which gets when one moves from to another point of , can be split into two parts as follows:
Here, and is a quantity tending to 0 along with .
The former part of is called the (total) differential or the exact differential of the function in the point and it is denoted by of briefly . In the special case , we see that and thus ; similarly and . Accordingly, we obtain for the general case the more consistent notation
where may be thought as independent variables.
We now assume conversely that the increment of a function in can be split into two parts as follows:
where the coefficients are independent on the quantities and are as in the above theorem. Then one can infer that the partial derivatives exist in the point and have the values , respectively. In fact, if we choose , then whence (3) attains the form
Similarly we see the values of and .
The last consideration showed the uniqueness of the total differential.
Definition. A function in , satisfying the conditions of the above theorem is said to be differentiable in the point .
Remark. The differentiability of a function of two variables in the point means that the surface has a tangent plane in this point.
- 1 Ernst Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset II. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1932).
|Date of creation||2013-03-22 19:11:24|
|Last modified on||2013-03-22 19:11:24|
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