The character of S depends on the eigenvalues of the matrix A:
If both eigenvalues are positive, S is bowl-shaped, opening up.
If both eigenvalues are negative, S is bowl-shaped, opening down.
If one eigenvalue is positive and the other is negative, S issaddle-shaped.
Here is a picture of S for a particular choice of a andb, together with the eigenvalues of the corresponding matrix A.This picture is a frame from a movie, showing how S changes as Avaries. Clicking on the image will bring up the movie.