If I have w(x, y, z) and take "dw = (∂w/∂x)dx + (∂w/∂y)dy + (∂w/∂z)dz"; doesn't that accurately describe how w changes as x, y and z change by an infinitesimal? Or does that only work for some special cases? I feel like if I take the total derivative I am actually describing how the function w changes with respect to an infinitesimal change in a parameter t. Or to one of the variables which all the other variables happen to depend on, not the variables x, y and z. Is that right? Also, does "dw = (∂w/∂x)dx + (∂w/∂y)dy" mean that if I go one unit in the direction of x and one unit in the direction of y I will have climbed "(∂w/∂x) + (∂w/∂y)" along the direction of w? Which is kind of like going one for x and one for y along the tangent plane?