math4250homework4 - v In fact there is a special vector ∇...

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Math 4250/6250 Homework #4 This homework assignment covers our notes on regular surfaces (7-8), tangent planes (9) and the first fundamental form (10). Please pick 5 of the following problems. Remember that under- graduate students should average one challenge problem per assignment, while graduate students should average two challenge problems per assignment. Everyone should complete problem #1 on page 99 as one of your five problems. This assignment will be due on March 22. 1. R EGULAR P ROBLEMS Do Carmo, p 67. #16, p 88. #2, #5, #10. p 99. #1. 2. C HALLENGE P ROBLEMS Do Carmo, p 99. #3, #11, #12, #14. #14 is a particularly important problem in understanding what the theory of differential geometry is for . We might remember that a function f ( x ) on R n has a directional derivative in any direction v given by the limit Df ( v ) = lim h 0 f ( x + hv ) - f ( x ) h . We learn in multivariable calculus that this directional derivative is a linear function of v , and so that Df is a linear functional on the space of direction vectors
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Unformatted text preview: v . In fact, there is a special vector ∇ f = ( ∂f/∂x 1 , . . . , ∂f/∂x n ) so that Df ( v ) = h v, ∇ f i . If the function f ( x ) is defined on a curved surface S , we still want to be able to understand what it means to differentiate the function. In fact, the directional derivative at p is a linear function of directions in the tangent plane T p S . This linear functional is now written as Df ( v ) = I p ( v, ∇ f ) . But what is the formula for ∇ f ? It is important to know it. But it is clearly not as simple as it used to be for functions defined on R n . Finding a formula for ∇ f will require us to understand the first fundamental form in some detail, using the theory we’ve developed. This leads us to an answer to our question “What is differential geometry for?”: Differential geometry tells you how to do calculus on a curved surface. 1...
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This note was uploaded on 03/12/2012 for the course MATH 4250 taught by Professor Staff during the Spring '08 term at UGA.

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