代微积拾级：修订间差异

{{from|1=A<span style{{=}}"font-size: 45%0.6rem;>RTICLE</span> (157.) I<span style{{=}}"font-size: 45%0.6rem;>N</span> the Differential Calculus, as in Analytical Geometry, there are two classes of quantities to be considered, viz., ''variables'' and ''constants''.<br><br>Variable quantities are generally represented by the last letters of the alphabet, ''x'', ''y'', ''z'', etc., and any values may be assigned to them which will satisfy the equations into which they enter.<br><br>Constant quantities are generally represented by the firstletters of the alphabet, ''a'', ''b'', ''c'', etc., and these always retain the same values throughout the same investigation.<br><br>Thus, in the equation of a straight line, <br><div style="text-align: center;"><math>y=ax+b</math>,</div>the quantities a and b have but one value for the same line, while ''x'' and ''y'' vary in value for every point of the line.<br><br>(158.) '''One variable is said to be a ''function'' of another variable, when the first is equal to a certain algebraic expression containing the second.''' Thus, in the equation of a straight line <br><div style="text-align: center;"><math>y=ax+b</math>,</div>''y'' is a function of ''x''. <br><br>So, also, in the equation of a circle, <br><div style="text-align: center;"><math>y=\sqrt{R^2-x^2}</math>;</div>and in the equation of the ellipse, <br><div style="text-align: center;"><math>y=\frac{B}{A}\sqrt{2Ax-x^2}</math>.</div>(159.) When we wish merely to denote that ''y'' is ''dependent'' upon ''x'' for its value, without giving the particular expression which shows the value of ''x'', we employ the notation<br><div style="text-align: center;"><math>y=F(x)</math>, or <math>y=f(x)</math>,<br><span style{{=}}"text-align: left; float: left;">or</span><math>x=F(y)</math>, or <math>x=f(y)</math>,</div>which expressions are read, ''y'' is a function of ''x'', or ''x'' is a function of ''y''.|2=''Elements of Analytical Geometry and of The Differential and Integral Calculus'', Differential Calculus, Section I, Definitions and First Principles}}

{{from|1=(170.) '''The rate of variation of a function or of any variable quantity is called its ''differential''''', and is denoted by the letter ''d'' placed before it. Thus, if<br><div style="text-align: center;"><math>u=x^3</math>,</div>then<br><div style="text-align: center;"><math>dx:du::1:3x^2</math>.</div>The expressions ''dx'', ''du'' are read differential of ''x'', differential of ''u'', and denote the rates of variation of ''x'' and ''u''.<br><br>If we multiply together the extremes and the means of the preceding proportion, we have<br><div style="text-align: center;"><math>du=3x^2dx</math>,</div>which signifies that the rate of increase of the function ''u'' is 3''x''² times that of the variable ''x''.<br><br>If we divide each member of the last equation by ''dx'', we have<br><div style="text-align: center;"><math>\frac{du}{dx}=3x^2</math>,</div>which expresses the ratio of the rate of variation of the function to that of the independent variable, and is called the ''differential coefficient'' of ''u'' regarded as a function of ''x''.|2=''Elements of Analytical Geometry and of The Differential and Integral Calculus'', Differential Calculus, Section I, Differentiation of Algebraic Functions, Proposition II － Theorem}}

{{from|1=A<span style{{=}}"font-size: 70%;>RTICLE</span> (291.) T<span style{{=}}"font-size: 70%;>HE</span> Integral Calculus is the reverse of the Differential Calculus, its object being to determine the expression or function from which a given differential has been derived.<br><br>Thus we have found that the differential of ''x''² is 2''xdx'', therefore, if we have given 2''xdx'', we know that it must have been derived from ''x'', or plus a constant term.<br><br>(292.) '''The function from which the given differential has been derived, is called its ''integral''.''' Hence, as we are not certain whether the integral has a constant quantity or not added to it, we annex a constant quantity represented by C, the value of which is to be determined from the nature of the problem. <br><br>(293.) Leibnitz considered the differentials of functions as indefinitely small differences, and the sum of these indefinitely small differences he regarded as making up the function; hence the letter S was placed before the differential to show that the sum was to be taken. As it was frequently required to place S before a compound expression, it was elongated into the sign ''∫'', which, being placed before a differential, denotes that its integral is to be taken. Thus,<br><div style="text-align: center;"><math>\int 2xdx = x^2+C</math>.</div>This sign ''∫'' is still retained ever by those whu reject the philosophy of Leibnitz. |2=''Elements of Analytical Geometry and of The Differential and Integral Calculus'', Integral Calculus, Section I, Integration of Monomial Differentials}}