Illustration of the General
Implicit Function Theorem

Here we use the following example to illustrate the content of the general implicit function theorem.

The image above shows the solution manifold of the first equation (green) and of the second equation (blue). The simultaneous solutions of both equations form the intersection (red) of the two surfaces. If the gradient vectors of f ₁ (green vectors) and f ₂ (yellow vectors) are not parallel on this intersection, then the intersection is itself a manifold - in this case of dimension 1.

Additional comment The level set of f₂ has a vertex at the origin. So the level set has no tangent plane there. This can only happen if the gradient vector f₂(0,0,0) is either (0,0,0) or is undefined (as in this particular case).

Return to the main document on the implicit function theorem.

Illustration of the General Implicit Function Theorem

Illustration of the General
Implicit Function Theorem