total differential

There is the generalisation of the theorem in the parent entry (http://planetmath.org/Differential) concerning the real functions of several variables; here we formulate it for three variables:

Theorem.  Suppose that $S$ is a ball in ${ℝ}^3$, the function  $f:S\to ℝ$  is continuous and has partial derivatives $f_x^{\prime },f_y^{\prime },f_z^{\prime }$ in $S$ and the partial derivatives are continuous in a point $(x,y,z)$  of $S$.  Then the increment

$$\mathrm{\Delta }f:=f(x+\mathrm{\Delta }x,y+\mathrm{\Delta }y,z+\mathrm{\Delta }z)-f(x,y,z),$$

which $f$ gets when one moves from  $(x,y,z)$  to another point $(x+\mathrm{\Delta }x,y+\mathrm{\Delta }y,z+\mathrm{\Delta }z)$  of $S$, can be split into two parts as follows:

$\mathrm{\Delta }f=[f_x^{\prime }(x,y,z)\mathrm{\Delta }x+f_y^{\prime }(x,y,z)\mathrm{\Delta }y+f_z^{\prime }(x,y,z)\mathrm{\Delta }z]+⟨\varrho ⟩\varrho .$(1)

Here,  $\varrho :=\sqrt{\mathrm{\Delta }{x}^2+\mathrm{\Delta }{y}^2+\mathrm{\Delta }{z}^2}$  and $⟨\varrho ⟩$ is a quantity tending to 0 along with $\varrho$.

The former part of $\mathrm{\Delta }x$ is called the (total) differential or the exact differential of the function $f$ in the point  $(x,y,z)$  and it is denoted by  $df(x,y,z)$  of briefly $df$.  In the special case  $f(x,y,z)\equiv x$,  we see that  $df=\mathrm{\Delta }x$  and thus  $\mathrm{\Delta }x=dx$;  similarly $\mathrm{\Delta }y=dy$  and $\mathrm{\Delta }z=dz$.  Accordingly, we obtain for the general case the more consistent notation

$df=f_x^{\prime }(x,y,z)dx+f_y^{\prime }(x,y,z)dy+f_z^{\prime }(x,y,z)dz,$(2)

where $dx,dy,dz$ may be thought as independent variables.

We now assume conversely that the increment of a function $f$ in ${ℝ}^3$ can be split into two parts as follows:

$f(x+\mathrm{\Delta }x,y+\mathrm{\Delta }y,z+\mathrm{\Delta }z)-f(x,y,z)=[A\mathrm{\Delta }x+B\mathrm{\Delta }y+C\mathrm{\Delta }z]+⟨\varrho ⟩\varrho$(3)

where the coefficients $A,B,C$ are independent on the quantities $\mathrm{\Delta }x,\mathrm{\Delta }y,\mathrm{\Delta }z$ and$\varrho ,⟨\varrho ⟩$ are as in the above theorem.  Then one can infer that the partial derivatives$f_x^{\prime },f_y^{\prime },f_z^{\prime }$ exist in the point  $(x,y,z)$  and have the values $A,B,C$, respectively.  In fact, if we choose  $\mathrm{\Delta }y=\mathrm{\Delta }z=0$, then  $\varrho =|\mathrm{\Delta }x|$  whence (3) attains the form

$$f(x+\mathrm{\Delta }x,y+\mathrm{\Delta }y,z+\mathrm{\Delta }z)-f(x,y,z)=A\mathrm{\Delta }x+⟨\mathrm{\Delta }x⟩\mathrm{\Delta }x$$

and therefore

$$A=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{f(x+\mathrm{\Delta }x,y+\mathrm{\Delta }y,z+\mathrm{\Delta }z)-f(x,y,z)}{\mathrm{\Delta }x}=f_x^{\prime }(x,y,z).$$

Similarly we see the values of $f_y^{\prime }$ and $f_z^{\prime }$.

The last consideration showed the uniqueness of the total differential.

Definition.  A function $f$ in ${ℝ}^3$, satisfying the conditions of the above theorem is said to be differentiable in the point  $(x,y,z)$.

Remark.  The differentiability of a function $f$ of two variables in the point  $(x,y)$  means that the surface  $z=f(x,y)$  has a tangent plane in this point.