In calculus, we learn that the derivative measures change. But in a deeper mathematical sense, differentiation acts as the opposite of taking a boundary. This idea is best understood using the language of differential forms and chains. At its core, differentiation isn't just about slopes—it's about how things flow across a space. And just as … Continue reading Derivative is the Opposite of the Boundary

To thoroughly grasp the concept of a gradient, one needs to think deeply. First, we must understand that the gradient represents the steepest slope, which can be illustrated with an analogy like standing at a position on a mountain. However, while analogies can be helpful, they are often not rigorous enough. Mathematics, on the other … Continue reading Is Gradient a Vector or 1-Form?

Differential Forms is a beauty! Multivariable calculus can be greatly simplified and generalized when applying differential forms.

I have written 4 blogs on MCP several months ago and am glad to see that it's integrated in Windsurf and Cursor now. Basically, MCP is a standardized interface so AI agents can communicate seamlessly to various servers such as servers of Github, Slack, Google map, so on and so forth and certainly could be … Continue reading MCP is Available in Windsurf and Cursor

A cylinder can be viewed as a fiber bundle: The Möbius band is a slightly more complicated example but still a fiber bundle. Here's how we can understand it:

Let's consider our familiar workflow of portfolio weighting and capping. Have you ever thought about the essence of weight capping? When we cap a position at 10% and redistribute the excess weight proportionally to the rest of the portfolio, what are we really doing? In essence, we're optimizing the portfolio weights by minimizing a weight … Continue reading A Simple Perceptron to Understand Neural Network

It's essential to understand Stokes' Theorem, why? first of all, it's a higher dimensional generalization of the fundamental theorem of calculus. to dive deep, first understand or go over basic concepts again: what is k form in R n? then what is derivative of this k form in Rn? Examples to make it more concrete:

what's the definition of differential forms? k-forms are those that can integrate on k-dimensional domains or manifolds. Given what discussed about pull back of function and push forward, it's the same concept just expanded to higher dimensions: the purpose is to pull back differential forms by passing through wedges: Contraction operator? it's written as below … Continue reading Gauge Theory by Tim 04 Differential Forms

A cotangent vector is a linear functional of a tangent vector at point p, i.e. it's a dual vector, by the way definition of function and functional is: A function takes numbers (or vectors) as inputs and produces numbers (or vectors) as outputs. A functional takes a function as input and produces a scalar (number) … Continue reading Gauge Theory by Tim 03b Cotangent Vector

In this session, he talks about differential geometry starting from coordinate charts: to have this clear picture starting from a punched plane is very helpful in later variable coordinate etc. transformation. then to understand the concept of pull back and push forward. pull back means pull back of functions, or change of variables. To understand … Continue reading Gauge Theory by Tim 03 Pull Back and Push Forward