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Calculus IIIChapter 5: Line IntegralsSection 5-5

Fundamental Theorem for Line Integrals — Question 3

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Question 3

Let f(x,y,z)=xy+z2,f(x,y,z)=xy+z^2, and let CC be any piecewise smooth closed curve contained in the domain of ff. Evaluate ∮C∇f⋅d𝒓.\oint_C\nabla f\cdot d\mathbf r.

Tasks

  1. Apply the endpoint form of the theorem to the closed curve.

  2. Explain the role of the curve’s initial and terminal points.

  3. State whether self-intersections affect the result.

Original worksheet page 1: question and worked solution for 5-5-003
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Question 3 – Solution

Strategy. Regard a closed curve as an oriented path whose terminal point equals its initial point.

Step 1: Name the common endpoint Let AA be the point at which the parametrization starts and ends. The theorem gives ∮C∇f⋅d𝒓=f(A)−f(A)=0.\oint_C\nabla f\cdot d\mathbf r=f(A)-f(A)=\boxed{0}.

See the diagram in the original worksheet below.

Step 2: Interpret The integral measures the accumulated change of ff during one traversal. Returning to the starting point returns to the same value of ff, so all increases and decreases cancel.

Step 3: Self-intersections Self-intersections do not alter the endpoint calculation. Each piece contributes the change in ff across that piece, and the intermediate values telescope when the pieces are added.

Verification Reversing the closed curve would negate its integral, but the negative of zero is still zero, consistent with the answer.

Original worksheet page 2: question and worked solution for 5-5-003

Original worksheet layout. Use Enlarge or open the PDF for a closer view.

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