Bendixson’s criterion The streamlines of a planar fluid floware the smooth curves traced by the fluid’s individual particles.The vectors F = M(x, y)i + N(x, y)j of the flow’s velocity fieldare the tangent vectors of the streamlines. Show that if the flowtakes place over a simply connected region R (no holes or missingpoints) and that if Mx + Ny 0 throughout R, then none of thestreamlines in R is closed. In other words, no particle of fluid everhas a closed trajectory in R. The criterion Mx + Ny ≠ 0 is calledBendixson’s criterion for the nonexistence of closed trajectories.
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:
Bendixson’s criterion The streamlines of a planar fluid flow
are the smooth curves traced by the fluid’s individual particles.
The
are the tangent vectors of the streamlines. Show that if the flow
takes place over a simply connected region R (no holes or missing
points) and that if Mx + Ny 0 throughout R, then none of the
streamlines in R is closed. In other words, no particle of fluid ever
has a closed trajectory in R. The criterion Mx + Ny ≠ 0 is called
Bendixson’s criterion for the nonexistence of closed trajectories.
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