Elements of the Differential and Integral CalculusWiley and Putnam, 1842 - 318 páginas |
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abscissa angle asymptote ax² axis becomes co-ordinates curvature curve cycloid d'u d'u d³u deduced denominator denoted differential coefficient differential equation du du dx dx dx dy dx² dxdy dy dx dy dy dy² dy³ dz dz ellipse equal Examples exponent factor formula fraction frustrum function fXdx give given point hence increased increment infinitely small integral least common multiple logarithms maximum or minimum multiplied negative normal nx² obtain ordinate osculatory circle parabola partial differential partial fractions particular value plane positive preceding article primitive equation quantity ratio reduces result revolution second differential coefficient second member solid solid of revolution spiral Substituting these values subtangent surface surface of revolution tang tangent tion whence dx whole number
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Página 23 - ... into the differential of the denominator, divided by the square of the denominator. Let...
Página 118 - Remark. It should be observed that X is the axis about which the curve is revolved. If it were revolved about the axis Y, it would be necessary to change x into y and y into x. 295. If a right angled triangle CAB be revolved about the perpendicular...
Página 34 - TAYLOR'S THEOREM. The object of Taylor's theorem is, to show how to develop a function of the algebraic sum of two variables into a series arranged according to the ascending powers of one of the variables, with'co-efficients which are functions of the othej.
Página 3 - Whenever the relation between the co-ordinates of the points of a line can be expressed by the ordinary operations of Algebra, that is, by addition, subtraction, multiplication, division...
Página 119 - The differential of a surface of revolution, is equal to the circumference of a circle perpendicular to the axis, multiplied by the differential of the arc of the generating curve. Let the curve APP...
Página 19 - Ans. PROPOSITION IX. — THEOREM. (180.) The differential of the product of two functions dependent on the same variable, is equal to the sum of the products obtained by multiplying each by the differential of the other. Let us designate two functions by u and v, and suppose them to depend on the same variable x ; then, when x is increased so as to become x+h, the new functions may be written, Art. 178, u'=u+Kh +BA", v'=v+A.'h+B'h'.
Página 151 - Let it be required to find the equation of the evolute of the common parabola. The equation of the involute is y'=2px; whence Also, dx y...
Página 18 - The differential of the sum or difference of any number of functions dependent on the same variable, is equal to the sum or difference of their differentials taken separately.
Página 20 - ... product, is equal to the sum of the quotients obtained by dividing the differential of each function by the function itself. Ex. 1. What is the differential of zy"!
Página 63 - ... the total differential of a function of any number of variables is equal to the sum of the partial differentials.