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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 dened 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 dened on R n . Finding a formula for f will require us to understand the rst fundamental form in some detail, using the theory weve 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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- Spring '08