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What direction should you travel to increase your height on a mountain as fast as possible? What direction should you travel to keep your height constant (i.e. travel on a contour aka a level curve)? In this video we discuss the math of this problem, assuming we had some nice function describing the height of the mountain. The gradiant vector, whose components are the respective partial derivatives gives us the answer to the direction of maximal increase. Indeed, we saw previously how directional derivatives could be written in terms of the gradient vector. At the end we look at all of this with an actual topographical map full of contours. ♡♡♡SUPPORT THE CHANNEL♡♡♡ ►Support on PATREON: / drtrefor ►MATH BOOKS I LOVE (affiliate link): https://www.amazon.com/shop/treforbazett ►CURIOSITY BOX: https://www.CuriosityBox.com/DrTrefor use CODE drtrefor for 25% off awesome STEM merch boxes 0:00 The Mountain Problem 2:24 Deriving the Gradient Formula 6:07 Directional Derivatives 10:54 Topographical Maps COURSE PLAYLISTS: ►DISCRETE MATH: • Discrete Math (Full Course: Sets, Logic, P... ►LINEAR ALGEBRA: • Linear Algebra (Full Course) ►CALCULUS I: • Calculus I (Limits, Derivative, Integrals)... ►CALCULUS II: • Calculus II (Integration Methods, Series, ... ►MULTIVARIABLE CALCULUS (Calc III): • Calculus III: Multivariable Calculus (Vect... ►VECTOR CALCULUS (Calc IV): • Calculus IV: Vector Calculus (Line Integra... ►DIFFERENTIAL EQUATIONS: • Ordinary Differential Equations (ODEs) ►LAPLACE TRANSFORM: • Laplace Transforms and Solving ODEs ►GAME THEORY: • Game Theory OTHER PLAYLISTS: ►Cool Math Series: • Cool Math Series ►Learning Math Series: • 5 Tips To Make Math Practice Problems Actu... ►LaTeX: • Intro to LaTeX **Full Tutorial** Part II ... SOCIALS: ►X/Twitter: http://X.com/treforbazett ►TikTok: / drtrefor ►Instagram (photography based): / treforphotography