Why is the total differential of a function just the sum of the components?

Question

In a multivariable calculus class, I was taught if $f$ is a function of $x, y, z$, the total differential of the function is: $$df = f_xdx + f_ydy + f_zdz$$ Intuitively, it seems that there are missing terms to me. For example, shouldn't there be some sort of interactive term to account for how $x$ and $y$ change together? Or do these terms just "go away" under a technical derivation? Or maybe these terms or never there to begin with? Any help is appreciated!

Answer 1

The total differential of $f$ at a point $p$, denoted by $df_p$, is the linear map best approximating $f$ around $p$, meaning that $$f(p+h) = f(p) + df_p(h) + o(\|h\|).$$ This implies that $df_p(h)$ must be the directional derivative of $f$ in direction $h$, and you get your definition (remembering that for example $dx$ is the linear map giving $1$ when eating the unit vector in direction $x$, and $x$ on the unit vectors in direction $y$ and $z$). The mixed terms you mention indeed appear in the higher derivatives.

Answer 2

The "interaction" terms are higher order in $\Delta x $ and $\Delta y $. For example $(x+h)(y+k)-xy=xk+yh+hk $; for small k,h and nonzero x,y the last term is much smaller than the first two, enough to vanish to first order (i.e in any directional derivative).