Tensor operator

Question

I have come across the following expression:

H:E

where, H = e(levi-cita symbol)*a constant which means a 3rd order tensor with 27 components

E = 2nd order tensor,

now, what does H:E mean? I know that the result has to be a vector (3 components).

Answer 1

Assuming that $E=E_{ij}\textbf{e}^i\textbf{e}^j$ is a symmetric tensor such that $E_{ij}=E_{ji}$, So you have: $$H^{ijk}=\alpha\cdot\epsilon^{ijk}$$ $$E=E_{ij}$$ In order to produce a first order tensor (vector) with three components, we arrange the tensors in such a way to produce a vector, with components $V^i$: $$H:E=H^{ijk}E_{jk}=V^i$$ So therefore: $$V^i=\alpha\cdot\epsilon^{ijk}E_{jk}=\alpha\cdot\epsilon^{ijk}E_{kj}$$