ROTATION OF THE COORDINATE AXES :
- In the preceding section vectors were defined or represented in two equivalent ways:
- (1) geometrically by specifying magnitude and direction, as with an arrow, and (2) algebraically
- by specifying the components relative to Cartesian coordinate axes. The second
- definition is adequate for the vector analysis of this chapter. In this section two more
- refined, sophisticated, and powerful definitions are presented. First, the vector field is defined
- in terms of the behavior of its components under rotation of the coordinate axes. This
- transformation theory approach leads into the tensor analysis of Chapter 2 and groups of
- transformations in Chapter 4. Second, the component definition of Section 1.1 is refined
- and generalized according to the mathematician’s concepts of vector and vector space. This
- approach leads to function spaces, including the Hilbert space.
- The definition of vector as a quantity with magnitude and direction is incomplete. On
- the one hand, we encounter quantities, such as elastic constants and index of refraction
- in anisotropic crystals, that have magnitude and direction but that are not vectors. On
- the other hand, our naïve approach is awkward to generalize to extend to more complex
- quantities. We seek a new definition of vector field using our coordinate vector r as a
- prototype.
- There is a physical basis for our development of a new definition.We describe our physical
- world by mathematics, but it and any physical predictions we may make must be
- independent of our mathematical conventions.
- In our specific case we assume that space is isotropic; that is, there is no preferred direction,
- or all directions are equivalent. Then the physical system being analyzed or the
- physical law being enunciated cannot and must not depend on our choice or orientation
- of the coordinate axes. Specifically, if a quantity S does not depend on the orientation of
- the coordinate axes, it is called a scalar.
- 3This section is optional here. It will be essential for Chapter 2.
-
Now we return to the concept of vector r as a geometric object independent of the
-
coordinate system. Let us look at r in two different systems, one rotated in relation to the
-
other.
-
For simplicity we consider first the two-dimensional case. If the x-, y-coordinates are
-
rotated counterclockwise through an angle ϕ, keeping r, fixed (Fig. 1.6), we get the following
-
relations between the components resolved in the original system (unprimed) and
-
those resolved in the new rotated system (primed):
-
x′ = x cosϕ +y sin ϕ,
-
y′ =−x sinϕ +y cos ϕ.
-
(1.8)
-
We saw in Section 1.1 that a vector could be represented by the coordinates of a point;
-
that is, the coordinates were proportional to the vector components. Hence the components
-
of a vector must transform under rotation as coordinates of a point (such as r). Therefore
-
whenever any pair of quantities Ax and Ay in the xy-coordinate system is transformed into
-
(A′x,A′y ) by this rotation of the coordinate system with
-
A′x = Ax cosϕ +Ay sin ϕ,
-
A′y =−Ax sinϕ + Ay cos ϕ,
-
(1.9)
-
we define4 Ax and Ay as the components of a vector A. Our vector now is defined in terms
-
of the transformation of its components under rotation of the coordinate system. If Ax and
-
Ay transform in the same way as x and y, the components of the general two-dimensional
-
coordinate vector r, they are the components of a vector A. If Ax and Ay do not show this
-
4A scalar quantity does not depend on the orientation of coordinates; S′ = S expresses the fact that it is invariant under rotation
-
of the coordinates.
-
form invariance (also called covariance) when the coordinates are rotated, they do not
-
form a vector.
-
The vector field components Ax and Ay satisfying the defining equations, Eqs. (1.9), associate
-
a magnitude A and a direction with each point in space. The magnitude is a scalar
-
quantity, invariant to the rotation of the coordinate system. The direction (relative to the
-
unprimed system) is likewise invariant to the rotation of the coordinate system (see Exercise
-
1.2.1). The result of all this is that the components of a vector may vary according to
-
the rotation of the primed coordinate system. This is what Eqs. (1.9) say. But the variation
-
with the angle is just such that the components in the rotated coordinate system A′x and A′y
-
define a vector with the same magnitude and the same direction as the vector defined by
-
the components Ax and Ay relative to the x-, y-coordinate axes. (Compare Exercise 1.2.1.)
-
The components of A in a particular coordinate system constitute the representation of
-
A in that coordinate system. Equations (1.9), the transformation relations, are a guarantee
-
that the entity A is independent of the rotation of the coordinate system.
-
To go on to three and, later, four dimensions, we find it convenient to use a more compact
-
notation. Let
-
x→x1
-
y→x2
-
(1.10)
-
a11 = cos ϕ, a12 = sin ϕ,
-
a21 =−sin ϕ, a22 = cos ϕ.
-
(1.11)
-
Then Eqs. (1.8) become
-
x′1 = a11x1 + a12x2,
-
x′2 = a21x1 +a22x2.
