Stokes' Theorem is a classic result from vector calculus.... It tells us that the line integral of a vector field over a loop is equal to the surface integral of the curl of the vector field over some enclosed surface.show more

The most beautiful formula in calculus! Master one of... vector calculus's most profound concepts with this stunning visualization of Stokes' Theorem. This video provides a clear, intuitive animation showing the equality between the line integral of a vector field over a closed loop (boundary circulation) and the surface integral of the curl over the enclosed surface (surface curl). Watch how the circulation around curve C perfectly balances the sum of the infinitesimal rotations within surface S. #calculus #math #stokestheorem #manim #mathematicsshow more

The Helmholtz decomposition is one of the fundamental results... of vector calculus. It says any well-behaved vector field can be split into two parts, one capturing sources and sinks through divergence, and one capturing rotation through curl.show more

Vector Database by Hand ✍️ Vector databases are revolutionizing... how we search and analyze complex data. They have become the backbone of Retrieval Augmented Generation (#RAG). How do vector databases work? [1] Given ↳ A dataset of three sentences, each has 3 words (or tokens) ↳ In practice, a dataset may contain millions or billions of sentences. The max number of tokens may be tens of thousands (e.g., 32,768 mistral-7b). Process "how are you" [2] 🟨 Word Embeddings ↳ For each word, look up corresponding word embedding vector from a table of 22 vectors, where 22 is the vocabulary size. ↳ In practice, the vocabulary size can be tens of thousands. The word embedding dimensions are in the thousands (e.g., 1024, 4096) [3] 🟩 Encoding ↳ Feed the sequence of word embeddings to an encoder to obtain a sequence of feature vectors, one per word. ↳ Here, the encoder is a simple one layer perceptron (linear layer + ReLU) ↳ In practice, the encoder is a transformer or one of its many variants. [4] 🟩 Mean Pooling ↳ Merge the sequence of feature vectors into a single vector using "mean pooling" which is to average across the columns. ↳ The result is a single vector. We often call it "text embeddings" or "sentence embeddings." ↳ Other pooling techniques are possible, such as CLS. But mean pooling is the most common. [5] 🟦 Indexing ↳ Reduce the dimensions of the text embedding vector by a projection matrix. The reduction rate is 50% (4->2). ↳ In practice, the values in this projection matrix is much more random. ↳ The purpose is similar to that of hashing, which is to obtain a short representation to allow faster comparison and retrieval. ↳ The resulting dimension-reduced index vector is saved in the vector storage. [6] Process "who are you" ↳ Repeat [2]-[5] [7] Process "who am I" ↳ Repeat [2]-[5] Now we have indexed our dataset in the vector database. [8] 🟥 Query: "am I you" ↳ Repeat [2]-[5] ↳ The result is a 2-d query vector. [9] 🟥 Dot Products ↳ Take dot product between the query vector and database vectors. They are all 2-d. ↳ The purpose is to use dot product to estimate similarity. ↳ By transposing the query vector, this step becomes a matrix multiplication. [10] 🟥 Nearest Neighbor ↳ Find the largest dot product by linear scan. ↳ The sentence with the highest dot product is "who am I" ↳ In practice, because scanning billions of vectors is slow, we use an Approximate Nearest Neighbor (ANN) algorithm like the Hierarchical Navigable Small Worlds (HNSW).show more

We dropped a new build just in time for... the holidays. One of my favorite additions is symmetry for tube and solid shapes in vector layers. This is great for greebling up sci-fi and hard surface stuff.show more

How is it possible that a black hole’s surface... tells us about its interior volume? This finding is a critical clue towards our understanding of the quantum nature of space-time.show more

Rub some oil on a napkin and rub the... inside brim of the pot. That’ll keep it from boiling over. The oil is hydrophobic and interrupts the surface tension of the boiling water. Learned this in culinary schoolshow more

❗️NEW footage shows moment when the US launched surface-to-surface... HIMARS missiles towards Iran, from Kuwaiti soil. The myth that Persian Gulf countries aren’t helping the US is officially over.show more

The Breather surface is the surface of Gaussian curvature... -1 that corresponds to the solution of the Sine-Gordon equationshow more

This is a real, high-resolution view of the Martian... surface. 140 million miles away from us!show more

The color of a star is a function of... its surface temperatureshow more

The color of a star is a function of... its surface temperatureshow more

She's going back to back! Shooting over 75% from... the field for the second game in a row, Madina Okot is your Albert Player of the Game.show more

the inside of the Surface Laptop Ultra is a... work of art 🖼️show more

The Sun holds plasma a hundred thousand kilometres above... its own surface using nothing but magnetic field lines. Those fields are not permanent. When one destabilises, everything it was holding has nowhere to go but down. This is a failed eruption coronal rain on a catastrophic scale. Millions of tonnes of plasma falling back at three hundred kilometres a second, hammering the surface it came from, each impact throwing up fountains that fall again. The Sun rained on itself. Fire is only supposed to rise. On a star, it doesn’t get the choice.show more

I've updated my blog post to walk through the... remaining technical details of our Surface Winding Numbers algorithm: now the calculus of the algorithm is explained a bit more in detail. The post, paper, code, etc. is all here:show more

The Gooner Purge is but the surface of a... system that has long spiralled out of our controlshow more

A pair of oarfish caught on video swimming at... the surface, these are fish that usually stay thousands of feet below the surface. Legend says they can predict when an earthquake is about to happen and move closer to the surface.show more

Generate a consistent series of vector illustrations while staying... in full control of every detail. Every QuiverAI output is a clean, editable SVG that scales without limits.show more

Tarp is on the field at Wrigley. The Crosstown... Classic is heading into a delay with a tie ballgame in the bottom of the 8th.show more