Noncoordinate basis vector fields

[desic]

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

 

[Chris Hillman]

Vector fields

Hi, desic,

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 $$\mathbold{R}^2$$ "translation along x" and "rotation around the origin" can be written
$\vec{T} = \partial_x$
$\vec{R} = \partial_\theta = -y \, \partial_x + x \, \partial_y$
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

 

[tacman]

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