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Spherical and cylindrical coordinates

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

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Spherical and cylindrical coordinates

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

  2. Math 200 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 Goals

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

  4. 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 Math 200 Surfaces

  5. 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 Math 200

  6. 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 φ θ θ Math 200 z ρ Spherical coordinates y x

  7. 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 φ θ θ Math 200 z ρ converting For φ, z is the adjacent side y z x

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

  9. 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! Math 200 Surfaces in Spherical

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

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

  14. Math 200 φ ρ θ

  15. 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 Math 200 Example 2: Converting Surfaces

  16. 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! Math 200 Example 3: Converting more surfaces

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