On a smooth manifold, vector lives in the tangent vector space, while 1-form lives in the cotangent space. Note lot of confusion comes from the understanding of the basis vector and basis covetor. Vectors transform with Jacobian, 1-forms transform with the inverse transpose Jacobian. On a non-flat surface (a manifold), basis vectors are no longer … Continue reading Vectors vs. 1-forms: Two Different Spaces

On group level, commutation can fail, but not associativity! so x(yz) = (xy)z! How and why? From the other perspective, introducing the function to evaluate we got

The crucial tool connecting all $k$-forms is the Exterior Derivative d. This derivative is the mathematical foundation for generalizing the integral theorems of vector calculus into one single theorem: Stokes' Theorem.

Imagine the sphere as Earth, and vector filed are arrows painted on the surface.

Snowpark is a powerful, developer-centric framework that lets you execute code written in popular languages like Python, Scala, or Java directly within Snowflake's data cloud. It moves computation closer to the data, drastically reducing data movement and leveraging Snowflake’s highly scalable, optimized engine. The single most important principle of Snowpark is client-side coding, server-side execution … Continue reading Making the Most from Snowflake Snowpark

Curvature κ\kappaκ is how fast the direction of the unit tangent vector changes with respect to arc-length:

A Banach space is a complete normed vector space. Here, the norm is a function that measures the “length” of a vector vvv, and it must satisfy certain properties like the triangle inequality. The distance between two vectors uuu and vvv is defined simply by the norm of their difference, ∥u−v∥\|u - v\|∥u−v∥. Importantly, Banach … Continue reading Hilbert Spaces and Banach Spaces: What’s the Difference?

The OWASP Top 10 1. Broken Access Control This is the “users touching things they shouldn’t” category.Example: A simple missing WHERE user_id = ? lets User A view or modify User B’s data.Access control is not UI logic—it is backend logic. 2. Cryptographic Failures Encryption done wrong is effectively no encryption.Example: Sending passwords over HTTP … Continue reading Security Matters in Building Safer Software

There is a wealth of excitement surrounding the recently launched Gemini 3, so I decided to tackle a famous Gaussian integral problem and explore its image creation capabilities. At first, I just used their default prompt and it's mind-blowing: "Create a comic with the title: Getting started with Nano Banana Pro 🍌. Panel 1: A … Continue reading Gemini 3 Power