Alternative construction of potential functions in ℝ2 Assume the vector field F is conservative on ℝ2, so that the line integral ∫C F ⋅ dr is independent of path. Use the following procedure to construct a potential function w for the vector field F = ⟨ƒ, g⟩ = ⟨2x - y, -x + 2y⟩ .a. Let A be (0, 0) and let B be an arbitrary point (x, y). Define φ(x, y) to be the work required to move an object from A to B, where φ(A) = 0. Let C1 be the path from A to (x, 0) to B, and let C2 be the path from A to (0, y) to B. Draw a picture.b. Evaluate ∫C1 F ⋅ dr = ∫C1 ƒ dx + g dy and conclude thatφ(x, y) = x2 - xy + y2.c. Verify that the same potential function is obtained by evaluatingthe line integral over C2.
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:
Alternative construction of potential functions in ℝ2 Assume the vector field F is conservative on ℝ2, so that the line
a. Let A be (0, 0) and let B be an arbitrary point (x, y). Define φ(x, y) to be the work required to move an object from A to B, where φ(A) = 0. Let C1 be the path from A to (x, 0) to B, and let C2 be the path from A to (0, y) to B. Draw a picture.
b. Evaluate ∫C1 F ⋅ dr = ∫C1 ƒ dx + g dy and conclude that
φ(x, y) = x2 - xy + y2.
c. Verify that the same potential function is obtained by evaluating
the line integral over C2.
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