Feb16-12

Feb16-12 - 2. Scalar Functions of Several Variables...

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Math 260, Spring 2012 Jerry L. Kazdan Class Outline: Feb. 16, 2012 1. Curves in Space We will consider the motion of a particle in R 3 , say its position is given by the vector function X ( t ). a) Concepts: The derivative, tangent vector, velocity vector, speed. b) Derivative of inner product: h ( t ) := h X ( t ) , Y ( t ) i . Special case: derivative of k X ( t ) k 2 . c) Arc Length of curve X ( t ), a t b : Arc Length = Z b a k X 0 ( t ) k dt. Special cases: Curve y = f ( x ) in R 2 , polar coordinates in R 2 . d) Skip (alas): curvature of curves. This has many important aspects that I find beautiful. Reading: Marsden and Tromba, Vector Calculus , Chapter 1, Chapter 2.4, Chapter 4.1 and 4.2.
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Unformatted text preview: 2. Scalar Functions of Several Variables Reading: Marsden and Tromba, Vector Calculus , Chapter 2. a) Pictures: plane, paraboloid, hyperboloid in R 3 . b) Continuity c) 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 denes a plane z = h b, X i in R 3 Quadratic polynomial g ( X ) := h X, AX i d) Best Linear Approximation [Best Ane Approximation] e) The Gradient [Last revised: February 16, 2012] 1...
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