Parameterizing Curves Lecture

Introduction

• Topic: Parameterizing a curve in 2D or 3D.
• Purpose: To describe all points along a curve using a single variable, essential for computing line integrals.

Examples of Parameterizing Curves

Part A: Line Segment in 3D

• Points: From (1, 2, 3) to (7, -6, 5).
• Parameterization Steps:
  1. X-coordinates: Starts at 1 and increases by 6.
     • Equation: x = 1 + 6t for t in [0, 1].
  2. Y-coordinates: Starts at 2 and decreases by 8.
     • Equation: y = 2 - 8t for t in [0, 1].
  3. Z-coordinates: Starts at 3 and increases by 2.
     • Equation: z = 3 + 2t for t in [0, 1].
• Visualization:
  • t = 0: starting point (1, 2, 3).
  • t = 1: ending point (7, -6, 5).
  • Result: Describes the line segment using the parameter t.

Part B: Parabola (y = x² + 1)

• Range: X ranges from 1 to 3.
• Parameterization:
  • x = t.
  • y = t² + 1.
  • t ranging from 1 to 3.
• Visualization:
  • Captures the portion of the parabola from x = 1 to x = 3.

Part C: Part of a Circle (x² + y² = 9) in the Second Quadrant

• Circle Details: Radius 3, Centered at the origin.
• Range: Second quadrant, angles from π/2 to π.
• Equation:
  • Using polar coordinates: x = 3cos(t), y = 3sin(t).
• Visualization:
  • Describes the part of the circle in the second quadrant using t in [π/2, π].

Comments on Parameterizing Curves

Path Matters

• Example: Cannot use a straight line to parameterize a curved path.

Non-Unique Parameterizations

• Example: Line segment (1, 2, 3) to (7, -6, 5)
  • Standard: t in [0, 1].
    • x = 1 + 6t, y = 2 - 8t, z = 3 + 2t.
  • Alternate: t in [0, 2].
    • x = 1 + 3t, y = 2 - 4t, z = 3 + t.
  • Point: Multiple valid ways to parameterize.

Direction Matters

• Context: Depending on the type of line integral.
  • Line Integral w.r.t Arc Length (F ds): Direction doesn't matter.
  • Line Integrals w.r.t Coordinates (P dx + Q dy + R dz): Direction matters.
  • Key: Check the type of integral to determine if direction matters.