The Following is from Fourier's paper in which he says:
It is easy now to generalize this result and to recognize that it exists in every case of the varied movement of heat expressed by the equation $v = F(x, y, z, t)$.
Let us in fact denote by $x', y', z'$, the co-ordinates of this point $m$, and its actual temperature by $v'$ . Let $x'+ξ,y'+η,z'+ζ$, be the co-ordinates of a point $μ$ infinitely near to the point m, and whose temperature is $w$; $ξ, η, ζ$ are quantities infinitely small added to the co-ordinates $x', y', z'$ ; they determine the position of molecules infinitely near to the point $m$, with respect to three rectangular axes, whose origin is at $m$, parallel to the axes of $x, y,$ and $z$.
Differentiating the equation
$v = F(x, y, z, t)$
and replacing the differentials by $ξ, η, ζ$, we shall have, to express the value of $w$ which is equivalent to $v+dv$, the linear equation $w = v'+$$\frac{du'}{dx}ξ+$$\frac{du'}{dy}η+$$\frac{du'}{dz}ζ$;
the coefficients $u'$,$\frac{du'}{dx}$,$\frac{du'}{dy}$,$\frac{du'}{dz}$ are functions of $x, y, z, t$, in which the given and constant values $x', y', z'$, which belong to the point $m$, have been substituted for $x, y, z$.
My question has to do with his differentiation. While he has a function with four variables he differentiate it only for the three of them $x,y,z$. It is correct? Why?
You can find the paper here: page 80