Calculus: Analytic Geometry and Calculus, with Vectors

106 Vectors and geometry in three dimensions When ,r/2 < 0 < 7r, the scalar triple product is the negative of the volume o ...

2.6 Vector products and changes of coordinates in E3 107 and therefore (5) x = x' cos 0 - y' sin 0 y = x' sin 0 + y' cos 4). Rem ...

108 Vectors and geometry in three dimensions 16 Prove that a line which is not completely contained in a quadric surface can int ...

2.6 Vector products and changes of coordinates in Ea 109 which says that Q1 is the point on the line P3P4 which lies in the plan ...

110 Vectors and geometry in three dimensions matrix. Therefore U'1 = UT. This is important; the inverse of a unitary matrix U is ...

Functions, 3 limits, derivatives 3.1 Functional notation As we progress in a study of a science, it is necessary to become famil ...

112 Functions, limits, derivatives This equation is read "z equals f of x and y and 0." It happens that the law of cosines, whic ...

3.1 Functional notation 113 positive because radii of disks are positive numbers), we can say that y is a function of x and writ ...

114 Functions, limits, derivatives The contraption in the central part of Figure 3.151 is guaranteed to make nearly everybody im ...

3.1 Functional notation 115 when someone says that the temperature u at the north pole of our earth is a function of the time t ...

116 Functions, limits, derivatives where n and ao, al, , anand bo, b1, .,b,a are constants, n being a nonnegative integer. A rat ...

3.1 Functional notation 117 such as h, k, p, q for Ax, Ay, tO, Az, but very often the extra labor involved in writing the more e ...

118 Functions, limits, derivatives (o)Ax + Ox)-AX) g(x + Ox) g(x) = 2x + Ax 6x Ax (p) 4(x + Ax) - ¢(x) -2x- Ax AX [1 + x^2 ][1 ...

3.1 Functional notation 119 7 If h(x) = x + 1/x when x 76 0, show that h(11t) = h(t) when t 0 0 and that [h(x)]2 = h(x2) + 2. Wo ...

120 Functions, limits, derivatives if -2 5 x S 2 and, for each such x, y(x) is one or the other of the two numbers -x - -\/3(4 - ...

3.1 Functional notation 121 The particular point (x,71) for which Mt # = 0 and .11v1', = 0 is called the cen- troidt (thing like ...

122 Functions, limits, derivatives called a black box and is sometimes called a transformer T. When an element x of the domain i ...

3.2 Limits 123 involve only simple words and may seem, at first sight, to be childishly simple. It is reasonable to suppose that ...

124 Functions, limits, derivatives same as that of the assertion (3.22) To each positive number e there corresponds a positive n ...

3.2 Limits 125 It is both easy and customary to adopt the absurd view that everybody has spent huge amounts of time squaring all ...