Does the System Define a Manifold and How to Find Tangent and Normal Spaces?

[tomelwood]

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.
$$f=x^{2}-y^{2}+xyz^{2}-11=0 g=x^{3}+y^{3}+z^{3}-xyz-30=0$$

2. Determine the absolute extremes of $$f(x,y,z) = x^{2}+y^{2}+z^{2}+2y-2$$ 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 $$\textbf{R}^{3}$$)
So the matrix is [tex]$$\left($$\stackrel{8 -1 6}{25 9 -6}$$\right)$$[/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

 

[mighty2000]

Homework Solution


Firstly, don't worry too much about not understanding Differential Manifolds at this stage. It can be a difficult concept to grasp, but with practice and guidance, I'm sure you will understand it better.

For the first question, you are correct in your approach to determine if the system defines a manifold. The gradient matrix is correct and has rank 2, which means that it is a 1-manifold. To find the tangent space, you can use the two equations you obtained by setting the gradient matrix multiplied by the column vector equal to 0. These two equations represent the two tangent vectors at the point (3,2,1). To find the normal space, you can use the fact that the normal vector is perpendicular to both tangent vectors. So, you can solve the system of equations formed by setting the dot product of the normal vector with each tangent vector equal to 0. This will give you a line, which represents the normal space at the point (3,2,1).

For the second question, you are correct in considering the interior and frontier separately. To find the absolute extremes, you can use the method of Lagrange multipliers. This involves finding the critical points of the function f(x,y,z) on the manifold A, and then evaluating the function at these points to determine the absolute extremes.

I hope this helps. Good luck on your exam! Remember to keep practicing and seeking clarification when needed. Science can be challenging, but with determination and effort, you can overcome any difficulties. Best of luck!