total differential

There is the generalisation of the theorem in the parent entry ( concerning the real functions of several variables; here we formulate it for three variables:

Theorem.  Suppose that S is a ball in 3, the functionMathworldPlanetmathf:S  is continuousMathworldPlanetmath and has partial derivativesMathworldPlanetmath 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+Δx,y+Δy,z+Δz)  of S, can be split into two parts as follows:

Δf=[fx(x,y,z)Δx+fy(x,y,z)Δy+fz(x,y,z)Δz]+ϱϱ. (1)

Here,  ϱ:=Δx2+Δy2+Δz2  and ϱ is a quantity tending to 0 along with ϱ.

The former part of Δx 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)x,  we see that  df=Δx  and thus  Δx=dx;  similarly  Δy=dy  and Δz=dz.  Accordingly, we obtain for the general case the more consistent notation

df=fx(x,y,z)dx+fy(x,y,z)dy+fz(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 3 can be split into two parts as follows:

f(x+Δx,y+Δy,z+Δz)-f(x,y,z)=[AΔx+BΔy+CΔz]+ϱϱ (3)

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


and therefore


Similarly we see the values of fy and fz.

The last consideration showed the uniqueness of the total differential.

Definition.  A function f in 3, satisfying the conditions of the above theorem is said to be differentiableMathworldPlanetmath 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.


  • 1 Ernst Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset II.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1932).
Title total differential
Canonical name TotalDifferential
Date of creation 2013-03-22 19:11:24
Last modified on 2013-03-22 19:11:24
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Definition
Classification msc 53A04
Classification msc 26B05
Classification msc 01A45
Synonym exact differential
Synonym differential
Related topic ExactDifferentialForm
Related topic ExactDifferentialEquation
Related topic Differential
Related topic DifferntiableFunction
Defines differentiable