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Overview
The treatment begins with a chapter on vectors and vector addition, followed by a chapter on products of vector. Two succeeding chapters on vector calculus cover a variety of topics, including functions of a vector; line, surface, and volume integrals; the Laplacian operator, and more. The text concludes with a survey of standard applications, including Poinsot's central axis, Gauss's theorem, gravitational potential, Green's theorems, and other subjects.
Product Details
| ISBN-13: | 9780486809236 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 01/14/2016 |
| Series: | Dover Books on Mathematics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 160 |
| File size: | 27 MB |
| Note: | This product may take a few minutes to download. |
About the Author
Read an Excerpt
Concise Vector Analysis
By C. J. Eliezer
Dover Publications, Inc.
Copyright © 2015 Dover Publications, Inc.All rights reserved.
ISBN: 978-0-486-80923-6
CHAPTER 1
Vectors and Vector Addition
1.1. Vectors
Mathematics has played an important part in the advance of science, and mathematical language has become essential for the formulation of the laws of science. Numbers, functions, vectors, tensors, spinors, matrices are examples of mathematical entities which occur in various branches of applied mathematics.
The physical quantities with which we are concerned here may be divided into two groups: (a) scalars, (b) vectors. A scalar requires only its magnitude for its specification. For example, temperature, mass, density, volume and energy are scalars. Vectors require both magnitude and direction for their specification. Force, displacement, velocity, acceleration and momentum are vectors.
Vectors may be classified in different ways. In one elementary classification we have three types of vectors distinguished by their effects:
(i) Unlocalized or free vector which has magnitude and direction but no particular position associated with it, e.g. the moment of a couple.
(ii) Sliding vector or vector localized along a straight line, e.g. force acting on a rigid body (though we speak of a force as acting at a point, by the principle of transmissibility of force, it is the fine of action and not the point of application that is needed to specify the force).
(iii) Tied vector or vector localized at a point, e.g. electric field.
In the early part of this book whenever we speak of a vector without stating its classification we mean a free vector. Two vectors of the same magnitude and direction, but acting along parallel lines (or along the same line) will be considered as equal or identical. Thereafter other types of vectors are considered.
Strictly speaking, every quantity which has a magnitude and a direction is not necessarily a vector. A vector is defined as a quantity with a magnitude and a direction, and which obeys the same addition rule as displacements, that is, the rule known as the parallelogram law of addition. (As an example of a quantity which has magnitude and direction but is not a vector, consider rotations of a rigid body through finite angles. These have magnitudes and directions, but do not obey the parallelogram law of addition, and are not vectors.)
A vector is denoted by symbols in bold type such as a, b, P, Q or by letters with arrows written above such as [??], [??].
1.2. Representation of vectors
A vector may be represented by a directed line segment, the direction of the line indicating the direction of the vector and the length of segment representing, in an appropriate scale of units, the magnitude of the vector. Parallel segments of the same length will represent equal or identical vectors. Since the exact position of vectors is not important and parallel segments represent equal vectors, it is convenient to represent vectors in a diagram by line segments starting from the same origin, say O. Thus the segment [??] represents a vector a, where the direction of [??] is the same as the direction of the vector a, and the length [??] in a suitable scale of units is equal to the magnitude of the vector a. We denote the magnitude of the vector a by the symbol a written in italics or by the symbol [absolute value of a] which is read as "modulus of a" or briefly as "mod a".
1.3. Vector addition
Let [??],[??] represent two vectors a and b respectively. We complete the parallelogram OACB, which has 'OA, OB as adjacent edges.
[??] = a, [??] = b. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let [??] = c. The line segment [??] represents the displacement of a particle from O to A, and [??] the displacement from A to C. The effect of these two displacements in succession is the same as that of a single displacement from O to C. Hence we write
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
that is a + b = c, (1)
where + denotes vectorial addition.
This is the triangle rule for vector addition.
We may state the rule in an alternative way, namely that the sum of two vectors a and b is the vector c, which is represented by the diagonal OC of the parallelogram of which OA and OB are adjacent edges. We write
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
that is, b + a = c,
we see that
a + b = b + a. (2)
This shows that vector addition is commutative, and the order of addition may be interchanged.
We consider also the associative law of addition. Take three vectors P, Q, R and represent them by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] respectively. Construct P + Q and Q + R and then (P + Q) + R and P + (Q + R).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Hence
(P + Q) + R = P + (Q + R). (3)
The brackets which have been used to show the order of the operations may therefore be omitted, and the sum on either side be written as
P + Q + R.
This result may be extended to define the sum of any number of vectors P, Q, ..., W.
Suppose we show in a diagram, Fig. 4(a), the vectors P, Q, ..., W drawn from an origin O. We draw a polygon AB ... K in which the vectors are represented in a suitable scale by directed segments [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which are placed in succession, with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Join AK. Then [??] represents in the same scale the vector sum
P + Q + ... + W.
If the sum is denoted by R, then R = [??]. R is called the resultant vector, and P, Q, ... are called the components of R.
1.4. – a, O, λa
The symbol – a is used to indicate a vector which has the same magnitude as a but is opposite in direction. If a = [??], then [??] = – a
A vector whose terminal points coincide is called a null vector and is denoted by the symbol O. We see that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)
Also
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)
The sum of the vectors a and –b, that is a + (–b), may be written a – b. In Fig. 2, [??] = –b, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], by the parallelogram law. Hence [??] represents a – b.
If λ is a positive scalar, and a a vector then λa is defined to be the vector which has the same direction as a but whose magnitude is λ times that of a. If [??] represents a, and [??] represents λa, then the points O, A, A' lie on a line and their distances are such that OA'/OA = λ.
