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Mathematics
Linear and Abstract Algebra
How do we divide one vector field by another?
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[QUOTE="zinq, post: 5460280, member: 462505"] Of course, two vector fields on a 1-dimensional manifold can be divided, as long as the denominator vector field is nowhere zero. [SIZE=4]Because, at each point they belong to the tangent space at that point, which is the real numbers. For example, on the circle S[SUP]1[/SUP] parametrized by t, 0 ≤ t ≤ 2π, we could have the vector fields [/SIZE] [INDENT][SIZE=4]V(t) = e d/dt = e[SUP]1[/SUP] d/dt[/SIZE][/INDENT] [SIZE=4] and [/SIZE] [INDENT][SIZE=4]W(t) = e[SUP]sin(t)[SUP]2[/SUP][/SUP] d/dt.[/SIZE][/INDENT] [SIZE=4] In this case, we have [/SIZE] [INDENT][SIZE=4]f(t) = V(t) / W(t) = e[SUP]1 - sin(t)[SUP]2[/SUP][/SUP],[/SIZE][/INDENT] [INDENT][SIZE=4] = e[SUP]cos(t)[SUP]2[/SUP][/SUP][/SIZE][/INDENT] [SIZE=4] which is just a scalar field, or in other words just a real-valued function. This is just saying that [/SIZE] [INDENT][SIZE=4]f(t) W(t) = V(t).[/SIZE][/INDENT] [SIZE=4] (The same thing is also possible if the space on which the two vector fields are defined is a Riemann surface M. Since then at each point of M the vector of each vector field lies in the complex numbers.)[/SIZE] [/QUOTE]
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Linear and Abstract Algebra
How do we divide one vector field by another?
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