Understanding the Math
To understand divergence's mathematical manifestation, first consider a differentiable vector function v (x, y, z) where x, y and z are Cartesian coordinates. Further, let v1, v2 and v3 be the components of v. The divergence of a vector field is the dot product between the divergence operator and the vector field function. The formula for divergence of the vector field v can therefore be defined as:
div v = (∂v1/∂x) + (∂v2/∂y) + (∂v3/∂z)
Divergence can be understood as the partial derivative of each component with respect to its Cartesian coordinate plane. Dot products yield scalar solutions. The divergence operator therefore yields a scalar solution from a vector field, suggesting div v to be a directionless magnitude indication.
One Major Assumption
The basic concept underlying divergence makes one large assumption, that in a function characterizing a physical or geometrical property, values are independent of the particular choice of coordinates. In fact, this is the case. The outward flux is assumed to be moving away from the source with relative uniformity. Divergence can be understood as a qualitative rate for this flux or flow.
Invariance of the Divergence
Values for div v depend upon the points in space and the associated mathematical function. Values are invariant with respect to coordinate transformation. Selecting a different choice for the Cartesian coordinates x*, y* and z* and corresponding components v1*, v2* and v3* for function v will result in the same equation. This invariance of the divergence remains an essential theorem associated with this particular operator.
With respect to any other coordinates in the vector field and their corresponding function components, the divergence calculation remains the same: The divergence is the dot product between the operator and the vector field, or the partial derivative of each component with respect to its Cartesian coordinate plane.
Taken to the Next Level
Divergence plays a major role in advanced calculus. The operation underlies one of the "big" integral theorems, which can be used to transform incredibly complex calculations into more reasonable problems. This procedure is known as the Divergence Theorem of Gauss.
Imagine a closed bounded region in space, called T, with a piecewise smooth surface S for its boundary. Suppose n is the outer unit normal vector of the surface S. Let the vector function F(x, y, z) both be continuous and have continuous first partial derivatives in some domain containing T. The Divergence Theorem of Gauss states the triple integral of the divergence of F over a volume can be equated to the double integral of the dot product between F and n over an area. Thus, complex volume integrals can be transformed into more manageable surface integrals through an understanding and extrapolation of the divergence of a vector field.