Vector Field Properties Clause Samples

Vector Field Properties. The vector ﬁelds on a manifold form a vector space over the ﬁeld of real numbers and a module over the ring of real-valued manifold functions. A module is like a vector space except that there is no multiplicative inverse operation on the scalars of a module. Man- ifold functions that are not the zero function do not necessarily 3The make-operator procedure takes a procedure and returns an operator. have multiplicative inverses, because they can have isolated zeros. So the manifold functions form a ring, not a ﬁeld, and vector ﬁelds must be a module over the ring of manifold functions rather than a vector space. Vector ﬁelds have the following properties. Let u and v be vector ﬁelds and let α be a real-valued manifold function. Then (u + v)(f) = u(f) + v(f) (3.9) (αu)(f) = α(u(f)). (3.10) Vector ﬁelds are linear operators. Assume f and g are functions on the manifold, a and b are real constants.4 The constants a and b are not manifold functions, because vector ﬁelds take derivatives. See equation (3.13). v(af + bg)(m) = av(f)(m) + bv(g)(m) (3.11) v(af)(m) = av(f)(m) (3.12) Vector ﬁelds satisfy the product rule (Leibniz rule). v(fg)(m) = v(f)(m) g(m) + f(m) v(g)(m) (3.13) Vector ﬁelds satisfy the chain rule. Let F be a function on the range of f.