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Calculus and Beyond Homework Help
Tangent and Normal Spaces, lagrange multiplier and Differentiable Manifolds question.
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[QUOTE="tomelwood, post: 3104104, member: 284364"] [h2]Homework Statement [/h2] OK I have a Differential Calculus exam next week and I do not understand about Differential Manifolds. We have been given some questions to practise, but I have no idea how to do them, past a certain point. For example 1. Study if the following system defines a manifold around (3,2,1). If so, calculate the tangent and normal spaces. [tex]f=x^{2}-y^{2}+xyz^{2}-11=0 g=x^{3}+y^{3}+z^{3}-xyz-30=0[/tex] 2. Determine the absolute extremes of [tex]f(x,y,z) = x^{2}+y^{2}+z^{2}+2y-2 [/tex] on the manifold [tex] A = \left\{(x,y,z) : 4x^{2}+2y^{2}+z^{2}-8\leq 0\right\} [h2]Homework Equations[/h2] [h2]The Attempt at a Solution[/h2] OK. To see if it defines a manifold, I take the gradient (in this case a 2x3 matriz) and evaluate it at the point, and if it has maximum rank=k, then it is a manifold of dimension 3-k (as we are in [tex]\textbf{R}^{3}[/tex]) So the matrix is [tex][tex]\left([/tex]\stackrel{8 -1 6}{25 9 -6}[tex]\right)[/tex][/tex] I believe. This has rank 2, maximal, so we have a 1-manifold. Now I was under the impression that to find the tangent space, you just multiply the gradient by the column vector (x,y,z) and set equal to 0. However this gives me 2 equations. Is this ok? What should I do? And about the normal space, I know it should be a line, since the tangent space is a plane, but I don't know how to do this, except that it's something to do with being parallel to the gradient, which is a 2x3 matrix, which is where I'm stuck at the moment. 2. I think I have to consider the interior and the frontier differently, but not entirely sure how to do it. Any pointers here at all would be great. Many Thanks [/QUOTE]
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Tangent and Normal Spaces, lagrange multiplier and Differentiable Manifolds question.
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