Editing Divergence (section)

===Cartesian coordinates===

In three-dimensional Cartesian coordinates, the divergence of a [[continuously differentiable]] [[vector field]] <math>\mathbf{F} = F_x\mathbf{i} + F_y\mathbf{j} + F_z\mathbf{k}</math> is defined as the [[scalar (mathematics)|scalar]]-valued function:

<math display="block">\operatorname{div} \mathbf{F} = \nabla\cdot\mathbf{F} = \left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right) \cdot (F_x,F_y,F_z) = \frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}.</math>

Although expressed in terms of coordinates, the result is invariant under [[Rotation matrix|rotations]], as the physical interpretation suggests. This is because the trace of the [[Jacobian matrix and determinant|Jacobian matrix]] of an {{math|''N''}}-dimensional vector field {{math|'''F'''}} in {{mvar|N}}-dimensional space is invariant under any invertible linear transformation{{clarification needed|date=January 2024|reason=Presumably this means the trace of L^-1 J L, where J is the original Jacobian and L is the invertible linear function R^N to R^N? As opposed to the trace of LJ? It is true that the trace is preserved under the former, but obviously not the latter, e.g. take L = 2I, I the identity matrix.}}.

The common notation for the divergence {{math|∇ · '''F'''}} is a convenient mnemonic, where the dot denotes an operation reminiscent of the [[dot product]]: take the components of the {{math|∇}} operator (see [[del]]), apply them to the corresponding components of {{math|'''F'''}}, and sum the results. Because applying an operator is different from multiplying the components, this is considered an [[abuse of notation]].