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6.1 Definition of a Vector Field

A vector field is a vector each of whose components is a scalar field, that is, a function of our variables. We use any of the following notations for one:

v(x, y, z) = (v1(x, y, z), v2(x, y, z), v3(x, y, z))

v(x, y, z) = v1(x, y, z)i  + v2(x, y, z)j + v3(x, y, z)k

v(x, y, z) = (L(x, y, z), M(x, y, z), N(x, y, z))


One important class of vector fields are those that are gradients of a scalar field. If f is a scalar field, its gradient,f, is a vector field, namely