Notes on Converting Between Coordinate Systems (Rectangular, Polar, Spherical)

Overview and Objectives

• Topic: Convert between systems of coordinates using vector operations and linear algebra principles.
• Focus areas: polar, rectangular (Cartesian), and spherical coordinates; converting equations into spherical coordinates; understanding 3D coordinates.
• Objectives (as stated):
  • Convert points from polar into rectangular coordinates, or vice versa.
  • In space, convert points from spherical into rectangular coordinates.
  • Convert equations into spherical coordinates.
• Practice references (from transcript):
  • Exercises 9.4 #1-2, 5-6, 7-12 (one representation only), 13, 15-17.
  • Exercises 13.7 #1-6, 7-14.
• Context: Examples are drawn from Calculus: Early Transcendental Functions, 5th ed., Smith, Minton, Rafhi (McGraw-Hill).

Rectangular (Cartesian) Coordinates

• In the plane:

  • Rectangular coordinates are the standard Cartesian coordinates, denoted as

  (x, y)

  • Visual: coordinate grid with x-axis (horizontal) and y-axis (vertical).

• Key idea: Points in the plane are identified by their x and y components.

Polar Coordinates

• Polar coordinates describe a point by:
  • r: distance from the origin to the point.
  • \theta: angle in standard position (measured from the positive x-axis).
• Relationship to Cartesian: a point with polar coordinates corresponds to Cartesian coordinates through
  • x = r\cos\theta
  • y = r\sin\theta
• In space, polar coordinates share the same interpretation for the 2D plane, with r > 0 and a principal angle \theta.

Angles, Radians, and Representations

• Angles are measured in radians in Calculus context:
  • 180^{\circ} = \pi radians.
  • Positive angles cause counterclockwise rotation.
• Usual restrictions for polar representation (to ensure uniqueness for all points except the origin):
  • r > 0
  • 0 \le \theta < 2\pi (alternatively, -\pi < \theta \le \pi).
• With these conventions, every point other than the origin has exactly one polar representation.
• Conceptual note: The constant factor in differentiation of degrees is non-arbitrary when changing units (e.g., differentiate sin x° w.r.t. x in degrees yields a factor \pi/180). This underscores that unit choice affects derivatives.
• Tip: When dealing with degrees, you effectively compose with a linear scaling from degrees to radians, which introduces constants in derivatives.

Practice and Visualization Hints

• Try Chapter 9 exercises on a graphing device to verify results and gain a dynamical picture of the relationships.
• Visual intuition: Use dynamic geometry tools to rotate and observe how (r, θ) map to (x, y).

From Polar into Rectangular

• Core conversions:
  • x = r\cos\theta
  • y = r\sin\theta
• These equations convert polar coordinates to rectangular (Cartesian) coordinates.

From Rectangular into Polar

• Core conversions:
  • r^2 = x^2 + y^2
  • \tan\theta = \frac{y}{x}
• If we restrict r > 0, then
  • r = \sqrt{x^2 + y^2}
• Important caution: The standard arctangent function \tan^{-1}(y/x) returns values in (-\tfrac{\pi}{2}, \tfrac{\pi}{2}), so a correction is needed when x < 0 to place the angle in the correct quadrant.

Quadrant Rules for (\theta) and Related Questions

• Explicitly, for 0 ≤ \theta < 2\pi:
  • If x > 0 and y > 0 (Quadrant I):
  • \theta = \tan^{-1}\left(\frac{y}{x}\right)
  • If x < 0 (Quadrants II, III):
  • \theta = \tan^{-1}\left(\frac{y}{x}\right) + \pi
  • If x > 0 and y < 0 (Quadrant IV):
  • \theta = \tan^{-1}\left(\frac{y}{x}\right) + 2\pi
• Query: Can you develop similar rules for -\pi < \theta \le \pi?
• Also, what about cases where x = 0?
• Tip: Do not memorize these rules; instead, draw a picture and interpret the quadrant placement each time.

Beyond the Exam and Visualization Hints

• Explore: Beyond the exam topics include: Pythagoras, multiple representations, and polar curves; use visual intuition to understand representations beyond standard forms.

In Space: Orientation and the Right-Hand Rule

• Rectangular coordinates in 3D:
  • The coordinate system can be rotated but not reflected.
  • Right-hand rule for orientation: If your right palm moves from the positive x-axis to the positive y-axis, your thumb points toward the positive z-axis.

