total differential (original) (raw)

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 ℝ3, the function f:S→ℝ is continuous and has partial derivatives fx′,fy′,fz′ in S and the partial derivatives are continuous in a point(x,y,z) of S. Then the increment

which f gets when one moves from (x,y,z) to another point(x+Δ⁢x,y+Δ⁢y,z+Δ⁢z) of S, can be split into two parts as follows:

Here, ϱ:=Δ⁢x2+Δ⁢y2+Δ⁢z2 and ⟨ϱ⟩ is a quantity tending to 0 along with ϱ.

The former part of Δ⁢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 d⁢f⁢(x,y,z) of briefly d⁢f. In the special case f⁢(x,y,z)≡x, we see that d⁢f=Δ⁢x and thus Δ⁢x=d⁢x; similarly Δ⁢y=d⁢y and Δ⁢z=d⁢z. Accordingly, we obtain for the general case the more consistent notation

where d⁢x,d⁢y,d⁢z may be thought as independent variables.

We now assume conversely that the increment of a function f in ℝ3 can be split into two parts as follows:

where the coefficients A,B,C are independent on the quantities Δ⁢x,Δ⁢y,Δ⁢z andϱ,⟨ϱ⟩ are as in the above theorem. Then one can infer that the partial derivativesfx′,fy′,fz′ exist in the point (x,y,z) and have the values A,B,C, respectively. In fact, if we choose Δ⁢y=Δ⁢z=0, then ϱ=|Δ⁢x| whence (3) attains the form

and therefore

Similarly we see the values of fy′ and fz′.

Definition. A function f in ℝ3, 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.

References

• 1 Ernst Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset II. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1932).