Solid Analytic Geometry (Dover Books on Mathematics) - Softcover

Albert, Abraham

Softcover

ISBN 10: 0486810267 ISBN 13: 9780486810263

Verlag: Dover Publications Inc., 2016

Inhaltsangabe

The first seven chapters of this concise text provide an exposition of the basic topics of solid analytic geometry and comprise the material for a one-semester course on the subject for undergraduate mathematics majors. The remaining two chapters offer additional material for longer courses or supplementary study.
Chapters 1 and 2 contain a treatment of the equations of lines and planes. Subsequent chapters offer an exposition of classical elementary surface and curve theory, a treatment of spheres, and an examination of the classical descriptions of quadric surfaces in standard position. An exploration of the theory of matrices follows, with applications to the three-dimensional case of quadric surfaces. The text concludes with a survey of spherical coordinates and elements of projective geometry.

Über diesen Titel

Über die Autorin bzw. den Autor

Abraham Adrian Albert (1905–72) taught at the University of Chicago for 40 years and was Chairman of the Mathematics Department from 1958–62. His other highly regarded texts include the Dover release An Introduction to Finite Projective Planes.

Von der hinteren Coverseite

The first seven chapters of this concise text provide an exposition of the basic topics of solid analytic geometry and comprise the material for a one-semester course on the subject for undergraduate mathematics majors. The remaining two chapters offer additional material for longer courses or supplementary study.
Chapters 1 and 2 contain a treatment of the equations of lines and planes. Subsequent chapters offer an exposition of classical elementary surface and curve theory, a treatment of spheres, and an examination of the classical descriptions of quadric surfaces in standard position. An exploration of the theory of matrices follows, with applications to the three-dimensional case of quadric surfaces. The text concludes with a survey of spherical coordinates and elements of projective geometry.
Dover republication of the edition originally published by the McGraw-Hill Book Company, Inc., New York, 1949.
www.doverpublications.com

Auszug. © Genehmigter Nachdruck. Alle Rechte vorbehalten.

Solid Analytic Geometry

By A. Adrian Albert

Dover Publications, Inc.

Copyright © 2016 Dover Publications, Inc.
All rights reserved.

ISBN: 978-0-486-81026-3

Contents

PREFACE, v, 
1. COORDINATES AND LINES, 1, 
2. PLANES, 24, 
3. SURFACES AND CURVES, 37, 
4. SPHERES, 54, 
5. QUADRIC SURFACES, 59, 
6. THEORY OF MATRICES, 77, 
7. ROTATIONS OF AXES AND APPLICATIONS, 103, 
8. SPHERICAL COORDINATES, 123, 
9. ELEMENTS OF PROJECTIVE GEOMETRY, 143, 
INDEX, 159, 

CHAPTER 1

COORDINATES AND LINES

1. Vectors. A sequence P = (x1 ..., xn) of n numbers xi is called an n-dimensional vector. The elements x1 ..., xn are called the coordinates of P and xi is the ith coordinate. We shall limit our attention to real vectors, i.e., to vectors whose coordinates are real numbers.

The vector whose coordinates are all zero is called the zero vector and will be designated by 0. A real vector may be interpreted as a representation, relative to a fixed coordinate system with O as origin, of a point in n-dimensional real Euclidean space.

It may also be interpreted as the line segment [??] directed from O to P. These interpretations have little intuitive significance except for the cases n = 3, and we shall carry out the details in this text for the case n = 3.

The sum P + Q of two vectors P = (x1 ..., xn) and Q = (y1 ..., yn) is the vector (x1 + y1, ..., xn + yn) whose ith coordinate is the sum xi + yi of the ith coordinate of P and the ith coordinate of Q. We leave the verification of the following simple results to the reader:

Lemma 1.Addition of vectors is commutative, that is, P + Q = Q + P for all vectors P and Q.

Lemma 2.Addition of vectors is associative, that is, (P + Q) + R = P + (Q + R) for all vectors P, Q, R.

Lemma 3.The zero vector 0 has the property that P + 0 = P for all vectors P.

Lemma 4.Let P = (x1, ..., xn). Then the vector - P = (- x1, ..., - xn) has the property that P + (-P) = 0.

Lemma 5.If P and Q are any vectors the equation P + X = Q has the solution X = Q + (- P). We call this vector thedifferenceof Q and P and write X = Q - P. Then the ith coordinate of Q - P is the difference of the ith coordinate of Q and the ith coordinate of P.

EXERCISE

Verify the five lemmas.

