Polar Coordinates
x = rcosθ y = rsinθ
 Polar Coordinates are another way of representing graph systems
 How to we apply polar coordinates to what we have been doing with double and triple integrals?
 There is a proof for this, but this is what we feed into our double and triple integrals for polar coordinates!
 You can use polar coordinates for functions that are not nice with regular coordinates.
 A classic example of such would be x^{2} + y^{2} = 1
Polar Coordinates for Double Integrals
Step One: Convert from f(x,y) to g(r, θ)
Step Two: Convert dx/dy to dArθ  Tip for this, it will ALWAYS be rdrdθ
Step Three: Convert limits (Plug in and work out)  This can be a bit tricky.
Step Four: Now, integrate as normal. USub may be needed
Polar Coordinates for Triple Integrals
Now, we are dealing with cylindrical coordinates! (Polar in 3D)
x = rcosθ y = rsinθ z = z
dVol ⇛ dxdydz ⇛ rdrdθdz
 Volume = ∫∫∫dVol
Spherical Coordinates (Globe)
x = rcosθsinϕ y = rsinθsinϕ z = rcosϕ
r^{2} = x^{2} + y^{2} +z^{2}
 With Spherical Coordinates, you are going to have an R like usual
 However, you are going to have TWO angles
 A polar angle along X and Y, and an azimuthal angle along Z
 The regular angle can stretch from 0 < θ < 2π
 The azimuthal angle can stretch from 0 < ϕ < π
 It is not necessary for the full 2π, as you would be double accounting for each position
 Remember with Usub, you can either change your limits in terms of U, or sub in for you after the integration with the original limits
END OF TEST 2 MATERIAL
Vector Spaces

A vector spaces is a set of vectors

Addition and Scalar multiplication can be used!

Vector spaces can be anything, vectors, polynomials, functions

If I have two polynomials of the first degree ,
u = x + 1
andv = 2x  3
, they both share the same vector space, because they are same degree 
If I add the polynomials
u = x + 1
andv = 2x  3
together, I getuv = 3x  2

The resulting
uv = 3x  2
is also a polynomial of degree one, so we’re still in that original vector space! 
The same is true if I multiplied
u = x + 1
by 5. I would get5u = 5x +5
.5u
is still a polynomial of degree one, so we’re still in that first degree polynomial space!
—
 The same is true of I have the two matrices
and
$v^=[−11 01 ]$ If I add and
v̂
, I getû + v̂
 This results in a 2D vector, meaning that we are staying in the same 2D vector space.
5û
would result in the same thing, another 2D vector. We’re good here.
—
These are known as the properties of closure.

Given
û
εV (û
is a vector in the vector space V) and scalarC
, thenCû
ε V (The multiplied scalar will also be in vector space V) 
Given
û
εV andv̂
εV , thenû + v̂
ϵ V (The new added vector is also in vector space V!)
The dimension of the vector is usually noted by ℝ^{n}. The amount of non zero rows in a vector will give you that.
—
What are Axioms? What are the 8 Axioms?
Axioms are a bunch of rules which hold true inside of a vector space. There are 10 of which hold true in any vector space V
With vectors û, v̂ and ŵ, and scalars a and b
Addition

û + v̂
=v̂ + û
This is known as the commutative property of addition! 
û + (v̂ + ŵ)
=(û + v̂) + ŵ
This is known as the associative property 
û + 0 = û + 0
There is a zero vector. This is known as the additive identity 
û + û'
For every vector in a vector space, there is another vector that can bring it back to zero. This property is known as the additive inverse! Multiplication 
For ûϵ**V*
1û = u
For every single vector in the vector space V, 1* û will equal û. This is known as the multiplicative identity 
a * (bu) = (ab) * u
This property does not seem to have a name, but is used to combine scalars. Scalars can be combined as such. 
(a + b)u = au + bu
This is known as distributivity of scalar multiplication with respect to field addition. Vectors can distribute like this! 
a(u+v) = au + av
This is known as distributivity of scalar multiplication with respect to vector addition. Just like the last property, this works as well!