1. Spherical and cylindrical coordinates
Math 200
Week 8 - Friday
Spherical and cylindrical coordinates

2. Math 200
Goals
Be able to convert between the three different coordinate systems in 3-Space: rectangular, cylindrical, spherical
Develop a sense of which surfaces are best represented by which coordinate systems

3. Cylindrical coordinates
Math 200
Cylindrical coordinates
Cylindrical coordinates are basically polar coordinates plus z
Coordinates: (r,θ,z)
x = rcosθ
y = rsinθ
z = z
r2 = x2 + y2
tanθ = y/x x y z θ r
Just like 2D polar

4. Math 200
Surfaces
Let’s look at the types of surfaces we get when we set polar coordinates equal to constants.
Consider the surface r = 1
This is the collection of all points 1 unit from the z- axis
Or, using our transformation equations, it’s the same as the surface x2+y2=1

5. Or, since tanθ = c, we have y/x = c
Math 200
How about θ=c?
This is the set of all points for which the θ component is fixed, but r and z can be anything.
Or, since tanθ = c, we have y/x = c
y = cx is a plane θ

6. Spherical coordinates
Math 200
Spherical coordinates
Coordinates: (ρ, θ, φ)
ρ: distance from origin to point
θ: the usual θ (measured off of positive x-axis)
φ: angle measured from positive z-axis x y z θ ρ φ

7. For φ, z is the adjacent side
Math 200
converting
Let’s start with ρ:
From the distance formula/Pythagorus we get ρ2=x2+y2+z2
We already know that tanθ=y/x
Lastly, since z = ρcosφ, we have x y z ρ φ θ z
For φ, z is the adjacent side

8. r is the opposite side to φ
Math 200
Going the other way around is a little trickier…
From cylindrical/polar, we have
r is the opposite side to φ x y z ρ r φ θ
Notice that r = ρsinφ. So,

9. Surfaces in Spherical Let’s start with ρ=constant
Math 200
Surfaces in Spherical
Let’s start with ρ=constant
What does ρ=2 look like?
It’s all points 2 units from the origin
Also, if ρ=2, then ρ2=4. So, x2+y2+z2=4
It’s a sphere!

10. From the conversion formula we have
Math 200
How about φ=constant?
Let φ = π/3.
From the conversion formula we have
Recall: z2=x2+y2 is a double cone
Multiplying the right-hand side by 1/3 just stretches it
Let’s simplify some

11. For spherical coordinates, we restrict ρ and φ ρ≥0 and 0≤φ≤π
Math 200
For spherical coordinates, we restrict ρ and φ
ρ≥0 and 0≤φ≤π
So, φ=π/3 is just the top of the cone

12. Example 1: Converting Points
Math 200
Example 1: Converting Points
Consider the point (ρ,θ,φ) = (5, π/3, 2π/3)
Convert this point to rectangular coordinates
Convert this point to cylindrical coordinates
Rectangular
In rectangular coordinates, we have

13. Math 200
Polar:
We already have z and θ:

14. Math 200 φ ρ θ

15. Example 2: Converting Surfaces
Math 200
Example 2: Converting Surfaces
Express the surface x2+y2+z2=3z in both cylindrical and spherical coordinates
Cylindrical
Using the fact that r2=x2+y2, we have r2+z2=3z
Spherical
Using the facts that ρ2=x2+y2+z2 and z = ρcosφ, we get that ρ2=3ρcosφ
More simply, ρ=3cosφ

16. Example 3: Converting more surfaces
Math 200
Example 3: Converting more surfaces
Express the surface ρ=3secφ in both rectangular and cylindrical coordinates
We can rewrite the equation as ρcosφ=3
This is just z = 3 (a plane)
Conveniently, this is exactly the same in cylindrical!