Question about total differentials

Question

Why is it true that if you move from point $(u,v,w)$ to point $(u+du,v+dv,w+dw)$, a scalar function $\phi(u,v,w)$ changes by an amount $$d\phi=\frac{\partial \phi}{\partial u}du+\frac{\partial \phi}{\partial v}dv+\frac{\partial \phi}{\partial w}dw$$

Answer 1

This is true up to first order. That is, let $p = (x,y,z)$ and let $h = (h_1,h_2,h_3)$ be a small perturbation. Then we have the Taylor expansion $$\phi(p+h) = \phi(p) +\nabla\phi(p)\cdot h +\text{ terms of higher order } $$

where the terms of higher order involve monomials of degrees $>1$ in $h_1,h_2,h_3$.