Line Integrals of Vector Fields
Calculating the work done by a force as an object moves along a path.
Introduction
If represents a force field, the line integral represents the Work done by the force on a particle moving along C.
: Force helps the motion (with the wind).
: Force opposes the motion (against the wind).
: Force is perpendicular to motion.
Evaluation Formula
Recipe
- Parameterize the curve .
- Compute Velocity: (this replaces ).
- Evaluate Field: .
- Dot Product & Integrate: .
Visualizing Work
Interactive: Work along Parabola
Worked Examples
Example 1: Work on Helix
Calculate work done by along for .
1. Components:
- Field on Curve: .
- Velocity: .
2. Dot Product:
.
3. Integrate:
.
Term 1 (IBP): . Evaluated: .
Term 2: .
Term 3: over full period.
Total Work: .
Example 2: Against Gravity
Find work done by moving a particle from (0,0,0) to (1,2,3) along a straight line.
1. Parameterize:
, .
.
2. Dot Product:
.
3. Integrate:
.
Concept Check: Work = Force x Distance (Vertical) = -mg(3).
Example 3: Circulation
Evaluate for along unit circle.
1. Setup:
.
.
.
2. Dot Product:
.
3. Integrate:
.
The force is perfectly aligned with the motion at all times.
Practice Quiz
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