Verifying path independence Consider the potential functionφ(x, y) = (x2 - y2)/2 and its gradient field F = ⟨x, -y⟩.• Let C1 be the quarter-circle r(t) = ⟨cos t, sin t⟩, for 0 ≤ t ≤ π/2, from A(1, 0) to B(0, 1).• Let C2 be the line r(t) = ⟨1 - t, t⟩, for 0 ≤ t ≤ 1, also from A to B.Evaluate the line integrals of F on C1 and C2, and show that both are equal toφ(B) - φ(A).
Arc Length
Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length.
Line Integral
A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.
Triple Integral
Examples:
Verifying path independence Consider the potential function
φ(x, y) = (x2 - y2)/2 and its gradient field F = ⟨x, -y⟩.
• Let C1 be the quarter-circle r(t) = ⟨cos t, sin t⟩, for 0 ≤ t ≤ π/2, from A(1, 0) to B(0, 1).
• Let C2 be the line r(t) = ⟨1 - t, t⟩, for 0 ≤ t ≤ 1, also from A to B.
Evaluate the line integrals of F on C1 and C2, and show that both are equal to
φ(B) - φ(A).
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