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Unformatted text preview: Lecture 26 - Wednesday June 3rd [email protected] 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 γ is an oriented curve in a fluid with velocity field f , then the line integral Z γ f · dr is referred to as the circulation of f around γ . It is negative if the fluid tends to flow in the opposite direction to the curve γ , and zero if the fluid tends to flow perpendicular to...
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- Spring '07
- Math, Manifold, Vector field, Stokes' theorem