Total differential operator - intuitive interpretation

Question

I am having doubt regarding the total differential operator.

Let us say function Φ be a function of variables x,y,z which are dependent on t i.e.

$$ Φ=(x(t),y(t),z(t)) $$

I have read about the definition of total differential operator and that it seems acceptable (We need to see change in Φ w.r.t. to t, so we calculate independent changes in x,y,z and sum it up). $$ dΦ=\frac{\partial Φ}{\partial x}\frac{dx}{dt}+\frac{\partial Φ}{\partial y}\frac{dy}{dt}+\frac{\partial Φ}{\partial z}\frac{dz}{dt} $$

But , do we have something that is more intuitive to explain this operator. Let us assume that x,y,z represent independent directions (in terms of say a vector space). Calculating the individual derivatives and adding them up doesn't provide me any additional direction?

Any other physical interpretation is appreciated from the community.

Answer 1

You've just written out the regular old derivative, $\frac{d}{dt}\Phi(x(t), y(t), z(t))$.

$\Phi$ has a one-dimensional output, which is why all three terms "lie in the same dimension" and can be added meaningfully.