Line Integral of a Vector Field

In this topic:

    (1) Define a line integral of a vector field and state how to evaluate it.
    
    (2) State the fact that the value of a line integrals does not depend on the parametrization of the given curve and show an example.
    

Definition (Line Integral of Vector Field) Let  



be a vector field, and let be a piecewise smooth orientable curve with parametric representation   



Using we define the line integral of along by  



  

  

  
  

Proposition (Line Integral of Vector Field) Let be a smooth curve and let be a continuous function with domain containing the trace of Then the value of the integrals   



depends only on the initial point terminal point and the trace of That is, two different parametrizations having the same trace from to yield the same values for these integrals.

Example (Line Integral of Vector Field) Consider the smooth curves



and



Both and are smooth curves from to with the same trace which is the portion of the parabola for For we have    and therefore   

  

For we have    and therefore   

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