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We want to divide field A by field B.

Do we take Ax/Bx , Ay/By and Az/Bz individually?

But in this case we might end up with Three different scalar fields.

What's the proper way to do this?

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- Thread starter tade
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- #1

- 552

- 18

We want to divide field A by field B.

Do we take Ax/Bx , Ay/By and Az/Bz individually?

But in this case we might end up with Three different scalar fields.

What's the proper way to do this?

- #2

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The answer to the question will come from reflecting on why you want to do this.

- #3

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Thanks. I've figured it out the context of the division.

The answer to the question will come from reflecting on why you want to do this.

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mathwonk

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Of course, two vector fields on a 1-dimensional manifold can be divided, as long as the denominator vector field is nowhere zero. Because, at each point they belong to the tangent space at that point, which is the real numbers.

For example, on the circle S^{1} parametrized by t, 0 ≤ t ≤ 2π, we could have the vector fields

and

In this case, we have

which is just a scalar field, or in other words just a real-valued function.

This is just saying that

(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.)

For example, on the circle S

V(t) = e d/dt = e^{1} d/dt

and

W(t) = e^{sin(t)2} d/dt.

In this case, we have

f(t) = V(t) / W(t) = e^{1 - sin(t)2},

= e^{cos(t)2}

which is just a scalar field, or in other words just a real-valued function.

This is just saying that

f(t) W(t) = V(t).

(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.)

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