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Unformatted text preview: immediately with dierentiable parametrizations. Defnition 3.5.6. A vector-valued 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 dene 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 independentthat...
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- Fall '08