Mar1-12 - Math 260 Spring 2012 Jerry L Kazdan Class Outline...

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Math 260, Spring 2012 Jerry L. Kazdan Class Outline: March 1, 2012 Functions of Several Variables Reading: Marsden and Tromba, Vector Calculus , Chapter 2, Chapter 3.1–3.4 and Sections 8.1-8.2 in the notes http://www.math.upenn.edu/ kazdan/260S12/notes/math21/math21-2012-2up.pdf Exam. 2 will be on Tuesday, March 13 during class. As usual, it will be closed book, no calculators or cell phones etc., but you may use one 3 × 5 card with notes on both sides. The Exam will cover the mater covered through class on Tuesday, Feb. 28, with emphasis on the material since Exam. 1. 1. The Chain Rule a) Simplest Case Say you have a function f ( X ) := f ( x,y,z ) which might givde the temperature at a point X = ( x,y,z ) in R 3 . If we have a curve X ( t ) = ( x ( t ) ,y ( t ) ,z ( t )), then h ( t ) := f ( X ( t )) gives the temperature at points of the curve. We want to compute dh/dt . So we need the chain rule . For h ( t ) = f ( x ( t ) ,y ( t ) ,z ( t )) it states dh ( t ) dt = ∂f ∂x dx dt + ∂f ∂y dy dt + ∂f ∂z dz dt = f ( X ( t )) · X ( t ) . Example f ( X ) := x 2 y + e 2 yz and the curve is X ( t ) = (cos t, sin t, 2 t ). If h ( t ) := f ( X ( t )) then f ( X ) = (2 xy,x 2 + 2 ze 2 yz , 2 ye 2 yz ) and X ( t ) = ( sin t, cos t, 1) . Thus, for instance at t = 0, X (0) = (1 , 0 , 2), X (0) = (0 , 1 , 1) and f ( X (0)) = (0 , 5 , 0) so h (0) = 5. Application Say we have a surface in R 3 on which f ( X ) is constant, f ( x,y,z ) = c . This is called a level surface of f (if f ( X ) specifies the temperature, this surface is often called an isotherm ). Let X ( t ) be a smooth curve on this surface, so f ( X ( t )) = c . Since c is a constant, if we take the derivative of this we find Theorem On any level surface, f ( X ) · X ( t ) = 0 so the vector f ( X ) is orthogonal to the surface . Proof Since X ( t ) is a curve on the surface, its tangent vector, X ( t ) is tan- gent to the surface. But f ( X ) · X ( t ) = 0 means that f ( X ) is orthogonal to all these tangent vectors and hence to the surface. Example Find the tangent plane to the surface z = x 2 + y 3 at the point X 0 := ( 1 , 1 , 2).
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