Unformatted text preview: immediately with di±erentiable parametrizations. Defnition 3.5.6. A vectorvalued function −→ p ( s,t ) = ( x 1 ( s,t ) ,x 2 ( s,t ) ,x 3 ( s,t )) of two real variables is diferentiable (resp. continuously diferentiable , or C 1 ) if each of the coordinate functions x j : R 2 → R is diFerentiable (resp. continuously diFerentiable). We know from Theorem 3.3.4 that a C 1 function is automatically diFerentiable. We de±ne the partial derivatives of a diFerentiable function −→ p ( s,t ) to be the vectors ∂ −→ p ∂s = p ∂x 1 ∂s , ∂x 2 ∂s , ∂x 3 ∂s P ∂ −→ p ∂t = p ∂x 1 ∂t , ∂x 2 ∂t , ∂x 3 ∂t P . We will call −→ p ( s,t ) regular if it is C 1 and at every pair of parameter values ( s,t ) in the domain of −→ p the partials are linearly independent—that...
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 Fall '08
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 Calculus, Derivative, Vectorvalued function, realvalued functions, (resp. continuously

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