Line Integrals of Vector Fields

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• Misc
  • GREAT Day
    • (Day of student research talks, art exhibits/performances, etc.)
    • Tuesday, April 19
    • No class(es)
    • Extra credit (up to 2 problem set points per summary) for math session summaries
      • 1 paragraph to 1 page discussing whatever connects the presentations in a session to each other
      • At most 3 sessions per person for extra credit
• Questions?
• Line Integrals of Vector Fields
  • 3rd and 4th parts of section 16.2
    • Tangent vector to parameterized path points “forward”
    • Line integral is integral of tangential component of field along curve
      
    • To evaluate:
      • Parameterize path
      • Plug parametric versions of x, y, z into field function
      • Integrate
        
    • Line integrals wrt dx, dy, dz
      • Can define vector with only 1 component in order to integrate wrt that variable
        
  • Arc length as a line integral
    
  • Questions
    • Are these and Tuesday’s scalar line integrals really the same thing?
      • Yes
        
    • How come vector field line integrals and integrals such as M(x,y,z) dx even get to be the same chapter?
      • Book formula for sum of integrals of M dx, N dy, and P dz is really just the integral of the dot product of F with derivative of r
      • So F dot dr and M dx + N dy + P dz are really just different notations for the same line integral
  • Examples
    • F(x,y,z) = ⟨ y, -x, z ⟩, C: straight line from (-1,-1,-1) to (1,1,1)
      • Field spirals away from XY plane
        
        
    • Integrate force of gravity (= ⟨0,0,-w⟩ where w is your weight) over spiral (r(t) = ⟨ 2cos(2πt), 2sin(2πt), 10t ⟩) and zigzag ( r₁(t) = ⟨0,10t,5t⟩, r₂(t) = ⟨0,10-10t,5+5t⟩ ) staircases from Tuesday
      
• Next
  • Applications of line integrals of vector fields
  • Read rest of section 16.2

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