Implicit functions, differentials, and their geometric interpretation

Consider the following equation:

Geometrically, this equation represents a sphere centered at with the radius . The technical term for this is implicit surface because the relationship between , , and is defined implicitly.

The surface above can be also defined as:

This equation has the following geometric interpretation.

Consider a tangent plane touching the implicitly defined surface at some point . The differential equation states that for any point on that plane, the following must hold:

or more concisely,

All such planes turn out to be tangent planes to spheres centered at , and the initial condition allows us to select the "correct" sphere of radius .

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