The Curl of a Vector Field
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The relationship of the rotation of a rigid body to curl
Equations are omitted for technical reasons - download the original pdf
Let S be a rigid body rotating about an axis of rotation w, such that the rotation is clockwise when looked at in the direction w. [Equation goes here - download the original pdf to see it.]Let [symbol] be the angular speed of rotation of this body, and let the length of w be equal to [symbol].What this means is that if P is a point on the body, and d is the distance of that point from the axis of rotation, w, then the tangential velocity, v, of P is given by[Equation goes here - download the original pdf to see it.] [Diagram goes here - download the original pdf to see it.] Now let p be the position vector of P relative to an origin O lying on the axis of rotation, w. Then [Equation goes here - download the original pdf to see it.] where [symbol] is the angle between w and p. [Diagram goes here - download the original pdf to see it.] This gives [Equation goes here - download the original pdf to see it.] If [Equation goes here - download the original pdf to see it.] is a unit vector in the direction of v, then [Equation goes here - download the original pdf to see it.] Thus, from the definition of the cross product, we see that [Equation goes here - download the original pdf to see it.] Now to demonstrate the relationship of the rotation of a rigid body to the curl of a vector v, suppose now that we have a right-handed Cartesian coordinate system in which the axis of rotation of a rigid body is the z-axis, so that [Equation goes here - download the original pdf to see it.] then [Equation goes here - download the original pdf to see it.] Thus [Equation goes here - download the original pdf to see it.] That is, in general [Equation goes here - download the original pdf to see it.]This states that for a rotating rigid body the direction of the curl of the velocity field is the same as the axis of rotation, and its magnitude is twice that of the angular speed of rotation.Thus, given a vector, w, describing the axis and speed of rotation of a rigid body, and a position vector, p, of a point, P, on the body, then the velocity of P is [Equation goes here - download the original pdf to see it.] The curl of v reverses this; given a velocity vector [Equation goes here - download the original pdf to see it.] then the axis of rotation is given by [Equation goes here - download the original pdf to see it.]
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Contents of
The Curl of a Vector Field
1 The curl of a vector field
2 The relationship of the rotation of a rigid body to curl
3 The curl of a scalar function
4 Invariance of curl
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