You have trouble with the chain rule?Not for IMPLICIT functions of coordinates, you dork. Check your books since you are too lazy to calculate a chain of of derivatives.
Ah - you keep editing!
You have trouble with the chain rule?
Provided Eugene's coordinates have a nowhere zero Jacobian when expressed in terms of standard Cartesians then they are a valid set of coordinates everywhere.
Erm, the coordinates are just functions... the derivatives of coordinates are just functions... You don't seriously need to be told how to use the chain rule do you?
Yes, I'm sure you are but I mention them because they are examples of non-linear transformations where tensor covariance is demonstrated. I doubt Guest thinks Eugene is onto anything, I certainly don't, he's instead talking about the general concept of coordinate transformations and tensors.Agreed, I am very familiar with the transformation between cartesian and polar coordinates.
I wouldn't trust your mathematical abilities as far as I can throw a 2 ton rhino so forgive me if I don't take your word for it and I'm not going to waste my time checking. The point Guest is talking about is entirely independent from the specifics of your nonsense.Eugene said:I have clearly demonstrated that my nonlinear transformation is invertible, see equation (46), (59) and (60), therefore, if it's differentiable, then the Jacobian exists and is never zero. Please keep in mind however that, in general, my transformation is so general that it's easy to come up with non-differentiable examples.
You can't differentiate that?
I wouldn't trust your mathematical abilities as far as I can throw a 2 ton rhino so forgive me
Yes, I'm sure you are but I mention them because they are examples of non-linear transformations where tensor covariance is demonstrated. I doubt Guest thinks Eugene is onto anything, I certainly don't, he's instead talking about the general concept of coordinate transformations and tensors.
If Eugene's coordinates have non-singular $$J^{a}_{b} = \frac{\partial \tilde{x}^{a}}{\partial x^{b}}$$ then they are 'valid coordinates' and expressing Maxwell's equations in terms of them would still give the form $$\nabla^{a}F_{ab} = 0$$, just F would be different from the F in the Cartesian form.
You said that you think Guest doesn't know what he's talking about. I can assure you that Guest's knowledge in the likes of tensor calculus is formidable, I doubt that many of even the PhDs in physics here would even come close, myself included. I would have thought you might get the hint when he talked about exterior derivatives of things in cotangent bundles.
Hey, you were worried about derivatives - not me! I know the chain rule.
I'm afraid I probably think you're a little bit whacko (what I'm having to explain to Tach is largely independent of whatever you've written - I'm talking about any coordinate transformation). On the up side, Tach thinks we have a future together. Call me?Hopefully Guest, after he's finished playing, will take the time and educate you also.
News flash: some coordinate transformations might be messy!But you get a garbled mess when you try getting the tensor transforms. You claim to be a mathematician but you do not understand the conditions under the formulas you are citing have been derived.
News flash: some coordinate transformations might be messy!
(what I'm having to explain to Tach is largely independent of whatever you've written - I'm talking about any coordinate transformation).
What bit can't you do? You say you're ok with the differentiation, so which bit is too tough?It is not ONLY that they are "messy" , it is that their application does not result into the standard forms of tensor transformation. Alpha says that you are an expert, why are you so reluctant to do a simple calculation to prove it to yourself?
Good for you.I understand that. I took Ted Frankel's graduate level geometry of physics course when I was a math student at UCSD.
I understand that. I took Ted Frankel's graduate level geometry of physics course when I was a math student at UCSD.