Background: Math Review Part I

Expected for this Course: Math Knowledge Expected for this Course: Differential & Integral Calculus Differential Equations Vector Calculus Physics Knowledge Newton’s Laws of Motion Energy & Momentum Conservation Elementary E&M

Definition of a Scalar: Consider an array of particles in 2 dimensions, as in the figure. Particle masses M are labeled by their x & y coordinates as M(x,y) Consider an array of particles in 2d, as in Fig. a. Particle masses labeled by their x, y coordinates M(x,y).

Now, consider rotating the coordinate axes, as in the figure Now, consider rotating the coordinate axes, as in the figure. Doing that, we find M(x,y)  M(x,y) That is, the masses are obviously unchanged by a rotation of coordinate axes. So, we say that the masses are Scalars! More generally, any quantity which is unchanged by an arbitrary 2D rotation is a Scalar (in this 2D space) Now, rotate coordinate axes, as in Fig. b. Masses labeled by M(x,y). But masses are obviously unchanged by transformation of axes: M(x,y) = M(x,y). Masses are scalars. General definition of a scalar: Any quantity which is invariant under a coordinate transformation.

2D Coordinate Transformations (Rotations in 2D) Consider an arbitrary point P in 3D space, labeled with Cartesian coordinates (x1,x2,x3). Rotate coordinate axes through angle  to (x1,x2,x3). The figure illustrates this in 2D Consider arbitrary point P in 3d space, labeled with Cartesian coordinates (x1,x2,x3). Now, rotate axes to (x1,x2,x3). See figure for 2d illustration. In 2D, it is easy to show that: x1  = x1cosθ + x2sin θ x2  = -x1sin θ + x2cos θ = x1cos(θ + π/2) + x2cosθ

x2  = -x1sinθ + x2cosθ = x1cos(θ +π/2) + x2cosθ Direction Cosines New Notation: The angle between the xi axis & the xj axis  (xi,xj) Define the Direction Cosine of the xi axis with respect to the xj axis: λij  cos(xi,xj) x1  = x1cosθ + x2sinθ x2  = -x1sinθ + x2cosθ = x1cos(θ +π/2) + x2cosθ So: λ11  cos(x1,x1) = cosθ λ12  cos(x1,x2) = cos(θ - π/2) = sinθ λ21  cos(x2,x1) = cos(θ + π/2) = -sinθ λ22  cos(x2,x2) = cosθ Direction cosine definitions. For 2d case, easily find relations shown.

General rotation of Axes in 3D: Rewrite the 2D Coordinate Rotation Relations in terms of direction cosines as: x1 = λ11 x1 + λ12 x2; x2 = λ21 x1 + λ22 x2 Or: xi = ∑j λij xj (i,j = 1,2) Generalize this to a General rotation of Axes in 3D: The angle between the xi axis & the xj axis is  (xi,xj). The Direction Cosine of the xi axis with respect to the xj axis: λij  cos(xi,xj) Direction cosine definitions. For 2d case, easily find relations shown.

xi = ∑j λijxj (i,j = 1,2,3) λij  cos(xi,xj) The Direction Cosine of the xi axis with respect to the xj axis: λij  cos(xi,xj) This gives: x1 = λ11x1 + λ12x2 + λ13x3 x2 = λ21x1+ λ22x2 + λ23x3 x3 = λ31x1 + λ32x2 + λ33x3 Or: xi = ∑j λijxj (i,j = 1,2,3) Direction cosine definitions. For 2d case, easily find relations shown.

[λ]  Transformation Matrix or It is convenient to arrange the direction cosines into a square matrix: λ11 λ12 λ13 [λ]  λ21 λ22 λ23 λ31 λ32 λ33 In this notation, the coordinate axes are represented as column vectors: x1 x1 [x]  x2 [x]  x2 x3 x3 In this notation, the general coordinate rotation is expressed by the relation: [x]  [λ][x] where [λ]  Transformation Matrix or Rotation Matrix Relation between 2 sets of axes becomes matrix relation, with direction cosines as square matrix & coordinate axes as column vectors.

Example   30° Work this example in detail! Work Example 1.1!!!!!!

Rotation Matrices  cosα, cosβ, cosγ x1, x2, x3 are  α,β,γ Consider a line segment, as in the figure  The angles between the line segment & the axes x1, x2, x3 are  α,β,γ The Direction Cosines of that line are clearly  cosα, cosβ, cosγ With trig manipulation, it can be shown that: cos2α + cos2β + cos2γ = 1 (a) Direction cosines for a general line segment.

cosα cosα +cosβcosβ +cosγcosγ (b) Now, consider 2 line segments, with direction cosines: cosα, cosβ, cosγ, & cosα, cosβ, cosγ as in the figure  Trig manipulation can be used to show that the angle θ between the line segments is related to the direction cosines by the formula: cosθ = cosα cosα +cosβcosβ +cosγcosγ (b) Direction cosines for a general line segment.

3 axes means 9 direction cosines: Arbitrary Rotations Now consider an arbitrary rotation from axes (x1,x2,x3) to (x1,x2,x3). Describe it by giving the direction cosines of all angles between original axes (x1,x2,x3) & the final axes (x1,x2,x3). 3 axes means 9 direction cosines: λij  cos(xi,xj) Math manipulation can be used to show that not all 9 are independent! It can be shown that: 6 relations exist between various λij: Giving only 3 independent λij. For proofs, see almost any mechanics book! Direction cosines for a general line segment.

(c)  Orthogonality condition. These results can be combined to show that: ∑j λij λkj = δik (c) δik Kronecker delta δik 1, (i = k); = 0 (i  k). (c)  Orthogonality condition. Transformations (rotations) which satisfy (c) are called ORTHOGONAL TRANSFORMATIONS Direction cosines for a general line segment.

ORTHOGONAL TRANSFORMATIONS If we consider the unprimed axes in the primed system, it can also be shown that: ∑i λij λik = δjk (d) It can also be shown that (c) (previous slide) & (d) are equivalent! ∑j λij λkj = δik (c) Direction cosines for a general line segment.

math & what use is it anyway? At this point, you may be wondering: Why do we need all of this (abstract) math & what use is it anyway? Answer We’ll soon be discussing Special Relativity in which space & time are treated on an equal footing in a 4D “Space Time” . Direction cosines for a general line segment.

“Lorentz Transformations” Lorentz Transformation When doing this, we’ll need to talk about “Lorentz Transformations” of coordinates in which space & time get mixed when going from one coordinate system to another. What we’ll see is that a Lorentz Transformation can be viewed (has the same mathematical form) as an Orthogonal Transformation (rotation) in 4 D Space time. Direction cosines for a general line segment.