Anti-symmetric part of an equation

[GR191511]

I'm reading "Introducing Einstein's Relativity_A Deeper Understanding Ed 2"on page 271:
$X_{a}\partial_{b}X_{c}=\lambda f_{,a}\lambda_{,b}f_{,c}+\lambda^2f_{,a}f_{,cb}$
Taking the totally anti-symmetric part of this equation and noting that the first term on the right is symmetric in a and c and the second term is symmetric in b and c,we see that their totally anti-symmetric parts vanish...
"their totally anti-symmetric parts vanish"Why?

 

[martinbn]

I'm reading "Introducing Einstein's Relativity_A Deeper Understanding Ed 2"on page 271:
$X_{a}\partial_{b}X_{c}=\lambda f_{,a}\lambda_{,b}f_{,c}+\lambda^2f_{,a}f_{,cb}$
Taking the totally anti-symmetric part of this equation and noting that the first term on the right is symmetric in a and c and the second term is symmetric in b and c,we see that their totally anti-symmetric parts vanish...
"their totally anti-symmetric parts vanish"Why?

May be you can start with reminding yourself what totally anti-symmetric part is.

 

[Orodruin]

Why?

Because:

the first term on the right is symmetric in a and c and the second term is symmetric in b and c

 

[GR191511]

Because:

Is the" totally anti-symmetric part"of that equation$X_{[a}\partial_{b}X_{c]}+X_{a[}\partial_{b}X_{c]}+X_{[a}\partial_{b]}X_{c}$?

 

[Orodruin]

No, it is $X_{[a}\partial_b X_{c]}$. Meaning that it should change sign whenever you interchange any of the three indices.

 

[GR191511]

No, it is $X_{[a}\partial_b X_{c]}$. Meaning that it should change sign whenever you interchange any of the three indices.

any of the three indices?Doesn't the notation $X_{[a}\partial_b X_{c]}$means that "a" exchanges with "c" only?

 

[Orodruin]

any of the three indices?Doesn't the notation $X_{[a}\partial_b X_{c]}$means that "a" exchanges with "c" only?

No. All your indices are within the square brackets.

 

[Ibix]

any of the three indices?Doesn't the notation $X_{[a}\partial_b X_{c]}$means that "a" exchanges with "c" only?

No. That would be $X_{[a|}\partial_bX_{|c]}$, where the $|$ symbols are used to exclude indices between them from round or square brackets.

 

[GR191511]

No. That would be $X_{[a|}\partial_bX_{|c]}$, where the $|$ symbols are used to exclude indices between them from round or square brackets.

...OK!$X_{[a}\partial_bX_{c]}=\frac 1 6(X_{a}\partial_bX_{c}+X_{c}\partial_aX_{b}+X_{b}\partial_cX_{a}-X_{a}\partial_cX_{b}-X_{b}\partial_aX_{c}-X_{c}\partial_bX_{a})$?

 

[Orodruin]

...OK!$X_{[a}\partial_bX_{c]}=\frac 1 6(X_{a}\partial_bX_{c}+X_{c}\partial_aX_{b}+X_{b}\partial_cX_{a}-X_{a}\partial_cX_{b}-X_{b}\partial_aX_{c}-X_{c}\partial_bX_{a})$?

Yes.