-
(1.12)
-
The coefficient aij may be interpreted as a direction cosine, the cosine of the angle between
-
x′i and xj ; that is,
-
a12 = cos(x′1, x2) = sin ϕ,
-
a21 = cos(x′2, x1) = cos
-
-
ϕ + π2
-
-
=−sin ϕ.
-
(1.13)
-
The advantage of the new notation5 is that it permits us to use the summation symbol
-
-
and to rewrite Eqs. (1.12) as
-
x′i =
-
2
-
j=1
-
aij xj, i= 1, 2. (1.14)
-
Note that i remains as a parameter that gives rise to one equation when it is set equal to 1
-
and to a second equation when it is set equal to 2. The index j , of course, is a summation
-
index, a dummy index, and, as with a variable of integration, j may be replaced by any
-
other convenient symbol.
-
5You may wonder at the replacement of one parameter ϕ by four parameters aij. Clearly, the aij do not constitute a minimum
-
set of parameters. For two dimensions the four aij are subject to the three constraints given in Eq. (1.18). The justification for
-
this redundant set of direction cosines is the convenience it provides. Hopefully, this convenience will become more apparent
-
in Chapters 2 and 3. For three-dimensional rotations (9 aij but only three independent) alternate descriptions are provided by:
-
(1) the Euler angles discussed in Section 3.3, (2) quaternions, and (3) the Cayley–Klein parameters. These alternatives have their
-
respective advantages and disadvantages.
-
The generalization to three, four, or N dimensions is now simple. The set of N quantities
-
Vj is said to be the components of an N-dimensional vector V if and only if their values
-
relative to the rotated coordinate axes are given by
-
V ′ i =
-
N
-
j=1
-
aijVj, i= 1, 2, . . . , N. (1.15)
-
As before, aij is the cosine of the angle between x′i and xj . Often the upper limit N and
-
the corresponding range of i will not be indicated. It is taken for granted that you know
-
how many dimensions your space has.
-
From the definition of aij as the cosine of the angle between the positive x′i direction
-
and the positive xj direction we may write (Cartesian coordinates)6
-
aij =
-
∂x′i
-
∂xj
-
. (1.16a)
-
Using the inverse rotation (ϕ→−ϕ) yields
-
xj =
-
2
-
i=1
-
aij x′i or
-
∂xj
-
∂x′i = aij . (1.16b)
-
Note that these are partial derivatives. By use of Eqs. (1.16a) and (1.16b), Eq. (1.15)
-
becomes
-
V ′ i =
-
N
-
j=1
-
∂x′i
-
∂xj
-
Vj =
-
N
-
j=1
-
∂xj
-
∂x′i
-
Vj . (1.17)
-
The direction cosines aij satisfy an orthogonality condition
-
-
i
-
aij aik = δjk (1.18)
-
or, equivalently,
-
-
i
-
ajiaki = δjk. (1.19)
-
Here, the symbol δjk is the Kronecker delta, defined by
-
δjk =1 for j = k,
-
δjk =0 for j
-
= k.
-
(1.20)
-
It is easily verified that Eqs. (1.18) and (1.19) hold in the two-dimensional case by
-
substituting in the specific aij from Eqs. (1.11). The result is the well-known identity
-
sin2 ϕ + cos2 ϕ = 1 for the nonvanishing case. To verify Eq. (1.18) in general form, we
-
may use the partial derivative forms of Eqs. (1.16a) and (1.16b) to obtain
-
-
i
-
∂xj
-
∂x′i
-
∂xk
-
∂x′i =
-
-
i
-
∂xj
-
∂x′i
-
∂x′i
-
∂xk =
-
∂xj
-
∂xk
-
. (1.21)
-
6Differentiate x′i with respect to xj . See discussion following Eq. (1.21).
-
The last step follows by the standard rules for partial differentiation, assuming that xj is
-
a function of x′1, x′2, x′3, and so on. The final result, ∂xj /∂xk , is equal to δjk, since xj and
-
xk as coordinate lines (j
-
= k) are assumed to be perpendicular (two or three dimensions)
-
or orthogonal (for any number of dimensions). Equivalently, we may assume that xj and
-
xk (j
-
= k) are totally independent variables. If j = k, the partial derivative is clearly equal
-
to 1.
-
In redefining a vector in terms of how its components transform under a rotation of the
-
coordinate system, we should emphasize two points:
-
1. This definition is developed because it is useful and appropriate in describing our
-
physical world. Our vector equations will be independent of any particular coordinate
-
system. (The coordinate system need not even be Cartesian.) The vector equation can
-
always be expressed in some particular coordinate system, and, to obtain numerical
-
results, we must ultimately express the equation in some specific coordinate system.
