Calculate the circulation, F. dr, in two ways, directly and using Stokes' Theorem. The vector field F = 8yi – 8xj and C is the boundary of S, the part of the surface z = 9 – x? y? above the xy-plane, oriented upward. Note that C is a circle in the xy-plane. Find a r(t) that parameterizes this curve. F(t) = with 出 くts (Note that answers must be provided for all three of these answer blanks to be able to determine correctness of the parameterization.) With this parameterization, the circulation integral is dt, where a and b are the endpoints you gave above. Evaluate your integral to find the circulation: F dr = Using Stokes' Theorem, we equate fF. dr Ss curl F dA. Find curl F = Noting that the surface is given by z = 9- x2 - y2, find HA = H dy da. With R giving the region in the ry-plane enclosed by the surface, this gives Ss curl F. dA = SR 田 dy da. Evaluate this integral to find the circulation:

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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: