Tangent and Normal Spaces, lagrange multiplier and Differentiable Manifolds question.

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Homework Statement


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\}

Homework Equations





The Attempt at a Solution


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
 

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