Can you take the derivative of a function? Meet the calculus of variations.
In short: In standard calculus, you derive numbers. In the calculus of variations, you derive functions. This MIT lecture explains how functionals map functions to numbers, allowing us to treat functions as vectors and apply derivative concepts in machine learning and beyond.
A functional, denoted as F[u], acts as a mathematical operator that takes an entire function u(x) as its input and returns a single numerical value, often through the process of integration. Because functions exist within normed vector spaces, they can be treated with the same rigor as traditional vectors.
This framework, known as the calculus of variations, enables the definition of derivatives F' with respect to functions. This approach is a core component of the MIT 18.S096 course, Matrix Calculus For Machine Learning And Beyond, taught by Alan Edelman and Steven G. Johnson during the 2023 IAP session.
Opens the Intellect+ feed with this card first. Swipe up for the next one.
Sources
- Summary by Intellect · source: Lecture 6 Part 2: Calculus of Variations and Gradients of Functionals
To cite this card, link to https://intellectplus.co.uk/app/ideas-lecture-6-part-2-calculus-of-variations-and-gradients-of-fun.
How we source, check and correct cards: Editorial standards.
General information, not professional advice. Spotted an error? Flag the card or see corrections and complaints.
Intellect Plus is a visual educational discovery platform for exploring fascinating facts and ideas. Brain Food, Not Brain Rot. What is Intellect Plus?
Related
More in Mathematics · More in Ideas & Philosophy · Latest · A–Z · All topics
