594 VIII. ON THE STRUCTURE OF MATHEMATICS |
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as we let
n increase without limit, this sum will approach a limit, which can be found by integrating the function, that is, by finding the function F{x) of which f(x) is the derivative, and by taking the integral between the limits ; that is, by taking the difference betweenIt must be noticed that in ouf first example, the case of the indefinite integral, we considered integration as the inverse of differentiation; in the second example, we considered the definite integral as the limit of a sum.
The symbol of the integral,, had its origin in the letter
S from the latin word 'summa', the integral being historically understood as the definite integral, or the limit of a sum.Section D. Further applications.1.
PARTIAL DIFFERENTIATIONWhen we have more than one independent variable, for example, two, we have to become acquainted with what is called partial differentiation. This process is important, as in practice we usually deal with several independent variables. It presents very little that is new from a structural and methodological point of view, but we give it here, simply to explain the meaning of the term, as the reader may find it used in other works.
If we have a function
z of two independent variables which geometrically represents a surface, we may differentiate with respect to one of the variables, let us say x, and hold the other variable y fast, that is, treat it as a constant. In this way we should then have a partial derivative of z with respect to *. Similarly, if we treat * as a constant and differentiate in respect to y, we should have the partial derivative of z with respect to y. The above definitions give us the rules for partial differentiationthat is, following the ordinary rules, considering each variable individually and treating all the other variables as constant.The notation for partial derivatives is similar to the ones explained before, except that the lower case letter
d is replaced by the script form d or a subscript is used to indicate the variable with respect to which the differentiation isperformed; for instance,, . Higher derivatives are
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