Anti-symmetric part of an equation

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In summary, taking the totally anti-symmetric part of the given equation results in vanishing terms because the first term on the right is symmetric in a and c and the second term is symmetric in b and c. This is due to the notation ##X_{[a}\partial_bX_{c]}##, where all three indices are within the square brackets and should change sign when interchanged.
  • #1
GR191511
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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?
 
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  • #2
GR191511 said:
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 totaly anti-symmetric part is.
 
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  • #3
GR191511 said:
Why?
Because:
GR191511 said:
the first term on the right is symmetric in a and c and the second term is symmetric in b and c
 
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  • #4
Orodruin said:
Because:
Is the" totaly anti-symmetric part"of that equation##X_{[a}\partial_{b}X_{c]}+X_{a[}\partial_{b}X_{c]}+X_{[a}\partial_{b]}X_{c}##?
 
  • #5
No, it is ##X_{[a}\partial_b X_{c]}##. Meaning that it should change sign whenever you interchange any of the three indices.
 
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  • #6
Orodruin said:
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?
 
  • #7
GR191511 said:
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.
 
  • #8
GR191511 said:
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.
 
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  • #9
Ibix said:
No. That would be ##X_{[a|}\partial_bX_{|c]}##, where the ##|## symbols are used to exclude indices between them from round or square brackets.
o_O...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})##?
 
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  • #10
GR191511 said:
o_O...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.
 
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FAQ: Anti-symmetric part of an equation

What is the anti-symmetric part of an equation?

The anti-symmetric part of an equation is a mathematical concept that refers to the portion of an equation that remains unchanged when the variables are replaced with their negative values. It is also known as the odd part of an equation.

How is the anti-symmetric part of an equation calculated?

The anti-symmetric part of an equation is calculated by taking the average of the original equation and its negative counterpart. This cancels out the symmetric (even) parts of the equation, leaving only the anti-symmetric (odd) parts.

What is the significance of the anti-symmetric part in physics?

In physics, the anti-symmetric part of an equation is important because it often represents physical quantities that are odd in nature, such as angular momentum and magnetic dipole moment. It also plays a role in understanding the symmetry properties of physical systems.

Can the anti-symmetric part of an equation be zero?

Yes, the anti-symmetric part of an equation can be zero. This means that the equation is symmetric, and the values of the variables do not change when they are replaced with their negative values. In other words, the equation is even.

How does the anti-symmetric part of an equation relate to the symmetric part?

The anti-symmetric part and the symmetric part of an equation are complementary to each other. This means that they add up to the original equation. In other words, the anti-symmetric part can be seen as the "opposite" of the symmetric part, as it is the part that remains when the symmetric part is removed.

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