Jacobi set

In Morse theory, a mathematical discipline, Jacobi sets provide a method of studying the relationship between two or more Morse functions.
For two Morse functions, the Jacobi set is defined as the set of critical points of the restriction of one function to the level sets of the other function.
The Jacobi set can also be defined as the set of points where the gradients of the two functions are parallel. 
If both the functions are generic, the Jacobi set is a smoothly embedded 1-manifold.
Definition.
Consider two generic Morse functions formula_1 defined on a smooth formula_2-manifold. Let the restriction of formula_3 to the level set formula_4 for formula_5 a regular value, be called formula_6; it is a Morse function. Then the Jacobi set formula_7 of formula_8 and formula_9 is formula_10,
Alternatively, the Jacobi set is the collection of points where the gradients of the functions align with each other or one of the gradients vanish (cite?), for some formula_11,
formula_12
Equivalently, the Jacobi set can be described as the collection of critical points of the family of functions formula_13, for some formula_11,
formula_15