Can the gradient exist for a function of n+1 variables?
For a function of variables can a gradient exist?
When I asked my professor this during class he said, "no, at most a gradient will exist for a function of three variables f(x,y,z) because there are only at most three standard basis vectors with which to represent a vector."
This is a calculus 3 class so perhaps this answer was given to keep the concept of the gradient within the scope of the class, but I suspect this isn't the whole story and there is more to this than my professor is telling.
The definition of the gradient for a function of two variables given during class was: Let z=f(x,y) be a function, then the gradient of f is defined as