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The streamlines of a planar fluid flow arc the smooth curves traccd by the fluid's individual particles. The vectors $\mathbf{F}=M(x, y) \mathbf{i}+N(x, y) \mathbf{j}$ of the flow's velocity field 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 $M_{x}+N_{y} \neq 0$ 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 $M_{x}+N_{y} \neq 0$ is called Bendixson's criterion for the nonexistence of closed trajectories.

Calculus 3

Chapter 16

Integrals and Vector Fields

Section 4

Green’s Theorem in the Plane

Vector Functions

Johns Hopkins University

Missouri State University

Campbell University

Idaho State University

Lectures

03:04

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x. The input of a function is called the argument and the output is called the value. The set of all permitted inputs is called the domain of the function. Similarly, the set of all permissible outputs is called the codomain. The most common symbols used to represent functions in mathematics are f and g. The set of all possible values of a function is called the image of the function, while the set of all functions from a set "A" to a set "B" is called the set of "B"-valued functions or the function space "B"["A"].

08:32

In mathematics, vector calculus is an important part of differential geometry, together with differential topology and differential geometry. It is also a tool used in many parts of physics. It is a collection of techniques to describe and study the properties of vector fields. It is a broad and deep subject that involves many different mathematical techniques.

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Bendixson's criterion…

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The $x$ component of veloc…

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The $y$ component of veloc…

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