Cartesian, Polar, and Cylindrical Coordinates Study Notes

Cartesian Coordinates

  • Definition: A coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are generally represented as (x,y).
  • Components:
    • x-axis
    • y-axis
  • Point P in Cartesian coordinates:
    • Coordinates: (x, y)
    • Position: First value represents the horizontal location, and the second value represents the vertical location on the graph.

Polar Coordinates

  • Definition: A coordinate system where each point on a plane is determined by a distance from a reference point and an angle from a reference direction.
  • Components:
    • r (radius): The distance from the origin to the point.
    • θ (theta): The angle measured from the positive x-axis, typically in radians.
  • Conversion from Polar to Cartesian:
    • Given polar coordinates (r, θ):
    • x = r cos(θ)
    • y = r sin(θ)
  • Conversion from Cartesian to Polar:
    • Given Cartesian coordinates (x, y):
    • r = √(x² + y²)
    • θ = tan⁻¹(y/x),
      • Adjust θ based on the quadrant the point lies in:
      • Quadrant I: (x > 0, y > 0)
      • Quadrant II: (x < 0, y > 0)
      • Quadrant III: (x < 0, y < 0)
      • Quadrant IV: (x > 0, y < 0)

Cylindrical Coordinates

  • Definition: An extension of polar coordinates to three-dimensional space.
  • Components:
    • r (radial distance): Same as in polar coordinates.
    • θ (theta): Same angle as in polar form.
    • z: The height above or below the xy-plane.
  • Point P in cylindrical coordinates:
    • Coordinates: (r, θ, z)
    • The projection of the point P onto the xy-plane is given by (r, θ).
  • Conversion from Cylindrical to Cartesian:
    • x = r cos(θ)
    • y = r sin(θ)
    • z = z
  • Conversion from Cartesian to Cylindrical:
    • r = √(x² + y²)
    • θ = tan⁻¹(y/x)
    • z = z

Conversions and Examples

  • Given cylindrical coordinates (2, 2π/3, 1):

    • Find rectangular coordinates:
    • θ = 2π/3 → 120 degrees.
    • r = 2, move in that direction.
    • z = 1 → Move up 1 unit from the xy-plane.
    • x = 2 cos(2π/3) = -1 ( (r = 2) )
    • y = 2 sin(2π/3) = √3 ( (r = 2) )

  • New Example: Given rectangular coordinates (3, -3, -7):

    • r² = x² + y² → r² = 18 ⇒ r = 3√2
    • θ = tan⁻¹(-3/3) → θ = -π/4 (Quadrant IV)
    • z = -7

  • Resulting cylindrical coordinates: (3√2, -π/4, -7)

Surfaces Described by Cylindrical Coordinates

  • Surfaces:
    • Circular Cylinder: Centered on the z-axis:
    • Description: x² + y² = c²
    • Ranges: θ ∈ [0, 2π], z ∈ (-∞, ∞)
    • Vertical Plane: θ = constant
    • Description: Describes a plane in 3D.
    • Allows for r (distance from the z-axis) and z (height) to vary freely.
    • Cone: When z = r
    • Appears as a conical surface radiating around z.

Triple Integrals in Cylindrical Coordinates

  • Setup: If you want to evaluate a triple integral in cylindrical coordinates:
    • Integral is of the form:
      \int \int \int_E f(x, y, z) \, dV
    • Use transformation:
    • dV = r \, dr \, d\theta \, dz
  • Volume Boundaries:
    • For solid regions:
    • Type one region where z is bound between functions of x and y in polar projections.
  • Example: Volume under paraboloid modeled as:
    • z = 4 - (x^2 + y^2)
    • Convert boundaries:
    • r ∈ [0, 2], θ ∈ [0, 2π], and 0 ≤ z ≤ 4 - r².

Integrating Example

  • Function integrated: x^2 becomes (r^2 cos^2(\theta)) when replacing variables.
  • Evaluate to find:
    • \int{0}^{2} \int{0}^{2\pi} f \cdot r \, d\theta \, dr
  • Results after simplifications lead to volumes related to specific constants.