The Fundamental Theorem of Gradients | Multivariable Calculus

To try everything Brilliant has to offer—free—for a full 30 days, visit https://brilliant.org/FoolishChemist. You’ll also get 20% off an annual premium subscription. In this video, we "derive" (or rather, intuitively explain) the formula for line integrals over vector fields and describe how to evaluate them with some examples. Then, we use that knowledge to build up to the fundamental theorem of line integrals, which tells us the the closed line integral of a gradient vector field (AKA conservative vector field) will always be zero. This turns out to just be an extension of the fundamental theorem of single-variable (1D) calculus into multiple dimensions (math is all about expanding previous knowledge into new domains, after all!). Enjoy! Chapters: 0:00 Intro 1:31 Prerequisites 1:56 Video Outline 2:21 Regular Functions, Vector Valued Functions, Vector Fields 4:22 Line Integrals over Vector Fields 12:16 Fundamental Theorem of Line Integrals Subscribe for more: https://www.youtube.com/@FoolishChemist?sub_confirmation=1 https://www.instagram.com/foolish.chemist/ https://www.tiktok.com/@foolishchemist Feel free to reach out to me at: [email protected] #maths #calculus #multivariable #multivariablecalculus #perspective #some #someπ #learn #learning #intuition #intuitive FTC Disclaimer: This video was sponsored by Brilliant