Consider the surface, S={x in R^n: h(x)=0}, where h is in C^1. A differentiable curve passing thru x* in S is {x(t): x(0)=x* and h(x(t))=0 for all t in (-e,e)}, for which the derivative, x'(t), exists, where e > 0. The tangent plane at x* is the set of all initial derivatives: {x'(0)}. (This is a misnomer, except in the special case of one func...
[n] - the plane that contains all the lines tangent to a specific point on a surface
noun the plane that contains all the lines tangent to a specific point on a surface