Three-Dimensional Plotting (Example)

• Example 2.1 (from Chapter 10): Plotting points in three dimensions.
  • Points to plot: (1, 2, 3), (3, -2, 4), (-1, 3, -2).
• Visualization notes: Each point lies in 3D space with coordinates corresponding to distances along the x-, y-, and z-axes respectively.

Spherical Coordinates

• Definition in space:
  • \rho: distance from the origin to the point P.
  • \phi: angle from the positive z-axis to the segment OP (0 ≤ φ ≤ π).
  • \theta: angle in standard position in the (x,y)-plane.
• Note: The pair $(\phi, \theta)$ together describe the direction (they are referred to as a solid angle in context).
• Usual restrictions (to ensure uniqueness):
  • \rho \ge 0
  • 0 \le \phi \le \pi
  • 0 \le \theta < 2\pi (alternatively, -\pi < \theta \le \pi).
• Visualization aid: Imagine measuring the distance \rho from the origin to P, placing a rod of length \rho on the positive z-axis, rotating it in the xz-plane to match the height of P to determine \phi, then rotating around the z-axis to align with the projection of P in the xy-plane to determine \theta.

From Spherical into Rectangular (Conversion Formulas)

• Rectangular coordinates in terms of spherical:
  • x = \rho \sin\phi \cos\theta
  • y = \rho \sin\phi \sin\theta
  • z = \rho \cos\phi
• Additional relationship:
  • r = \rho \sin\phi
• Useful note: It helps to denote r = \rho \sin\phi when converting to cylindrical coordinates or interpreting cross-sections.

From Rectangular to Spherical (and Order Considerations)

• In the transcript, a caution is given about the order of variables when using spherical coordinates in problems:
  • The triple (\rho, \phi, \theta) is to be read in that order.
  • When solving for the coordinates, you typically need two independent equations to determine the three Cartesian coordinates $(x, y, z)$ from the spherical ones.
• This serves as a reminder to keep track of which variable corresponds to which coordinate in multi-variable conversions.

Practice Hints and Study Tips

• Use Chapter 9 for practice on polar and rectangular conversions and for visualizations.
• Draw pictures to interpret angle placements rather than simply memorizing formulas (especially for cases where $x$ is negative or zero).
• Explore additional resources beyond the core material, such as topics on multiple representations and polar curves, to deepen understanding.

Quick Reference: Key Formulas and Concepts

• Rectangular to polar (2D):
  • x = r\cos\theta, y = r\sin\theta
  • r^2 = x^2 + y^2, \tan\theta = \frac{y}{x}
• Polar to rectangular (2D):
  • x = r\cos\theta, y = r\sin\theta
• Polar angle domain conventions (2D):
  • 0 \le \theta < 2\pi or -\pi < \theta \le \pi
• 3D (Rectangular) orientation: right-hand rule; rotation allowed without reflection
• Spherical coordinates:
  • x = \rho \sin\phi \cos\theta, y = \rho \sin\phi \sin\theta, z = \rho \cos\phi
  • r = \rho \sin\phi
• Spherical domain restrictions:
  • \rho \ge 0, 0 \le \phi \le \pi, 0 \le \theta < 2\pi (or -\pi < \theta \le \pi)

Exercises and References

• Exercises to consult for practice:
  • 9.4: #1-2, 5-6, 7-12 (one representation only), 13, 15-17
  • 13.7: #1-6, 7-14
• Note: Some examples are excerpted from the textbook; use the provided formulae and conversions to verify results.

Summary of Key Points

• Polar and rectangular coordinates are two representations of the same plane; they convert via x = r\cos\theta, \ y = r\sin\theta.
• Every non-origin point has a unique polar representation under the standard restrictions; origin has multiple representations depending on how r and θ are chosen.
• In 3D, rectangular coordinates extend to spherical coordinates with the definitions above; orientation in 3D follows the right-hand rule, and spherical coordinates use the distance \rho, the angle from the z-axis \phi, and the azimuthal angle \theta in the xy-plane.
• Conversions between spherical and rectangular require three equations and are sensitive to the order of the variables $(\rho, \phi, \theta)$; two independent equations are often needed to determine Cartesian coordinates from spherical ones.
• Visual aids and dynamic graphing tools are recommended to deepen understanding of the relationships among coordinate systems.