Vectors (Dated: August 24 2017)

I. BRIEF REVIEW OF VECTORS

Scalar product of ~u and ~v:

~u · ~v = |u||v| cos θ, (1) where θ is the angle between ~u and ~v.

Three (two, four, n analogously) components of vectors (u1, u2, u3), (v1, v2, v3)

~u = u1xˆ + u2yˆ + u3zˆ (2)

~v = v1xˆ + v2yˆ + v3zˆ (3)

Herex ˆ ,y ˆ,z ˆ deﬁne orthonormal Cartesian coordinate triad. (Mutually orthogonal, and each has unit length). In terms of components of vectors,

3 X ~u · ~v = u1v1 + u2v2 + u3v3 = uivi = uivi (4) i=1

Omitting summation sign and meaning summation over repeating indices : Einstein convention.

Orthogonality of vectors: cosθ = 0 or uivi = 0 Application of the concept of the scalar product: Energy conservation Law.

d~v F~ = m · ~a = m (5) dt

d~v m dv2 ~v · F~ = m · ~v = i (6) dt 2 dt

d mv2 d m~v · ~v ~v · F~ = ( i = ( ) (7) dt 2 dt 2 2

m~v·~v mv2 Here 2 = 2 is the kinetic energy Ek Discussion : m here is a scalar quantity, assumed time-independent (No reactive motion). Generally m is a tensor. In uniaxial crystals growing along the axis z, the electron mass describing transport properties of a crystal along that direction could diﬀer from the mass in perpendicular direction.

Z b Z b d mv2 ~v · F~ dt = dt = Ek(b) − Ek(a) (8) a a dt 2

Z b Z b d~r Z b ~v · F~ dt = dtF~ · = F~ · d~r = Ep(a) − Ep(b) (9) a a dt a Here we used the deﬁnition of potential energy (note the minus sign). Finally, because the right and the left hand side are equal, we have energy conservation law

Ep(a) + Ek(a) = Ep(b) + Ek(b) (10)

II. CHANGE OF BASIS AND ORTHOGONAL TRANSFORMATIONS

Choice of any particular coordinate system is arbitrary. Consider two Cartesian cordinate systems K and K0 with the same origin. The vector ~v is deﬁned in each of the coordinate systems:

~v = v1xˆ + v2yˆ + v3zˆ in K (11)

0 0 0 0 0 0 0 ~v = v1xˆ + v2yˆ + v3zˆ in K (12)

Note that

~v · xˆ = v1 + 0 + 0 (13)

Therefore we can also write

~v = (~v · xˆ)ˆx + (~v · yˆ)ˆy + (~v · zˆ)ˆz in K (14)

~v = (~v · xˆ0)ˆx0 + ~v · yˆ0)ˆy0 + (~v · zˆ0)ˆz0 in K0 (15)

We now take ~v =x ˆ0, ~v =y ˆ0 and ~v =z ˆ0. We obtain the law of rotation from K to K0 :

xˆ0 = (ˆx0 · xˆ)ˆx + (ˆx0 · yˆ)ˆy + (ˆx0 · zˆ)ˆz (16)

yˆ0 = (ˆy0 · xˆ)ˆx + (ˆy0 · yˆ)ˆy + (ˆy0 · zˆ)ˆz (17)

zˆ0 = (ˆz0 · xˆ)ˆx + (ˆz0 · yˆ)ˆy + (ˆz0 · zˆ)ˆz (18) 3

The nine quantities in brackets are called direction cosines. The equations (16-18) can be written in a matrix form       xˆ0 (ˆx0 · xˆ) (ˆx0 · yˆ (ˆx0 · zˆ) xˆ        0   0 0 0     yˆ  =  (ˆy · xˆ) (ˆy · yˆ (ˆy · zˆ)   yˆ  (19)       zˆ0 (ˆz0 · xˆ) (ˆz0 · yˆ) (ˆz0 · zˆ) zˆ

Denoting transformation matrix R

  a11 a12 a13     R =  a21 a22 a23  (20)   a31 a32 a33 we can brieﬂy write χ0 = Rχ, where χ0 is the vector showing orientation of unit vectors in primed coordinate system, and χ is the vector showing orientation of unit vectors in an unprimed coordinate system.

Out of 9 coefficients aij defining matrix R in three dimensions only 3 are independent. In two dimensions, only one coefficient of the analogous 2x2 matrix is independent. The physical meaning of these independent coefficients are cosines of angles between unprimed and primed ort vectors (three such angles in 3 dimensions and one such angle in two dimensions. In three dimensions, there are 6 equations for 9 coefficients that constrain their values and make only 3 of them independent. These equations are: three normalization conditionsx ˆ0 · xˆ)0 = 1, yˆ0 · yˆ)0 = 1 andz ˆ0 · zˆ)0 = 1 and three orthogonality conditions:x ˆ0 · yˆ)0 = 0,x ˆ0 · zˆ)0 = 0 andy ˆ0 · zˆ)0 = 0

2 In general in n dimensions, there are n normalization conditions and Cn = n(n − 1)/2 orthogonality conditions, yielding n(n − 1)/2 independent directed cosines.

III. TRANSFORMATIONS OF ARBITRARY VECTORS UNDER ROTATIONS

Once aij are determined, which ﬁxes the rotation of coordinate system, we can determine how any other vector transforms.

0 0 0 0 0 0 ~v = v1xˆ + v2yˆ + v3zˆ

0 0 0 = v1(a11xˆ + a12yˆ + a12zˆ) + v2(a21xˆ + a22yˆ + a23zˆ) + v3((a31xˆ + a32yˆ + a33zˆ)

0 0 0 0 0 0 0 0 0 = (a11v1 + a21v2 + a31v3)ˆx + (a12v1 + a22v2 + a32v3)ˆy + (a13v1 + a23v2 + a33v3). (21) 4

Therefore we obtain

0 0 0 v1 = a11v1 + a21v2 + a31v3 (22)

0 0 0 v2 = a12v1 + a22v2 + a32v3 (23)

0 0 0 v3 = a13v1 + a23v2 + a33v3 (24)

Unifying this relations, we get

0 vi = anivn (25)

We can invert these relations multiplying Eq.() by ami , summing over i and using condition of orthonormality amiani = δnm:

X X 0 X 0 0 amivi = amianivn = vnδnm = vm (26) i i,n n Application- Theorem: scalar product of two vectors is invariant under rotation of a coordinate system:

0 0 0 0 0 0 0 0 ~u · ~v = uivi = (amium)(anivn) = δnmumvn = unvn = ~u · ~v (27)

IV. VECTOR PRODUCT

Cross product, outer product. Given ~v, ~u deﬁne

~w = ~u × ~v (28)

Magnitude

|~w| = |~u||~v| sin θ (29)

In vector components:

~w = ~u × ~v =x ˆ(u2v3 − v3u2) +y ˆ(u3v1 − v3u1) +z ˆ(u1v2 − v1u2) (30)

Direction is determined by the right hand rule.