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EEE 340Lecture 061 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)

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EEE 340Lecture 062 (2-135)

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EEE 340Lecture 063 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.

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EEE 340Lecture 064 or Cartesian coordinates Cylindrical coordinates (2-136) (2-138) (2-135)

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EEE 340Lecture 065 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)

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EEE 340Lecture 066 Example 2-21 Show that if a). b). Solution a). Cylindrical coordinates

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EEE 340Lecture 067 b). Spherical coordinates

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EEE 340Lecture 068 Irrotational or Conservative field.

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EEE 340Lecture 069 2-10 Stokes’s theorem Proof: 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. (2-143)

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EEE 340Lecture 0610 Example 2-22 Given verify Stoke’s theorem over a quarter-circular disk with a radius 3 in the first quadrant Solution therefore

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EEE 340Lecture 0611 In Example 2-14

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EEE 340Lecture 0612 2-11 Two Null Identities Identity 1 Gravity field Electro static field Identity 2

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EEE 340Lecture 0613 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.

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