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Course: Multivariable calculus > Unit 4

Lesson 3: Line integrals in vector fields
Math
Multivariable calculus
Integrating multivariable functions
Line integrals in vector fields
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Line integrals in conservative vector fields

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Problem

Define a scalar field φ(x,y)=x−y−x2+y2. Let the curve C be the perimeter of a quarter circle traversed once counterclockwise.
Find the line integral of the gradient of φ around the curve C.
∫C∇φ⋅ds=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi