**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

1.2.2 : Gradient

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

E No idea, review please

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

What about these two?

+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?

how was the reading?

**I clickers**

"Can you explain tensor a little better?"

"Could you go over gradients?"

"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

What is the gradient of

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

Gradient

**Vector operations of Del**

**Review**

Vector algebra

Components

Triple product

Separation vector

Transforms

Differential Calculus

Gradient

Del Operator

- easy, dot, cross

- again, but in components

- look them up

- useful later

- bit of math, but nice to have

- what a derivative is

- partial derivative, def. of Grad

- vector operator

"I’ve never learned about triple products or tensors"