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Hi all!

I read about tensor analysis and came about following expressions, where also a questions arose which I cannot explain to me. Perhaps you could help me:

I: Consider the following expressions:

[tex]d\vec v=dc^k e^{(k)}[/tex]

[tex]d\vec v=dc^k e_{(k)}[/tex]

where:

[tex]dc^k=dv^k+v^t\Gamma_{wt}^k dx^w[/tex]

[tex]dc_k=dv_k-v_t\Gamma_{wk}^t dx^w[/tex]

Now, consider the covariant derivatives:

[tex]\frac{\partial c^k}{\partial x^q}=\frac{\partial v^k}{\partial x^q}+v^t\Gamma_{wt}^k \frac{\partial x^w}{\partial x^q}=\frac{\partial v^k}{\partial x^q}+v^t\Gamma_{wt}^k \delta^w_q=\frac{\partial v^k}{\partial x^q}+v^t\Gamma_{qt}^k[/tex]

analagous:

(1)[tex]\frac{\partial c_k}{\partial x^q}=\frac{\partial v_k}{\partial x^q}-v_t\Gamma_{qk}^t [/tex]

So far so good, here I start transforming:

[tex]\frac{\partial c_k}{\partial x^q}=\frac{g_{kl}\partial c^l}{\partial x^q}=g_{kl}\frac{\partial c^l}{\partial x^q}=\frac{g_{kl}\partial v^l}{\partial x^q}+v^t\Gamma_{qt}^l g_{kl}=\frac{\partial v_l}{\partial x^q}+v^t\Gamma_{qtk}[/tex]

As the second term looks different from the one above we continue transforming it:

[tex]v^t\Gamma_{qtk}=v^t\Gamma_{qk}^s g_{ts}=(t\rightarrow s, g_{ts}=g_{st})=v^s\Gamma_{qk}^t g_{ts}=v_t\Gamma_{qk}^t[/tex]

so, we finally get:

(2)[tex]\frac{\partial c_k}{\partial x^q}=\frac{\partial v_k}{\partial x^q}+v_t\Gamma_{qk}^t[/tex]

By comparing (1) and (2) I miss a minus sign!

I suspect that the Christoffel symbol of first kind is antisymmetric and indices permute just like they do in the epsilon tensor and thereby generate a minus but I am not sure...

II: In both of the above daces of derivatives one uses dx^q as differential which is contravariant. Does it make sence to also use a covariant dx_q? Is in general differentiation of covariant vectors with respect to a covariant variable defined? (I suppose it must be, since you also differentiate a contravariantvector w.r.t. a contravariant variable)

III: And another question: Is the Kronecker delta symmetric in non-orthogonal coordinates

[tex]\delta^i_j=\delta^j_i[/tex] ???

If not, then which one of the two definitions is correct: (I´ve seen both in the net)

[tex]e^{i}e_j=\delta^j_i[/tex]

or

[tex]e^je_i=\delta^j_i[/tex]

I have also seen two types in which you define covariant vectors:

[tex]\vec v=v_ke^k[/tex] and [tex]\vec v=v_ke_k[/tex]

Which one is correct, or do they just represent the same covariant vector once in the covariant and in the contravariant basis?

IV: And the last one: I haven´t seen a classification of the Christoffel symbol of this kind:

[tex]\Gamma^{kl}_m[/tex] Is it also symmetric in the upper indices?

Thanks a lot, I really appreciate your help!

marin

I read about tensor analysis and came about following expressions, where also a questions arose which I cannot explain to me. Perhaps you could help me:

I: Consider the following expressions:

[tex]d\vec v=dc^k e^{(k)}[/tex]

[tex]d\vec v=dc^k e_{(k)}[/tex]

where:

[tex]dc^k=dv^k+v^t\Gamma_{wt}^k dx^w[/tex]

[tex]dc_k=dv_k-v_t\Gamma_{wk}^t dx^w[/tex]

Now, consider the covariant derivatives:

[tex]\frac{\partial c^k}{\partial x^q}=\frac{\partial v^k}{\partial x^q}+v^t\Gamma_{wt}^k \frac{\partial x^w}{\partial x^q}=\frac{\partial v^k}{\partial x^q}+v^t\Gamma_{wt}^k \delta^w_q=\frac{\partial v^k}{\partial x^q}+v^t\Gamma_{qt}^k[/tex]

analagous:

(1)[tex]\frac{\partial c_k}{\partial x^q}=\frac{\partial v_k}{\partial x^q}-v_t\Gamma_{qk}^t [/tex]

So far so good, here I start transforming:

[tex]\frac{\partial c_k}{\partial x^q}=\frac{g_{kl}\partial c^l}{\partial x^q}=g_{kl}\frac{\partial c^l}{\partial x^q}=\frac{g_{kl}\partial v^l}{\partial x^q}+v^t\Gamma_{qt}^l g_{kl}=\frac{\partial v_l}{\partial x^q}+v^t\Gamma_{qtk}[/tex]

As the second term looks different from the one above we continue transforming it:

[tex]v^t\Gamma_{qtk}=v^t\Gamma_{qk}^s g_{ts}=(t\rightarrow s, g_{ts}=g_{st})=v^s\Gamma_{qk}^t g_{ts}=v_t\Gamma_{qk}^t[/tex]

so, we finally get:

(2)[tex]\frac{\partial c_k}{\partial x^q}=\frac{\partial v_k}{\partial x^q}+v_t\Gamma_{qk}^t[/tex]

By comparing (1) and (2) I miss a minus sign!

I suspect that the Christoffel symbol of first kind is antisymmetric and indices permute just like they do in the epsilon tensor and thereby generate a minus but I am not sure...

II: In both of the above daces of derivatives one uses dx^q as differential which is contravariant. Does it make sence to also use a covariant dx_q? Is in general differentiation of covariant vectors with respect to a covariant variable defined? (I suppose it must be, since you also differentiate a contravariantvector w.r.t. a contravariant variable)

III: And another question: Is the Kronecker delta symmetric in non-orthogonal coordinates

[tex]\delta^i_j=\delta^j_i[/tex] ???

If not, then which one of the two definitions is correct: (I´ve seen both in the net)

[tex]e^{i}e_j=\delta^j_i[/tex]

or

[tex]e^je_i=\delta^j_i[/tex]

I have also seen two types in which you define covariant vectors:

[tex]\vec v=v_ke^k[/tex] and [tex]\vec v=v_ke_k[/tex]

Which one is correct, or do they just represent the same covariant vector once in the covariant and in the contravariant basis?

IV: And the last one: I haven´t seen a classification of the Christoffel symbol of this kind:

[tex]\Gamma^{kl}_m[/tex] Is it also symmetric in the upper indices?

Thanks a lot, I really appreciate your help!

marin

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