Calculus: Analytic Geometry and Calculus, with Vectors

448 Trigonometric functions

In case m = n, and in other cases where

lim m=0,

m,n-.roW2

lim S,nn = 27rah,

m,n- m

the right member being the number usually considered to be the correct area of

the cylinder. In case m = n2, we obtain

(6) lira Smn= 21rah 1

+ (222a)2

1Je

the right member being a number which is not usually considered to be the correct

area of the cylinder. Other remarks can be made. The above calculations were

made by a German mathematician Schwarz, and they constitute the Schwarz

paradox. The paradox shows that the triangulation idea provides a precarious

basis for definitions of area of curved surfaces. The theory of these areas is

extremely complicated.

17 It is sometimes useful as well as interesting to have information about the

things we see. A polynomial P in x and y is the sum of a finite set of terms of the

form cxtyk, where c is a constant and j and k are nonnegative integers. The poly-

nomial has degree n if j + k = n for at least one term in the sum having a non-

zero coefficient and j + k S n for each term in the sum having a nonzero coeffi-

cient. A polynomial in which the coefficients are all zero is said to be trivial;

it does not have a degree. Thus the polynomials having values

(1) x' + xy + y' - 34, (x2 - y2 - 1)(x2 + y2 - 4)

both have degree 4. A nontrivial polynomial is irreducible if it is not the product

of two polynomials of lower degree. An algebraic equation of degree n is an equa-

tion of the form P(x,y) = 0, where P is a nontrivial polynomial of degree n in x

and y. The graph of an algebraic equation is an algebraic graph. A function f

of one variable x is said to be an algebraic function if there is a nontrivial poly-

nomial P(x,y) such that

(2) P(x, f(x)) = 0

for each x in the domain of f. Since each nontrivial polynomial in x and y can

be put in the form

(3) Qo(x) + Qi(x)y + Q2(x)y2 +.. + Q,(x)y°,

where Qo, Q1, -, Q, are polynomials in x at least one of which is nontrivial,
it follows that a function f is an algebraic function if and only if there exist poly-
nomials Qo, Qi, , Q. in x such that Q. is nontrivial and

(4) Qo(x) + Qi(x)f(x) + Q2(x)[f(x)J2 + ... + QR(x)Lf(x)Jn= 0

for each x in the domain of f.Functions that are not algebraic functions are
said to be transcendental functions, the old idea being that polynomials lieat the