The gradient is just the jacobian matrix correct
also, I went to a couple differential geometry talks before and they were referring to nabla notation, were they referring to the gradient?
The gradient of a function of two variables is the vector . I'm not sure I would call that the "Jacobian matrix". The "nabla" is the operator which can be thought of a kind of "symbolic vector": or, in three dimensions .
The gradient can be thought of a product of that "vector" with a scalar function: .
But you can also take the dot product of the "vector" operator and a vector valued function: . That is the "divergence of the vector valued function" or .
Or you can take the cross product of the "vector" operator and a vector valued function: .
That is the "curl of the vector valued function" or .