amsaarel is correct in (his? her?) writeup, however, I felt that it needed to be expanded:
If you have an n+1 dimensional
manifold M of paramaters t_1, t_2, ..., t_n, then you have an n dimensional
vector space which has all the tangent vectors in it. To find the vector with greatest slope (dM, the complete differential) use the vector differential operator
del, defined as:
del:=<@/@t_1,@/@t_2,@/@t_3,...,@/@t_n>
where @/@x is the partial with respect to x operator. del(M) will yield a vector in the vector space, and by dotting del(M) with unit vectors,
directional derivatives can be found in those directions. By using a multidimensional cross product between all n vectors found, the
normal to the vector space can be found, this an equation for the vector space can be found.
I now feel tempted to write "
The proof is left as an exercise to the reader" here since it annoys me so!