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2-9 Curl of a vector

2-9 Curl of a vector It is an axial vector whose magnitude is the maximum circulation of per unit area as the area tends to zero and whose direction is the normal direction of the area when the area is oriented so as to make the circulation maximum.

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2-9 Curl of a vector

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  1. 2-9 Curl of a vector • It is an axial vector whose magnitude is the maximum circulation of per unit area as the area tends to zero and whose direction is the normal direction of the area when the area is oriented so as to make the circulation maximum. • curl  a measure of the circulation or how much the field curls around P. (2-125) Lecture 06

  2. (2-135) Lecture 06

  3. In order to attach some physical meaning to the curl of a vector, we will employ the small “paddlewheel”. Let the vector field be a fluid velocity field. Place the small paddlewheel in this velocity field. The paddlewheel axis should be oriented in all possible directions. The maximum angular velocity of the paddlewheel at a point is proportional to the curl, while the axis points in the direction of the curl according to the right-hand rule. If the paddlewheel does not rotate, the vector field is irrotational, or has zero curl. Lecture 06

  4. or Cartesian coordinates (2-136) (2-135) Cylindrical coordinates (2-138) Lecture 06

  5. In spherical coordinates • Properties of the curl • 1) The curl of a vector is another vector • 2) The curl of a scalar V, V, makes no sense • 3) • 4) • 5) • 6) (2-139) Lecture 06

  6. Example 2-21 Show that if • a). • b). • Solution • a). Cylindrical coordinates Lecture 06

  7. b). Spherical coordinates Lecture 06

  8. Irrotational or Conservative field. Lecture 06

  9. 2-10 Stokes’s theorem • Proof: (2-143) The circulation of around a closed path L is equal to the surface integral of the curl of over the open surface S bounded by L, provided that and are continuous on S. Lecture 06

  10. Example 2-22 Given • verify Stoke’s theorem over a quarter-circular disk with a radius 3 in the first quadrant • Solution • therefore Lecture 06

  11. In Example 2-14 Lecture 06

  12. 2-11 Two Null Identities • Identity 1 • Gravity field • Electro static field • Identity 2 Lecture 06

  13. 2-12 Helmholtz’s Theorem • Helmholtz’s Theorem: A vector field (vector point function) is determined to within an additive constant if both its divergence and its curl are specified everywhere. Lecture 06

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