Noncoordinate basis vector fields

In summary, the conversation discusses the concept of vector fields and their relationship to flows and first order linear homogeneous partial differential operators. The focus is on using these concepts to show that a given basis is non-coordinate, or that the commutator is non-zero. The conversation also mentions the use of formulas and undetermined functions to compute the commutator.
  • #1
desic
4
0
I'm self-studying Schutz Geometric methods of mathematical physics, having problems with ex. 2.1. page 44.

r=cos(theta)x+sin(theta)y
theta=-sin(theta)x+cos(theta)y

show this is non-coordinate basis, i.e. show commutator non-zero.

I try to apply his formula 2.7, assuming

V1=cos(theta), V2=sin(theta)
W1=-sin(theta), W2=cos(theta)
x(r)=r cos(theta)
y(r)=r sin(theta)
x(theta)=cos(theta)
y(theta)=sin(theta)

I believe the component of x should be (sin(theta))/r, however I get (sin(theta) - r sin(theta))/r.

would appreciate any help
 
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  • #2
Vector fields

Hi, desic,

desic said:
I'm self-studying Schutz Geometric methods of mathematical physics, having problems with ex. 2.1. page 44.

r=cos(theta)x+sin(theta)y
theta=-sin(theta)x+cos(theta)y

show this is non-coordinate basis, i.e. show commutator non-zero.

Vector fields, flows (in the sense of manifold theory), and first order linear homogeneous partial differential operators, are all tighly inter-related concepts, and it helps to able to freely intertranslate between these various representations.

E.g. in [tex]\mathbold{R}^2[/tex] "translation along x" and "rotation around the origin" can be written
[itex] \vec{T} = \partial_x[/itex]
[itex] \vec{R} = \partial_\theta = -y \, \partial_x + x \, \partial_y [/itex]
Now you can just compute [tex] \vec{T} \vec{R} f - \vec{R} \vec{T} f[itex] for an undetermined function f, from which you can read off the commutator.

See Olver, Equivalence, Invariants, and Symmetry for more about the virtues of this style of thinking/computing.

Chris Hillman
 
Last edited:
  • #3



First of all, great job on attempting to apply the formula 2.7 from Schutz's book! However, there are a few mistakes in your calculations.

First, let's define the vector fields V1 and V2 as V1 = cos(theta)x + sin(theta)y and V2 = -sin(theta)x + cos(theta)y, just like in the problem statement.

Next, let's calculate the components of these vector fields in terms of the basis vectors x and y. We have:

V1 = cos(theta)x + sin(theta)y
= cos(theta)(cos(theta)x+sin(theta)y) + sin(theta)(-sin(theta)x+cos(theta)y)
= cos^2(theta)x + sin^2(theta)y - sin(theta)cos(theta)x + sin(theta)cos(theta)y
= (cos^2(theta)-sin(theta)cos(theta))x + (sin^2(theta)+sin(theta)cos(theta))y
= (cos^2(theta)-sin(theta)cos(theta)+sin^2(theta)+sin(theta)cos(theta))x + (sin^2(theta)+sin(theta)cos(theta))y
= x + (sin^2(theta)+sin(theta)cos(theta))y

Similarly, we can calculate V2 as:

V2 = -sin(theta)x + cos(theta)y
= -sin(theta)(cos(theta)x+sin(theta)y) + cos(theta)(-sin(theta)x+cos(theta)y)
= -sin^2(theta)x - cos^2(theta)y + sin(theta)cos(theta)x - sin(theta)cos(theta)y
= (-sin^2(theta)+sin(theta)cos(theta))x + (-cos^2(theta)-sin(theta)cos(theta))y
= (-sin^2(theta)+sin(theta)cos(theta)+cos^2(theta)+sin(theta)cos(theta))x + (-cos^2(theta)-sin(theta)cos(theta))y
= x + (-cos^2(theta)-sin(theta)cos(theta))y

Now, we can use these components to calculate the commutator [V1, V2] as:

[V1,V2] = V1(V2) - V2(V1)
= (x + (sin^2(theta)+sin(theta)cos(theta))y)(x + (-cos^2(theta)-sin(theta)cos(theta))y) - (x + (-cos^2(theta)-sin(theta)cos
 

Related to Noncoordinate basis vector fields

1. What is a noncoordinate basis vector field?

A noncoordinate basis vector field is a set of vectors that are defined at each point in a space, but the basis vectors themselves do not vary with position. This is in contrast to a coordinate basis vector field, where the basis vectors change with position.

2. How is a noncoordinate basis vector field different from a coordinate basis vector field?

A noncoordinate basis vector field is different from a coordinate basis vector field in that the basis vectors are constant throughout the space, whereas the basis vectors in a coordinate basis vector field vary with position. Additionally, the components of a vector in a noncoordinate basis vector field do not necessarily have a clear geometric interpretation, unlike in a coordinate basis vector field.

3. What are some common examples of noncoordinate basis vector fields?

Some common examples of noncoordinate basis vector fields include the tangent and normal vectors on a curved surface, the spherical basis vectors in spherical coordinates, and the cylindrical basis vectors in cylindrical coordinates.

4. What is the significance of using noncoordinate basis vector fields?

Noncoordinate basis vector fields are often used in more complex systems or geometries where a coordinate basis vector field may not be well-defined or may be difficult to work with. Noncoordinate basis vector fields can also provide a more general and abstract framework for understanding vector fields.

5. How are noncoordinate basis vector fields used in scientific research?

Noncoordinate basis vector fields have various applications in scientific research, particularly in fields such as physics, engineering, and mathematics. They are used to describe and analyze vector fields in non-Cartesian coordinate systems, which are often necessary for solving complex problems in these fields. Noncoordinate basis vector fields are also used in theoretical and computational models, such as in fluid mechanics and electromagnetism.

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