Critical points of a vector field are key to their characterization. Not only their positions but also their indexes are crucial for understanding vector fields. We first review the role of singularities in vector fields including index, Euler characteristic, separatrices and so forth. This is demonstrated on interactive vector field generation software.
Considerable work exists in 2D for computing singularities, but very little is available for 3D or higher dimensions. Geometric Algebra is a derivative of Clifford Algebra that not only enables a succinct definition of the index of a critical point in higher dimension; it also provides insight and computational pathways for calculating the index. We describe the 3D problems in terms of Geometric Algebra and present octtree based solution using the algebra for finding critical points and their index in a 3D vector field.