Scalar vs Vector Fields
From the book
Chapter 11: Matrix Calculus. In the chapter mind map this icon labels Scalar vs Vector Fields. The discussion below is excerpted and lightly edited from § Scalar and Vector Fields in Mathematics for AI and Machine Learning.
Before diving into matrix calculus, we introduce the fundamental concepts of scalar fields and vector fields, which are central to vector calculus and will be important in later chapters on score functions and diffusion models.
What this drawing shows
What you see. Contrasts $f:\mathbb{R}\to\mathbb{R}$ (one scalar output) with $\mathbf{F}:\mathbb{R}^d\to\mathbb{R}^d$ (d-component vector); animation sweeps input $x$ so the yellow scalar box and three vector components update differently.
In the mind map. Chapter 11 — Scalar vs Vector Fields. See From the book above for definitions, figures, and worked examples.
Where to read next
Read the full definitions, figures, and worked examples in Chapter 11: Matrix Calculus — see the mind-map node Scalar vs Vector Fields.