201 Divergence
A divergence is a scalar value calculated from the derivative of a mapping of vectors that is used to calculate the change in a variable value within an open set.
Note: Also written \(\operatorname{div} \mathbf{v}\). Also called the divergence of the vector field defined by \(\mathbf{v}\).
definition [d] (Divergence = div) From Kreyszig: let \(\mathbf{v}(x,y,z) = [v_{1}, v_{2}, v_{3}] = v_{1}\,\mathbf{i} + v_{2}\,\mathbf{j} + v_{3}\,\mathbf{k}\) be a differentiable vector function, where \(x\), \(y\), \(z\) are Cartesian coordinates and \(v_{1}\), \(v_{2}\), \(v_{3}\) are the components of \(\mathbf{v}\). Then the function
- \(\displaystyle \operatorname{div} \mathbf{v} = \dfrac{\partial v_{1}}{\partial x} + \dfrac{\partial v_{2}}{\partial y} + \dfrac{\partial v_{3}}{\partial z}\)
is called the divergence of \(\mathbf{v}\) or the divergence of the vector field defined by \(\mathbf{v}\).
where
- \(\mathbf{v}\) is a differentiable vector function.
- \(v_{1}\), \(v_{2}\), \(v_{3}\) are the Cartesian components of \(\mathbf{v}\).
- \(x\), \(y\), \(z\) are Cartesian coordinates.
- \(\operatorname{div} \mathbf{v}\) is the divergence of \(\mathbf{v}\).