1. The problem statement, all variables and given/known data I am working in "Intro to PDEs with Applications" on page 6. Gradients come up in discussions of surfaces expressed as F(x,y,z). In discussing such matters, the buildup includes the assumption that grad F is not equal to the zero vector. A later line reads, "Under the assumption [above], the set of points (x,y,z) in the domain which satisfy the equation F(x,y,z) = c for some appropriate value of c, is a surface is the domain. 2. Relevant equations 3. The attempt at a solution My question is, what precautions does the gradient not being zero entail? The only answer I've been able to come up with so far is that they mean smooth surface when they simply say surface (since points where grad F = 0) seem to correspond to 'corners' or non-differentiate points. Any deeper or more accurate insight?