20e-lecture26

20e-lecture26 - Lecture 26 - Wednesday June 3rd...

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Unformatted text preview: Lecture 26 - Wednesday June 3rd jacques@ucsd.edu Key words: Circulation, vorticity vector, Stokes Theorem 26.1 Stokes Theorem There is a second interpretation of the curl of a vector field when the vector field represents the velocity field of a fluid. For such a vector field f , the curl f at each point is exactly twice the angular velocity vector of a solid body which approximates the motion of the fluid near that point. If the vector field is conservative so that curl f = 0 then the vector field is also called irrotational , which means that a small solid body floating in the fluid tends only to exhibit lateral motion and no angular motion. More generally, ( f ) n denotes the tendency of the fluid to rotate around a normal axis n : precisely it is the circulation of the fluid per unit area at a given point on a surface perpendicular to the axis n . The curl is often called the vorticity vector . If is an oriented curve in a fluid with velocity field f , then the line integral Z f dr is referred to as the circulation of f around . It is negative if the fluid tends to flow in the opposite direction to the curve , and zero if the fluid tends to flow perpendicular to...
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This note was uploaded on 02/12/2010 for the course MATH 20E taught by Professor Enright during the Spring '07 term at UCSD.

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20e-lecture26 - Lecture 26 - Wednesday June 3rd...

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