1. Background: Math Review Part I

2. 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

3. 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).

4. 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.

5. 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

6. 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.

7. 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.

8. 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.

9. [λ]  Transformation Matrix or
It is convenient to arrange the direction cosines into a square matrix: λ λ12 λ13
[λ]  λ λ22 λ23
λ λ32 λ33
In this notation, the coordinate axes are represented as column vectors:
x x1
[x]  x2 [x]  x2
x 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.

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

11. 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 = (a)
Direction cosines for a general line segment.

12. 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.

13. 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.

14. (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.

15. 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.

16. 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.

17. “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.