In Mathematical methods for physicists (sixth edition) by Arfken and Weber the triple scalar product was defined as (in page 147-148):
For $\vec A, \vec B, \vec C$ with components $A^i, B^j, C^k$ and a tensor of rank 3 is defined as the direct product of $\vec A, \vec B, \vec C$ or $V^{ijk}=A^iB^jC^k$. The pseudo-scalar $V$ which is the triple product would be,
$$V=\frac{1}{3!}\varepsilon_{ijk}V^{ijk}$$ Where $\varepsilon_{ijk}$ is the Levi-civita symbol.
I have calculated for $V$ in detail but the factor $\frac{1}{3!}$ remains which should not.
Also for 4 dimensional space 4 dimensional Levi-civita $\varepsilon_{ijkl}$ was defined in the same page of the book and using this the 4 dimensional volume element is pseudo-scalar is proven as follows:
Let $$\vec A = (dx^0, 0, 0, 0)$$ $$\vec B = (0, dx^1, 0, 0)$$ $$\vec C = (0, 0, dx^2, 0)$$ $$\vec D = (0, 0, 0, dx^3)$$ Now a 4th rank tensor is defined as: $$H^{ijkl}=A^iB^jC^kD^l$$ and the dual pseudo-scalar as: $$H=\frac{1}{4!}\varepsilon_{ijkl}H^{ijkl}$$ And the psedo-scalar is the volume element or $H=dx^0dx^1dx^2dx^3$ as written in the book.
I have calculated $H$ in detail but the factor $\frac{1}{4!}$ remains, which seems to be ignored for some reason.

I also checked that instead of $V^{ijk}=A^iB^jC^k$ if we make a anti-symmetric tensor by adding all even permutation of $ijk$ and subtracting all odd permutation of $ijk$ and set $V$ equals to that in that case it gives the correct results. Also in the case of 4-dimensional one.
My question is are factors ignored in the book just to make it to the point or the direct product of vectors is taken to be anti-symmetric by definition.
Please if anyone would like to share some insight that would be great.


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