The Curl of a Vector Field
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The curl of a vector field
Let [Equation goes here - download the original pdf to see it.] be a differentiable vector function, where [Equation goes here - download the original pdf to see it.] are a right-handed Cartesian coordinate system; then the function [Equation goes here - download the original pdf to see it.] is called the curl of the vector field v. (Note that the vector, v, is explicitly a tangent vector, that is a vector that has two parts (i)A point of application, p; (ii)A vector v; So it should be written, [Equation goes here - download the original pdf to see it.] . The vector field, v, is a function that maps each point of Euclidean space, [Equation], to the vector, [Equation goes here - download the original pdf to see it.] . It is usual to omit the subscript, p, for the sake of avoiding cumbersome notation.) For a left-handed coordinate system, the curl has a minus sign in front by convention. The curl of a vector field is sometimes called its rotation. [Example 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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