Monthly Archives: September 2010

Classical Relativity

Classical Relativity
    Stars and Planets are in constant motion
    And so instead of absolute units we use relative units.

Someone on Earth and someone standing on moving Jupiter watching me run along will see me moving in a different directions. We use coordinates to describe what is the positions Vs time relative to the earth, This can be described using a space time graph, with time on the vertical and position on the horizontal. The line formed by observing this is called the "World Line".

Earth (at Rest)           Jupiter (at v)
t                                    t’
x                                   x’
u                                   u’
a                                   a’
F                                  F’
PE                               PE’
KE                              KE’

The rules of physics are the same in both places, we just need to relate the coordinates.
t’  = t
x’ = x-vt
u  = u-v
F’ = F
a’ = a

Those were the rules until around the 1800’s, but E&M don’t work with this, because once a two charges have velocity, it acquires Magnetic Force, besides the old Electric Force. If you divide the Magnetic force by the Electric force, you are left with the speed of light as the exception. So until your up near the speed of light, this is not much of a problem.
GRAPHIC 1

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This is Calculus 3++…

We reviewed Vectors today, how vectors can be used to describe a line in 2D and 3D space. <1,3> + t<1,1> or (1+t)i + (3+t)j and in 3D 2si + (2+2s)j Why the s and the t? They are just variables describing the scaling factor of your place in the system.

For describing planes, this type of generalization works as well. There are two direction vectors and one initial vector pointing to the zero-zero location of the plane and two direction vectors describing the tilt of the plane.

For example, the Equation of three points, as a plane is:
POINTS: (1,2,3) , (5,5,5) , (-1,6,9)
Form a line with a starting zero zero location of <1,2,3> plus a scaling in t direction and s direction.
<1,2,3> + t<4,3,2> + s<-2,4,6>

Vector Rules
u · u > 0 unless u = 0
u · v = v · u
(u + v) ·w = (u ·w) + (v ·w)
r(u · v) = (ru)  · v = u ·(rv)
*****r is a scalar

DOT PRODUCT
u · v = |u| |v| cos θ
u · v = u1v1+u2v2+…unvn

Why does the dot product work this way?? Because of the Law of Cosines, which states:

c2 = a2 + b2 – 2ab cos θ

and if we work out a triangle, eventually we will see
a = u
b = v
c = u – v
|u – v|2 = |u|2 + |v|2 – 2|u||v| cos θ

If we want to find the angle between two vectors, we can use the dot product.
arccos (a∙b / (|a|*|b|) )
That is regular multiplication of the length of a and b.

Now we can project b onto A by using |b| cos θ

And we can describe a plane, by saying (x – p) ∙n = 0
So we are describing the plane with a normal n and a point (is that p?)

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Introduction to Multi-Var Calc and Lin Alg

Previously in calculus we did things like y=mx+b with m being the slope and for x not varying much.

The goal of this class is to understand things like z = x2+y2, a three dimensional shape, similar to a bowl. One dimensional curves in space and vector fields, like the flow of water in a river.

The best way to do this is through Vectors and linear transformations (Matrices).

The right hand coordinate system, has x emerging from the paper, y traveling to the right and z to the top of the page. The left hand coordinate system swaps y and x.

A vector has length and direction. For this class we will write in notation R3 meaning the real number system in three dimensions. We can describe locations in Rn by position vectors. Points are different than vectors. You can’t add points, but it vectors can be added or subtracted.

For the purposes of this class [#,#] is a vector Square Brackets, (0,0) is a point.

There is a special vector, the zero vector [0,0]. The length of vector v is the absolute value of v. I should write little hats on top of my vectors, but I can’t easily type those in.

Describe a Line in a plane
At some time t, the position of the vector passing through (0,0) of the plane and with a slope of one, the line can be described as t[1,1]

For some other line with a slope of one-half and passing through the point (0,2) this line can be described as [ 2t , t+2 ]

Line = starting point + t(direction)
Line = u +tv
u
and v are vectors. t is not. u is the zero vector for this point.

To describe a plane, use two vectors
[1,0,0] + t[-1,1,0]+s[-1,0,1]

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