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# F16 PH 333 1.1-1.2.3

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## Richard Datwyler

on 18 September 2018

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#### Transcript of F16 PH 333 1.1-1.2.3

1.1 - 1.2.3
Topics
1.1.1 : Vector Operations
1.1.2 : Vector Algebra
1.1.3 : Triple Products
1.1.4 : Position, Displacement and Separation Vectors
1.1.5 : Vector Transformations
1.2.1 : "Ordinary Derivatives

1.2.3 The Del Operator
Vector operations
Cross product
Dot product
Distributive
Commutative
Associative
A
B
Turn to your neighbor and work these out.
30 seconds....
Vector Algebra: Components
What is the angle between these two vectors?
recall
A 64
B 154
C 52
D I could do it, but no calculator
Triple product
Just go look them up, here are some
one of your homework will come from here.
Position, Displacement, and Separation vectors
-Q
+Q
x
What is the direction of force of attraction between these two charges?
0
-x
+Q
-Q
x
+Q
-Q
and these?
z
y
x
(x,y,z)
z
y
x
(x`,y`,z`)
z`
y`
x`
the vector between
them
r`
r
In Cartesian coordinates this is messy. but the representation is the same
Vectors Got it!?!
Define (length and direction)
But, components of what, and direction with regard to what?
Define (made of 3 components in 3D)
The key to a vector is to transform from one coordinate system to another.
z
y
y
o
z
rotate by o
Welcome back
how was the quiz?
I clickers
"Can you explain tensor a little better?"
"Would you give us some of the application of cross product and dot product ?."
"For the separation vector, is it easier to use cartesian coordinates?"
"I have a hard time remembering that r is the point at which we're measuring and r' is the point where the source is located. I tend to mix them up. Any suggestions on how I can keep these two straight"
"It was hard for me to understand how matrices related to vector transformation. Also, I think that I would like to review in more detail the del operator. "
A
0
0
Practice
X axis
Y axis
Z axis
z to y & -y to z

y to z & -z to y
x to z & -z to x

z to x & -x to z

y to x & -x to y

x to y & -y to x

1
7
6
5
4
3
2
1,1,1 axis
x to z, y to x, z to y
z to x, y to z, x to y
Datwyler
Find me the transform matrix that takes rotations around these axis:
Differential Calculus
Things change.

but when they do, they can cause other things to change.
when anything changes we call it
if it is a variable (like x) then

if they are associated with each other the derivative, ,tells you the proportionality factor.
if it is a function (like f) then
Giving
Example position,velocity, and time
Changes in a Vector
What if that function depends on 3 (or more) quantities. If one of them change, how does it affect the function.
Example: elevation.
The rate of change in elevation and hence the change in elevation itself, depends upon the direction you take
This is where partial derivatives come in, and they aren't that bad.
Partial derivative.
We define this term to be called the gradient.
note, interpretation
points in direction of biggest change
magnitude gives the rate of change
Practice
E Not sure, please review some more
A
D
C
B
Nabla
The nabla symbol is called the Del operator, it 'acts' on things, and is not a vector by itself.
We define vector multiplication in three ways
scalar product
dot product
cross product
The del operator is the same
Curl
Divergence
Vector operations of Del
Review
Vector algebra
Components
Triple product
Separation vector
Transforms
Differential Calculus