L24 Math 233 Lecture Notes - Lecture 19: Tangent Space, Directional Derivative, Unit Vector
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L24 Math 233 Lecture Notes - Lecture 18: Directional Derivative, Differentiable Function, Unit Vector
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L24 Math 233 Lecture Notes - Lecture 19: Tangent Space, Directional Derivative, Unit Vector
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L24 Math 233 Lecture Notes - Lecture 20: Talking Lifestyle 1278, Implicit Function, Partial Derivative
Document Summary
Math 233 lecture 19 reason behind how to find maximum directional. Why is the directional derivative maximized in the direction of the gradient of f ?: say f(x,y,z), x(t), y(t), z(t) are all differentiable. Pick a curve, c, on the surface, defined by r(t) = < x(t), y(t), z(t)> Then r"(t) is the tangent vector to c at p. Pick a point p(x0,y0,z0), x0 = x(t0), y0 = y(t0), z0 = z(t0) On c, f is a function of t: let"s compute df/dt, method 1. On c, f is the constant function k df/dt = 0: method 2 (we are getting the same answer, but it can give us some insight) Using the chain rule, we get df/dt = f/ x*dx/dt+ f/ y*dy/dt+ f/ z*dz/dt. = f r"(t: we can conclude: f r"(t) = 0. F(x0,y0,z0) is perpendicular to r"(t0), thus perpendicular to c at p. Since c was chosen arbitrarily, we see that f(x0,y0,z0) is perpendicular to every.
