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.
- Spring '08