Computing tangent spaces of implicitly defined manifolds

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  • #1
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Hi there,

Is there an "easy" way to find a tangent space at a specific point to an implicitly defined manifold? I am thinking of a manifold defined by all points x in R^k satisfying f(x) = c for some c in R^m. Sometimes I can find an explicit parametrization and compute the Jacobian matrix, sometimes I can compute the normal vector to the manifold (when c is just a real number), but that's where I am running out of ideas. I am hoping that there might be some sort of implicit differentiation trick that I have not figured out yet.
 

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  • #2
709
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Hi there,

Is there an "easy" way to find a tangent space at a specific point to an implicitly defined manifold? I am thinking of a manifold defined by all points x in R^k satisfying f(x) = c for some c in R^m. Sometimes I can find an explicit parametrization and compute the Jacobian matrix, sometimes I can compute the normal vector to the manifold (when c is just a real number), but that's where I am running out of ideas. I am hoping that there might be some sort of implicit differentiation trick that I have not figured out yet.
i am not sure if I am telling you something that you already know but the gradient of f is perpendicular to the tangent space of f(x) = c. So the equation for it is gradf(x).v = 0
 

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