# A Course of Higher Mathematics. Volume I by V. I. Smirnov and A. J. Lohwater (Auth.)

By V. I. Smirnov and A. J. Lohwater (Auth.)

Similar elementary books

Understanding elementary algebra with geometry: a course for college students, 6th Edition

Hirsch and Goodman provide a mathematically sound, rigorous textual content to these teachers who think scholars may be challenged. The textual content prepares scholars for destiny research in higher-level classes by way of steadily construction scholars' self belief with no sacrificing rigor. to assist scholars flow past the "how" of algebra (computational talent) to the "why" (conceptual understanding), the authors introduce issues at an user-friendly point and go back to them at expanding degrees of complexity.

Extra info for A Course of Higher Mathematics. Volume I

Sample text

We confine ourselves to the case of the product xy of two variables. We suppose t h a t x and y vary simultaneously, tending respectively to limits a and b, and we show t h a t xy tends to the limit ab. We have by hypothesis: x = a + a, y = b + β , 28] 53 BASIC THEOBEMS where a and ß are infinitesimals; hence: xy = (a + a) (b + ß) =ab + {aß + ba + aß). Using both of the properties of infinitesimals from [26], we see t h a t the sum in the bracket on the right of this equation is an infinitesimal, and hence we have: lim (xy) = ab = lim x · lim y.

We write, in fact: m = Obviously: and also: ^1 VAl + Bl ' χ = ^1 VAl + Bi ' A = ]/Al + Bi· -4X = m^4, B1 = nA , (20) m2 + n2 = 1 , \m\ < 1, | n | < 1, so that, from trigonometry, an angle bx can always be found such that: cos b1 = m , sin b1 = n. e. y = A (cos δ χ · sin a±x + sin 6X · cos axx) , y = Asm {ax x -f- δχ) 24· Inverse trigonometric, or circular, functions· These functions are obtained by inversion of the trigonometric functions: y = sin x, cos x, tan x, cot x , their symbols being respectively: y = arc sin x, arc cos #, arc t a n x, arc cot # ; t h e s e symbols are simply abbreviated forms of description for the angle (or arc), of which the sine, cosine, tangent or cotangent is respectively equal to x.

A variable x is said to be bounded, if there exists a positive number M, such that \ x | < M for all values of x. We can take x = sin a as an example of a bounded magnitude, where the angle a varies in any manner. Here, M can be taken as any number greater than unity. We now consider the case when the point K is displaced successively, and indefinitely approaches the origin. More precisely, we suppose 26] INFINITESIMALS 45 t h a t successive displacements of point K bring it inside any previously assigned small section S'S of t h e axis OX with centre 0, and t h a t it remains inside this section on further displacement.