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I am looking for a discrete counter part of a continuous variable.

the continuous version of energy term in a liquid crystal is given by [\vec{n}\cdot(\nabla\times\vec{n})]^2. This is a square of a dot product between a vector 'n' and its curl field. My question is what is the exact discrete version of this energy term?

In a literature it is given as [\hat{k}\cdot(\vec{n}_{i}\times\vec{n}_{i+1})]^2, it is mentioned that \hat{k} is a unit vector, \vec{n}_{i} is a vector at lattice point 'i' and \vec{n}_{i+1} is a vector at the lattice point 'i+1'.

For your clarification I have attached a picture file. See second term in Eq. (1) and below theory for justification in the attached file.

Thanks in advance.

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# Vector calculus

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