MATH 2374 Lecture Notes - Lecture 1: Directional Derivative, Tangent Space, Level Set
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Apparently this upside down triangle is named after a phoenician harp. Could be smith, maxwell, tait, or gibbs (1830s). The chain rule at a point gives the same answer for curves having the same tangent vector at the point. Let a be the unit vector in r^(n) (reals). Elsewhere, one can take directional derivatives with respect to non unit vectors. Compute g"(0) two ways to get left and right hand sides of conclusion. The direction is the unit vector in the direction of the v vector. To indicate the directional derivative of f in the direction u at x0. Suppose we want the maximum value of duf(x1,x2). Answer: cos is maximized at =0, so duf(x1,x2) is maximized if u points in the direction of gradient f. Prop:the directional derivative of f at the point x. If is a function, x in r^n (reals), then the gradient f(x) is orthogonal to the level set of f passing through x.