-
2. This definition is subject to a generalization that will open up the branch of mathematics
-
known as tensor analysis (Chapter 2).
-
A qualification is in order. The behavior of the vector components under rotation of the
-
coordinates is used in Section 1.3 to prove that a scalar product is a scalar, in Section 1.4
-
to prove that a vector product is a vector, and in Section 1.6 to show that the gradient of a
-
scalar ψ, ∇ψ, is a vector. The remainder of this chapter proceeds on the basis of the less
-
restrictive definitions of the vector given in Section 1.1
-
Summary: Vectors and Vector Space :
-
It is customary in mathematics to label an ordered triple of real numbers (x1, x2, x3) a
-
vector x. The number xn is called the nth component of vector x. The collection of all
-
such vectors (obeying the properties that follow) form a three-dimensional real vector
-
space. We ascribe five properties to our vectors: If x = (x1, x2, x3) and y = (y1, y2, y3),
-
1. Vector equality: x = y means xi = yi , i = 1, 2, 3.
-
2. Vector addition: x +y = z means xi +yi = zi, i = 1, 2, 3.
-
3. Scalar multiplication: ax↔(ax1, ax2, ax3) (with a real).
-
4. Negative of a vector: −x = (−1)x↔(−x1,−x2,−x3).
-
5. Null vector: There exists a null vector 0↔(0, 0, 0).
-
Since our vector components are real (or complex) numbers, the following properties
-
also hold:
-
1. Addition of vectors is commutative: x+ y = y+ x.
-
2. Addition of vectors is associative: (x+ y)+ z = x+ (y+ z).
-
3. Scalar multiplication is distributive:
-
a(x +y) = ax+ ay, also (a + b)x = ax+bx.
-
4. Scalar multiplication is associative: (ab)x = a(bx).
-
Further, the null vector 0 is unique, as is the negative of a given vector x.
-
So far as the vectors themselves are concerned this approach merely formalizes the component
-
discussion of Section 1.1. The importance lies in the extensions, which will be considered
-
in later chapters. In Chapter 4, we show that vectors form both an Abelian group
-
under addition and a linear space with the transformations in the linear space described by
-
matrices. Finally, and perhaps most important, for advanced physics the concept of vectors
-
presented here may be generalized to (1) complex quantities,7 (2) functions, and (3) an infinite
-
number of components. This leads to infinite-dimensional function spaces, the Hilbert
-
spaces, which are important in modern quantum theory. A brief introduction to function
-
expansions and Hilbert space appears in Section 10.4.
-
-
Exercises ;
-
1.2.1 (a) Show that the magnitude of a vector A, A = (A2x
-
+ A2y
-
)1/2, is independent of the
-
orientation of the rotated coordinate system,
-
-
A2x
-
+A2y
-
1/2
-
=
-
-
A′2
-
x +A′2
-
y
-
1/2
-
,
-
that is, independent of the rotation angle ϕ.
-
This independence of angle is expressed by saying that A is invariant under
-
rotations.
-
(b) At a given point (x, y), A defines an angle α relative to the positive x-axis and
-
α′ relative to the positive x′-axis. The angle from x to x′ is ϕ. Show that A = A′
-
defines the same direction in space when expressed in terms of its primed components
-
as in terms of its unprimed components; that is,
-
α′ = α −ϕ.
-
1.2.2 Prove the orthogonality condition
-
-
i ajiaki = δjk. As a special case of this, the direction
-
cosines of Section 1.1 satisfy the relation
-
cos2 α +cos2 β +cos2 γ = 1,
-
a result that also follows from Eq. (1.6).
Now we return to the concept of vector r as a geometric object independent of the
coordinate system. Let us look at r in two different systems, one rotated in relation to the
other.
For simplicity we consider first the two-dimensional case. If the x-, y-coordinates are
rotated counterclockwise through an angle ϕ, keeping r, fixed (Fig. 1.6), we get the following
relations between the components resolved in the original system (unprimed) and
those resolved in the new rotated system (primed):
x′ = x cosϕ +y sin ϕ,
y′ =−x sinϕ +y cos ϕ.