2. Scalar multiplication. If a is a number and P = (x1, ..., xn) is a vector, we define the scalar product of P by a to be

aP = (ax1, ..., axn).

Evidently 1P = P, (-1)P = -P, 0 P = 0. The reader should verify that

a(bP) = (ab)P, (a + b)P = aP + bP, a(P + Q) = aP + aQ

for all scalars a and b and all vectors P and Q.

A sum

P = a1P1 + ··· amPm,

of scalar products ajPj of vectors Pj by scalars aj, is called a linear combination of P1, ..., Pm. We shall say that P1, ..., Pm are linearly independent vectors if it is true that a linear combination a1P1 + · + amPm = 0 if and only if a1, ..., am are all zero. If P1, ..., Pm are not linearly independent, we shall say that P1, ..., Pm are linearly dependent.

Let Ei be the vector whose ith coordinate is 1 and whose other coordinates are all zero. Then

P = (x1, ..., xn) = x1E1 + ··· + xnEn.

Thus every vector is a linear combination of E1, ..., En. If x1E1 + ··· + xnEn = P = 0, then (x1, ..., xn) = 0, that is, x1 = x2 = ··· = xn = 0. It follows that E1, ..., En are linearly independent.

EXERCISES

1. Show that if P = (x1, ..., xn) and Q = (y1, ..., yn) are not zero then P and Q are linearly dependent if and only if Q is a scalar multiple of P.

2. Show that if P1, ..., Pm are linearly independent and Pm+1 is another vector then P1, ..., Pm, Pm+1 are linearly dependent if and only if Pm+1 is a linear combination of P1, ..., Pm.

3. Compute the following linear combinations of P1 = (1, -1, 2, 3), P2 = (0, 1, -1, 2), P3 = (-2, l, -1, 2).

(a) 2P1 + P2 + P3

(b) P1 + 3P2 - 2P3

(c) 3P1 + 2P2 - 4P3

4. Use Exercises 1 and 2 in determining which of the following sets of three vectors are linearly independent sets.

(a) (1, -1, 2), (1, 1, 0), (0, -1, 1)

(b) (2, 1, 1), (1, -1, 1), (5, 4, 2)

(c) (1, 0, -2), (2, -1, 2), (4, -3, 10)

(d) (1, -1, 1), (-1, 2, 1), (-1, 2, 2)

(e) (1, 0, -1, 1), (0, -1, 1, -1), (4, -1, -3, 4)

(f) (5, 1, -2, -6), (1, 1, 0, -2), (2, -1, -1, 0)

(g) (1, 0, 0, 0), (1, 1, 1, 1), (3, 1, 1, 1)

(h) (1, 1, -1, 2), (2, 2, -2, 3), (3, 3, -2, 6)

5. Prove that any three two-dimensional vectors are linearly dependent.

6. Prove that any four three-dimensional vectors are linearly dependent.

3. Inner products. If P = (x1, ..., xn) and Q = (y1, ..., yn) are any two vectors, we shall call the number

(1) P · Q = x1y1 + ··· + xnyn

the inner product of P and Q. Evidently, P · Q = Q · P.

The norm of a vector P is defined to be the inner product

(2) P · P = x12 + ··· xn2

If P is any real vector, the number P · P = 0 and has a nonnegative square root

(3) t = vP · P = v x12 + ··· + xn2,

which we shall call the length of P.

A vector P is called a unit vector if P · P = 1. Thus a real unit vector is a vector whose length (and whose norm) is 1.

Lemma 6.Every real nonzero vector is a scalar multiple of exactly two unit vectors. These are the vectors U = t-1 P and -U, where t is the length of P. Then if P = tU, where t > 0 and U is a unit vector, the number t is the length of P.

For proof we first let P = tU where U = (u1, ..., un) is a unit vector. Then P · P = (tu1)2 + ··· + (tun)2 = t2(u12 + ··· + ut2) = t2, and t = [+ or -] vP · P; t = v P · P if t = 0. Conversely, let U = t-1P, where t = vP · P. Then U · U = (t-1x1)2 + ··· + (t-1xn)2 = t-2(x12 + ··· +...

Details zur Ausgabe

• Verlag: Dover Publications Inc.
• Erscheinungsdatum: 2016
• Sprache: Englisch
• ISBN 10: 0486810267
• ISBN 13: 9780486810263
• Einband: Taschenbuch
• Auflage: 1
• Anzahl der Seiten: 176
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: Nicht verfügbar