The Implicit Function Theorem is discussed and proved using the local linear space of differentials. Here we discuss implicit functions defined by a function on R^3. This theorem is useful in the theory of Manifolds. It allows sub-manifolds to be defined and allows lower dimensional manifolds to be embedded in the original manifold M3. The equation of state in Thermodynamics defines an implicit function and a sub-manifold of a Thermodynamic state space. A simplified notation for partial derivatives is introduced and the three dimensional Taylor expansion formula and the implicit function formula are written in this new notation.