(1.8)
We saw in Section 1.1 that a vector could be represented by the coordinates of a point;
that is, the coordinates were proportional to the vector components. Hence the components
of a vector must transform under rotation as coordinates of a point (such as r). Therefore
whenever any pair of quantities Ax and Ay in the xy-coordinate system is transformed into
(A′x,A′y ) by this rotation of the coordinate system with
A′x = Ax cosϕ +Ay sin ϕ,
A′y =−Ax sinϕ + Ay cos ϕ,
(1.9)
we define4 Ax and Ay as the components of a vector A. Our vector now is defined in terms
of the transformation of its components under rotation of the coordinate system. If Ax and
Ay transform in the same way as x and y, the components of the general two-dimensional
coordinate vector r, they are the components of a vector A. If Ax and Ay do not show this
4A scalar quantity does not depend on the orientation of coordinates; S′ = S expresses the fact that it is invariant under rotation
of the coordinates.
form invariance (also called covariance) when the coordinates are rotated, they do not
form a vector.
The vector field components Ax and Ay satisfying the defining equations, Eqs. (1.9), associate
a magnitude A and a direction with each point in space. The magnitude is a scalar
quantity, invariant to the rotation of the coordinate system. The direction (relative to the
unprimed system) is likewise invariant to the rotation of the coordinate system (see Exercise
1.2.1). The result of all this is that the components of a vector may vary according to
the rotation of the primed coordinate system. This is what Eqs. (1.9) say. But the variation
with the angle is just such that the components in the rotated coordinate system A′x and A′y
define a vector with the same magnitude and the same direction as the vector defined by
the components Ax and Ay relative to the x-, y-coordinate axes. (Compare Exercise 1.2.1.)
The components of A in a particular coordinate system constitute the representation of
A in that coordinate system. Equations (1.9), the transformation relations, are a guarantee
that the entity A is independent of the rotation of the coordinate system.
To go on to three and, later, four dimensions, we find it convenient to use a more compact
notation. Let
x→x1
y→x2
(1.10)
a11 = cos ϕ, a12 = sin ϕ,
a21 =−sin ϕ, a22 = cos ϕ.
(1.11)
Then Eqs. (1.8) become
x′1 = a11x1 + a12x2,
x′2 = a21x1 +a22x2.
(1.12)
The coefficient aij may be interpreted as a direction cosine, the cosine of the angle between
x′i and xj ; that is,
a12 = cos(x′1, x2) = sin ϕ,
a21 = cos(x′2, x1) = cos
ϕ + π2
=−sin ϕ.
(1.13)
The advantage of the new notation5 is that it permits us to use the summation symbol
and to rewrite Eqs. (1.12) as
x′i =
2
j=1
aij xj, i= 1, 2. (1.14)
Note that i remains as a parameter that gives rise to one equation when it is set equal to 1
and to a second equation when it is set equal to 2. The index j , of course, is a summation
index, a dummy index, and, as with a variable of integration, j may be replaced by any
other convenient symbol.
5You may wonder at the replacement of one parameter ϕ by four parameters aij. Clearly, the aij do not constitute a minimum
set of parameters. For two dimensions the four aij are subject to the three constraints given in Eq. (1.18). The justification for
this redundant set of direction cosines is the convenience it provides. Hopefully, this convenience will become more apparent
in Chapters 2 and 3. For three-dimensional rotations (9 aij but only three independent) alternate descriptions are provided by:
(1) the Euler angles discussed in Section 3.3, (2) quaternions, and (3) the Cayley–Klein parameters. These alternatives have their
respective advantages and disadvantages.
The generalization to three, four, or N dimensions is now simple. The set of N quantities
Vj is said to be the components of an N-dimensional vector V if and only if their values
relative to the rotated coordinate axes are given by
V ′ i =
N
j=1
aijVj, i= 1, 2, . . . , N. (1.15)
As before, aij is the cosine of the angle between x′i and xj . Often the upper limit N and
the corresponding range of i will not be indicated. It is taken for granted that you know
how many dimensions your space has.
From the definition of aij as the cosine of the angle between the positive x′i direction
and the positive xj direction we may write (Cartesian coordinates)6
aij =
∂x′i
∂xj
. (1.16a)
Using the inverse rotation (ϕ→−ϕ) yields
xj =
2
i=1
aij x′i or
∂xj
∂x′i = aij . (1.16b)
Note that these are partial derivatives. By use of Eqs. (1.16a) and (1.16b), Eq. (1.15)
becomes
V ′ i =
N
j=1
∂x′i
∂xj
Vj =
N
j=1
∂xj
∂x′i
Vj . (1.17)
The direction cosines aij satisfy an orthogonality condition
i
aij aik = δjk (1.18)
or, equivalently,
i
ajiaki = δjk. (1.19)
Here, the symbol δjk is the Kronecker delta, defined by
δjk =1 for j = k,
δjk =0 for j
= k.