This multiplication of a vector by a scalar obeys the distributive law.
That is,
λ( a + b) = λa + λb. (6)
This may be verified by using properties of similar triangles. Let [??] = a, [??] = b, then [??] = a + b. Let A' and B' be such that
OA'/OA = OB'/OB = λ.
Then A'B' is parallel to AB, and the two triangles OAB, OA'B' are similar.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Also
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Hence we obtain eqn. (6).
We see that the rules for addition and subtraction of vectors, and multiplication by a scalar of vectors agree with the laws of ordinary algebra.
1.5. Resolution of a vector
Let Ox, Oy, Oz be three lines mutually at right angles. These lines may be used as the three axes of a system of Cartesian rectangular coordinates or as a frame of reference.
From a given point P, draw PM normal to the xy-plane (that is, the plane containing the axes Ox, Oy). Draw MA perpendicular to Ox. Let OA = x, AM = y, MP = z. Complete the rectangular solid with sides of length x, y and z shown in the diagram. OA, OB, OC, that is, x, y and z, are the projections of OP on the three axes, respectively, x, y, z are said to be the coordinates of the point P.
Yectorially,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the components of [??].
When a vector [??] is resolved into three mutually perpendicular components, such as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (or into two perpendicular components [??], [??] when OP is in the plane AOB), then each component is called the resolved part or resolute of the vector in the corresponding direction.
It is convenient to have a notation for the three unit vectors (that is vectors of unit magnitude) along the three axes. Let i, j, k denote unit vectors along the x-, y- and z- axes respectively. Then [??] = xi, [??] = yj, and [??] = zk, are the resolutes of OP along the three axes. [??] is called the position vector of the point P relative to the origin O, or simply, the position vector of P. If we denote [??] by r,
r = xi + yj + zk. (7)
Also OP2 = OA2 + AP2 (OA, AP being at right angles) = OA2 + AM2 + MP2. Hence
[absolute value of r2] = x2 + y2 + z2. (8)
If OP makes angles α, β, γ with the three axes respectively, cos α, cos β, cos γ are called the direction cosines of OP. We see that x = r cos α, y = r cos β, z = r cos γ, and the direction cosines of OP are x/r, y/r, z/r.
Suppose the projections of a vector a on the 3 axes are ax, ay, az. ax, ay, az are scalars. The projections may be represented in magnitude and direction by axi, ayj, azk. Then
a = axi + ayj + azk. (9)
The magnitudes of the resolutes along the three axes are
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
where (a, x) denotes the angle between the vector a, and the x-axis. Also, the magnitude of the vector is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
and the direction cosines of a are
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)
If a vector a has resolutes (ax, ay, az) and b has resolutes (bx, by, bz), then a = axi + ayj + azk, b = bxi + byi + bzk. Using the rules of eqn. (3) and (6), we obtain
a + b = (ax + bx)i + (ay + by)j + (az + bz)k (13)
showing that vectors may be compounded by the rule that the resolved part of (a + b) along any direction is the sum of the resolved parts along the same direction of a and b.
Example. Points A and B have position vectors
a = a1i + a2j + a3k, b = b1i + b2j + b3k.
Find the vector [??], the length of AB and the direction cosines of AB.
If O is the origin,
[??] = a, [??] = b,
and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Direction cosines of AB are
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
1.6 Point dividing AB in the ratio m :n
If a, b are the position vectors of two points A and B, the point R which divides AB in the ratio m:n has position vector
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)
Proof:
[??] = r – a, [??] = b – r. Since AR/RB = m/n, and AR, RB are in the same direction, n]??] = m]??]. Hence
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
As a particular case, we obtain that the position vector of the mid-point of AB is
½(a + b). (15)
Example. Ifa, b, care the position vectors of the vertices A, B, C of a triangle, show that the centroid of the triangle has position vector
1/3(a + b + c). (16)
If D is the mid-point of BC, the centroid G divides AD in the ratio 2:1.
The position vector of D is ½(b + c).
Using formula (14), G has position vector
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
1.7. Centroid or mean centre of n points
Given n points A1, A2, ..., An with position vectors r1r2, ..., rn, their centroid or centre of mean position is defined as the point G with position vector
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)
If m1, m2, ..., mn are n real numbers, the point G whose position vector is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)
is defined as the centroid of the given points A1, A2, ..., An with associated numbers m1, m2, ..., mn respectively. The point is also called the weighted mean centre, the numbers m1, m2, ..., mn being the associated weights.
For two points A and B, the point dividing AB in the ratio m:n is the centroid with the associated weights n at A and m at B. (Compare eqn. (14)).
We can show that the centroid is independent of the particular origin O used in defining the vectors. Let O' be a point whose position vector, relative to O is a. Relative to O' as origin, the position vectors of A1, A2, ..., An are r1 – a, r2 – a, ..., rn – a, and the centroid G is then defined by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Hence G and G' coincide, showing that the centroid is independent of the choice of origin.
The centroid of a curve, area or volume may be defined by dividing into a large number of small elements, finding the centroid with associated numbers proportional to the length, area or volume respectively, of each element, and finding the limiting position as n increases indefinitely and each element tends to a point.
Thus
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)
for a curve,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Example 1. Show that the centroid G defined by eqn. (18) satisfies the relation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (20)
(Continues...)
Excerpted from Concise Vector Analysis by C. J. Eliezer. Copyright © 2015 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
Contents
Preface, v,Chapter 1. Vectors and Vector Addition,
Chapter 2. Products of Vectors,
Chapter 3. Vector Calculus,
Chapter 4. Vector Calculus (Continued),
Chapter 5. Some Applications,
Index, 151,