Tangent Space for a Function
The tangent space for a function
The equation represents a first-order Taylor approximation of the function around
One-Dimensional Case
For a one-dimensional function
Two-Dimensional Case
For a two-dimensional function
Example: find the tangent plane to
at
Tangent Space for Implicitly Surface
For an implicit defined surface
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
find the tangent plane of the surface
at Find all the first order partials
Putting together, we get