Tangent Space for a Function

The tangent space for a function at is

The equation represents a first-order Taylor approximation of the function around .

One-Dimensional Case

For a one-dimensional function , the tangent space reduces to a tangent line. The equation of the tangent line at a point is:

Two-Dimensional Case

For a two-dimensional function , the tangent space becomes a tangent plane. The equation of the tangent plane at a point is: tangent_plane.png

Tangent Space for Implicitly Surface

See: implicit differentiation

For an implicit defined surface at a point , the tangent is

The intuition for the above equation is that the gradient is the normal for implicit surfaces:

Three-Dimensional Case

In 3D, The tangent plane of surface at a point is

See also