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

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

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

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

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

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

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

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,

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!

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

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

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

Math 200 Polar: We already have z and θ:

Math 200 φ ρ θ

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

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!