Consider two generic Morse functions ${\displaystyle f,g:M\to \mathbb {R} }$  defined on a smooth ${\displaystyle d}$ -manifold. Let the restriction of ${\displaystyle f}$  to the level set ${\displaystyle g^{-1}(t)}$  for ${\displaystyle t\in \mathbb {R} }$  a regular value, be called ${\displaystyle f_{t}:g^{-1}(t)\to \mathbb {R} }$ ; it is a Morse function. Then the Jacobi set ${\displaystyle J}$  of ${\displaystyle f}$  and ${\displaystyle g}$  is ${\displaystyle J=cl{\{x\in M\mid x{\mbox{ is critical point of }}f_{t}\}}}$ ,

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 ${\displaystyle \lambda \in \mathbb {R} }$ , ${\displaystyle J=\{x\in M\mid \nabla {f(x)}+\lambda \nabla {g(x)}=0{\mbox{ or }}\lambda \nabla {f(x)}+\nabla {g(x)}=0\}.}$ 

Equivalently, the Jacobi set can be described as the collection of critical points of the family of functions ${\displaystyle f+\lambda g}$ , for some ${\displaystyle \lambda \in \mathbb {R} }$ , ${\displaystyle J=\{x\in M\mid x{\mbox{ is a critical point of }}f+\lambda g{\mbox{ or }}\lambda f+g\}.}$