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Unformatted text preview: Math 260, Spring 2012 Jerry L. Kazdan Class Outline: Feb. 21, 2012 Scalar Functions of Several Variables Reading: Marsden and Tromba, Vector Calculus , Chapter 2. 1. Directional Derivative special case: partial derivatives. Example: Linear polynomial f ( X ) := h b, X i , where b ∈ R 2 is a constant vector (not depending on X ). This defines a plane z = h b, X i in R 3 Quadratic polynomial g ( X ) := h X, AX i 2. The Gradient f ( X ) = ∇ f ( x, y ) = grad f ( x, y ) = ( f x ( x, y ) , f y ( x, y ) . Theorem The directional derivative at a point X in the direction V is ∇ V f ( X ) · V = h∇ f ( X ) , V i . In particular, if V = e is a unit vector, since h∇ f ( X ) , e i = k∇ f ( X ) kk e k cos θ = k∇ f ( X ) k cos θ, where θ is the angle between ∇ f and e , we see that at a point X the direction in which f increases fastest is ∇ f ( X ). This allows us to interpret the gradient as a vector....
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