Computing tangent spaces of implicitly defined manifolds

[sin123]

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.

 

[wofsy]

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