(1.20)
It is easily verified that Eqs. (1.18) and (1.19) hold in the two-dimensional case by
substituting in the specific aij from Eqs. (1.11). The result is the well-known identity
sin2 ϕ + cos2 ϕ = 1 for the nonvanishing case. To verify Eq. (1.18) in general form, we
may use the partial derivative forms of Eqs. (1.16a) and (1.16b) to obtain
i
∂xj
∂x′i
∂xk
∂x′i =
i
∂xj
∂x′i
∂x′i
∂xk =
∂xj
∂xk
. (1.21)
6Differentiate x′i with respect to xj . See discussion following Eq. (1.21).
The last step follows by the standard rules for partial differentiation, assuming that xj is
a function of x′1, x′2, x′3, and so on. The final result, ∂xj /∂xk , is equal to δjk, since xj and
xk as coordinate lines (j
= k) are assumed to be perpendicular (two or three dimensions)
or orthogonal (for any number of dimensions). Equivalently, we may assume that xj and
xk (j
= k) are totally independent variables. If j = k, the partial derivative is clearly equal
to 1.
In redefining a vector in terms of how its components transform under a rotation of the
coordinate system, we should emphasize two points:
1. This definition is developed because it is useful and appropriate in describing our
physical world. Our vector equations will be independent of any particular coordinate
system. (The coordinate system need not even be Cartesian.) The vector equation can
always be expressed in some particular coordinate system, and, to obtain numerical
results, we must ultimately express the equation in some specific coordinate system.
2. This definition is subject to a generalization that will open up the branch of mathematics
known as tensor analysis (Chapter 2).
A qualification is in order. The behavior of the vector components under rotation of the
coordinates is used in Section 1.3 to prove that a scalar product is a scalar, in Section 1.4
to prove that a vector product is a vector, and in Section 1.6 to show that the gradient of a
scalar ψ, ∇ψ, is a vector. The remainder of this chapter proceeds on the basis of the less
restrictive definitions of the vector given in Section 1.1
Summary: Vectors and Vector Space :
It is customary in mathematics to label an ordered triple of real numbers (x1, x2, x3) a
vector x. The number xn is called the nth component of vector x. The collection of all
such vectors (obeying the properties that follow) form a three-dimensional real vector
space. We ascribe five properties to our vectors: If x = (x1, x2, x3) and y = (y1, y2, y3),
1. Vector equality: x = y means xi = yi , i = 1, 2, 3.
2. Vector addition: x +y = z means xi +yi = zi, i = 1, 2, 3.
3. Scalar multiplication: ax↔(ax1, ax2, ax3) (with a real).
4. Negative of a vector: −x = (−1)x↔(−x1,−x2,−x3).
5. Null vector: There exists a null vector 0↔(0, 0, 0).
Since our vector components are real (or complex) numbers, the following properties
also hold:
1. Addition of vectors is commutative: x+ y = y+ x.
2. Addition of vectors is associative: (x+ y)+ z = x+ (y+ z).
3. Scalar multiplication is distributive:
a(x +y) = ax+ ay, also (a + b)x = ax+bx.
4. Scalar multiplication is associative: (ab)x = a(bx).
Further, the null vector 0 is unique, as is the negative of a given vector x.
So far as the vectors themselves are concerned this approach merely formalizes the component
discussion of Section 1.1. The importance lies in the extensions, which will be considered
in later chapters. In Chapter 4, we show that vectors form both an Abelian group
under addition and a linear space with the transformations in the linear space described by
matrices. Finally, and perhaps most important, for advanced physics the concept of vectors
presented here may be generalized to (1) complex quantities,7 (2) functions, and (3) an infinite
number of components. This leads to infinite-dimensional function spaces, the Hilbert
spaces, which are important in modern quantum theory. A brief introduction to function
expansions and Hilbert space appears in Section 10.4.
Exercises ;
1.2.1 (a) Show that the magnitude of a vector A, A = (A2x
+ A2y
)1/2, is independent of the
orientation of the rotated coordinate system,
A2x
+A2y
1/2
=
A′2
x +A′2
y
1/2
,
that is, independent of the rotation angle ϕ.
This independence of angle is expressed by saying that A is invariant under
rotations.
(b) At a given point (x, y), A defines an angle α relative to the positive x-axis and
α′ relative to the positive x′-axis. The angle from x to x′ is ϕ. Show that A = A′
defines the same direction in space when expressed in terms of its primed components
as in terms of its unprimed components; that is,
α′ = α −ϕ.
1.2.2 Prove the orthogonality condition
i ajiaki = δjk. As a special case of this, the direction
cosines of Section 1.1 satisfy the relation
cos2 α +cos2 β +cos2 γ = 1,
a result that also follows from Eq. (1.6).
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