is a very important concept in dealing with motion, shape, and simulating physics. It is about finding the rate of change and the slope of a graph for a given function. When handling differentiation on a computer, it is often more practical to use numerical approximation or "numerical differentiation," which involves using the difference between values. Math textbooks may not touch upon this, but it is a useful technique to treat differentiation as subtraction. In fact, this technique is used in various applications, possibly without you even realizing it.