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Calculus Differentialand Integral

Calculus, Differential and Integral, two of the higher branches of pure mathematics, with very far-reaching applications in all branches of exact physical science. Their introduction may be said to date from the time of Newton. They relate essentially to infinitely small quantities, and their ratios. Leibnitz came to certain of the facts of the differential calculus by the method of infinitesimals, i.e. by studying the small quantities themselves. Newton arrived at the same facts by the method of fluxions, i.e. by studying the limiting values of the ratios of these small quantities. To exemplify what is meant by infinitesimals and their ratios, we may consider a square with side of given length. The area of this square depends on the length of the side, that is to say it is a function of the side, and if the length be altered the area will alter to a definite extent. If the side is increased by a very small quantity, the area will only increase by a very small quantity; and an infinitesimal change in one corresponds to an infinitesimal change in the other. But the small increase in area is seen geometrically to be a rectangle of length, equal to twice the length of the side of the square, and of width equal to the small increment in the side. Hence the ratio of the increment of the area to the increment of the side must always be twice the length of the side when these increments are taken infinitely small. This ratio is known as the differential coefficient of the area of the square with regard to the side, and might be called the rate of change of area when the side is chosen as our independent variable quantity. So similarly we have the limiting ratio in the case of a cube with regard to its side always as three times the area of one face. For any function of any variable there is always a definite differential coefficient with regard to that variable, and this differential coefficient is known as the first derived function. It is in the province of the differential calculus to obtain such derived functions from the primitive, whereas the integral calculus supplies us with the primitive when the derived function is given. The latter is, therefore, the inverse process of the former, and requires the recognition of a derived function as corresponding to a certain primitive. To effect this recognition considerable change of form is sometimes at first necessary. Sometimes the integral cannot be solved on account of its form being entirely unlike any of the standard derived functions, and new realms in pure mathematics are opened up by the study of these new forms.