Planos

It was no accident that the transformation describing a rotation in_R2was orthogonal, by which we mean that the matrix effecting the transformation was an orthogonal matrix.

An instructive way of writing the transformationSis, returning toEq. (3.20), to rewrite those equations as

ˆex= (ˆe0x· ˆex)ˆex0 + (ˆe0y· ˆex)ˆe0y, ˆey= (ˆe0x· ˆey)ˆe0x+ (ˆe0y· ˆey)ˆe0y. (3.27) This corresponds to writingˆex and ˆey as the sum of their projections on the orthogonal vectorsˆe0_xandˆe0_y. Now we can rewriteSas

S= 

ˆe0

x· ˆex ˆe0x· ˆey ˆe0

y· ˆex ˆe0y· ˆey 

. (3.28)

This means that each row ofScontains the components (in the unprimed coordinates) of a unit vector (eitherˆe0_x orˆe0_y) that is orthogonal to the vector whose components are in the other row. In turn, this means that the dot products of different row vectors will be zero, while the dot product of any row vector with itself (because it is a unit vector) will be unity. That is the deeper significance of an orthogonal matrixS; theµν element ofSST _{is the} dot product formed from theµth row ofSand theνth column ofST _{(which is the same as} theνth row ofS). Since these row vectors are orthogonal, we will get zero ifµ 6= ν, and because they are unit vectors, we will get unity ifµ = ν. In other words,SST _{will be a unit} matrix.

Before leavingEq. (3.28), note that its columns also have a simple interpretation: Each contains the components (in the primed coordinates) of one of the unit vectors of the unprimed set. Thus the dot product formed from two different columns of Swill van- ish, while the dot product of any column with itself will be unity. This corresponds to the fact that, for an orthogonal matrix, we also haveST_{S= 1.}

Summarizing part of the above,

The transformation from one orthogonal Cartesian coordinate system to another Carte- sian system is described by an orthogonal matrix.

In Chapter 2 we found that an orthogonal matrix must have a determinant that is real and of magnitude unity, i.e.,±1. However, for rotations in ordinary space the value of the determinant will always be+1. One way to understand this is to consider the fact that any rotation can be built up from a large number of small rotations, and that the determinant must vary continuously as the amount of rotation is changed. The identity rotation (i.e., no rotation at all) has determinant+1. Since no value close to +1 except +1 itself is a permitted value for the determinant, rotations cannot change the value of the determinant.

ArfKen_Ch03-9780123846549.tex

136 Chapter 3 Vector Analysis

Reflections

Another possibility for changing a coordinate system is to subject it to a reflection operation. For simplicity, consider first the inversion operation, in which the sign of each coordinate is reversed. In_R3_{, the transformation matrixS will be the 3× 3 analog of} Eq. (3.28), and the transformation under discussion is to set ˆe0_µ= −ˆe_µ, withµ = x, y, and z. This will lead to

S=   −1 0 0 0 −1 0 0 0 −1  ,

which clearly results in detS= −1. The change in sign of the determinant corresponds to the change from a right-handed to a left-handed coordinate system (which obviously cannot be accomplished by a rotation). Reflection about a plane (as in the image produced by a plane mirror) also changes the sign of the determinant and the handedness of the coordinate system; for example, reflection in the x y-plane changes the sign ofˆez, leaving the other two unit vectors unchanged; the transformation matrixSfor this transformation is

S=   1 0 0 0 1 0 0 0 −1  . Its determinant is also−1.

The formulas for vector addition, multiplication by a scalar, and the dot product are unaffected by a reflection transformation of the coordinates, but this is not true of the cross product. To see this, look at the formula for any one of the components of A× B, and how it would change under inversion (where the same, unchanged vectors in physical space now have sign changes to all their components):

Cx: AyBz− AzBy −→ (−Ay)(−Bz) − (−Az)(−By) = AyBz− AzBy. Note that this formula says that the sign of Cx should not change, even though it must in order to describe the unchanged physical situation. The conclusion is that our transforma- tion law fails for the result of a cross-product operation. However, the mathematics can be salvaged if we classify B× C as a different type of quantity than B and C. Many texts on vector analysis call vectors whose components change sign under coordinate reflec- tion polar vectors, and those whose components do not then change sign axial vectors. The term axial doubtless arises from the fact that cross products frequently describe phe- nomena associated with rotation about the axis defined by the axial vector. Nowadays, it is becoming more usual to call polar vectors just vectors, because we want that term to describe objects that obey for allSthe transformation law

A0=SA (vectors), (3.29)

(and specifically without a restriction toSwhose determinants are+1). Axial vectors, for which the vector transformation law fails for coordinate reflections, are then referred to as pseudovectors, and their transformation law can be expressed in the somewhat more complicated form

C0= det(S)SC (pseudovectors). (3.30)

A z x B B y A x A A y z x

FIGURE3.6 Inversion (right) of original coordinates (left) and the effect on a vector A and a pseudovector B.

The effect of an inversion operation on a coordinate system and on a vector and a pseu- dovector are shown inFig. 3.6.

Since vectors and pseudovectors have different transformation laws, it is in general with- out physical meaning to add them together.3It is also usually meaningless to equate quan- tities of different transformational properties: in A= B, both quantities must be either vectors or pseudovectors.

Pseudovectors, of course, enter into more complicated expressions, of which an example is the scalar triple product A·B×C. Under coordinate reflection, the components of B×C do not change (as observed earlier), but those of A are reversed, with the result that A· B × C changes sign. We therefore need to reclassify it as a pseudoscalar. On the other hand, the vector triple product, A× (B × C), which contains two cross products, evaluates, as shown inEq. (3.18), to an expression containing only legitimate scalars and (polar) vectors. It is therefore proper to identify A× (B × C) as a vector. These cases illustrate the general principle that a product with an odd number of pseudo quantities is “pseudo,” while those with even numbers of pseudo quantities are not.