Tangent Space and Manifold of a Cubic Surface

[atomqwerty]

Homework Statement

In which points the surface $$\{\left(x,y,z\right)\in\Re^{3}|x^{3}-y^{3}+xyz-xy=0\right\}$$ is a differentiable manifold (subvariedad diferenciable in spanish). Calculate its tangent space in the point (1,1,1).

Homework Equations

NA

The Attempt at a Solution

I've been several problems with the definition of Subvariedad - I don't know if it's said Manifols in english

Thank you

 

[hunt_mat]

Write the surface in the form z=f(x,y) and look for points where z becomes undefined.
Do they want the tangent space or tangent plane? For the latter, it is just the plane perpendicular to $$\nabla (x^{3}-y^{3}+xyz-xy)$$, fot the former it's slightly more complicated

 

[atomqwerty]

It was the space. Thank you so much :)