Intermediate Calculus

Analytic geometry in three dimensions makes essential use of coordinate systems. To introduce a coordinate system, we consider triples (a,b, c) of real numbers, and we call the set of all such triples of real numbers the three-dimensional number space. We denote this space by R’. Each individual triple is a point in R ³ . The three elements in each number triple are called its coordinates. We now show how three-dimensional number space may be represented on a geometric or Euclidean three-dimensional space.

Let A and B be two points in a plane. The length of the line segment joining A and B is denoted |AB|.

Suppose f and F are real-valued functions defined on some interval of R ¹ containing the number a. If f(x) and F(x) both approach 0 as x tends to the value a, the quotient may approach a limit, may become infinite, or may fail to have any limit. In the definition of derivative it is the evaluation of just such expressions that leads to the usual differentiation formulas. We are aware that the expression is in itself a meaningless one, and we use the term indeterminate form for the ratio 0/0.

A function f is a mapping which takes each element of the domain D into an element of the range S. We write f: D → S. If the domain D consists of ordered pairs of numbers, then we have a function on R ^e or a function of two variables. We employ the notation

z=f(x,y) or f:(x,y) → S

to indicate a typical function on R ² when the elements of the range are real numbers (elements of R’).

Let F be a region of area A situated in the xy plane. We shall always assume that a region includes its boundary curve. Such regions are sometimes called closed regions in analogy with closed intervals on the real line that is, ones which include their endpoints. We subdivide the xy plane into rectangles by drawing lines parallel to the coordinate axes. These lines may or may not be equally spaced (Fig. 5-1). Starting in some convenient place (such as the upper left-hand corner of F), we systematically number all the rectangles lying entirely within F. Suppose there are n such and we label them r ₁ r ₂,…,r _n .. We use the symbols A(r ₁ ), A(r ₂ ),…, A(r _n ) for the areas of these rectangles. The collection of n rectangles {r₁, r ₂ ,…, r _n } is called a subdivision A of F. The norm of the subdivision denoted by \(\left\| \Delta \right\|\), is the length of the diagonal of the largest rectangle in the subdivision Δ.

In the study of infinite series, the functions 1,x,x²,…,xⁿ,… play a central role. Most of the elementary functions of algebra, trigonometry,and calculus may be expanded in series which are sums of powers of x, that is, in power series. The coefficients in such a Taylor or Maclaurin series are the successive derivatives of the given function evaluated at a point.

An equation such as represents arelationbetweenxand y. A pair of numbers which satisfies this equation corresponds to a point in the xy-plane. In general, the totality of points in quation of the form is called thegraphof the equation. In the study of analytic geometry in the plane, the work with graphs consists mostly of the study of arcs or simple smooth curves. However, it is not at all obvious what the graph might be of an equation such as(1)above. Even equations which are quite simple in appearance sometimes have unusual graphs. For example, the equation is in the form (2) but has no graph, since the sum of positive quantities can never be zero. The equation^(x—^{1)2 + (y + 2)2 =} 0 is satisfied only when x = 1, y = —2, and the graph is a single point. The equation also in the form of (2), has an interesting graph. Since sin xi < 1 and secy~ >1,the equation can hold only when sin x = 1 and sec y. = —1 or when sinx = — 1 and secy =1.The graph consists of the isolated points and as shown in Fig.7-1.

We recall the elementary integration formulavalid for any n >-1 Since need not be an integer, we employ the variable x and write x>–1.

A vector function v(P) assigns a specific vector to each element P in a given domain S. The range of such a function is the collection of vectors which correspond to the points in the domain. In Chapter 2, Section 9, we discussed vector functions with domain a portion (or all) of R ¹ and with range a collection of vectors in R ² and R ³ . For example, if a vector function is defined on the interval a ≤ t ≤ b,then we may represent such a function in the form v (t) = f(t) i + g(t) j,where i and j are the customary unit vectors. The real-valued functions f and g are defined on the interval a ≤ t ≤ b.

The Fundamental Theorem of Calculus states that differentiation and integration are inverse processes. An appropriate extension of this theorem to double integrals of functions of two variables is known as Green’s Theorem. Suppose that P and Q are smooth (i.e., continuously differentiable) functions defined in some region R of the plane. A simple closed curve is a curve that can be obtained as the union of two arcs which have only their endpoints in common. Thus a circle is the union of two half circles. Of course, any two points on a simple closed curve divide it into two arcs in this way. It is intuitively clear that a simple closed curve in the plane divides the plane into two regions, constituting the “interior” and the “exterior” of the curve. This fact which is surprisingly hard to prove, is not used in the proofs of any theorems. A smooth simple closed curve is one which has a parametric representation x = x(t), y = y(t), a < t < b,in which x, y, x,and y are continuous and [x(t)]² + [y(t)] ² > 0 and x(b) = x(a),x(b) = x (a), y(b)= y(a), y(b) = y(a). If T is a smooth simple closed curve which, together with its interior G, is in R, then the basic formula associated with Green’s Theorem is The symbol on the right represents the line integral taken in the counterclockwise sense, so that F is traversed with the interior of G always on the left.