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Abstract: According to celebrated Hurwitz theorem, there exists four division algebrasconsisting of R real numbers, C complex numbers, H quaternions and Ooctonions. Keeping in view the utility of octonion variable we have tried toextend the three dimensional vector analysis to seven dimensional one. Startingwith the scalar and vector product in seven dimensions, we have redefined thegradient, divergence and curl in seven dimension. It is shown that the identitynn-1n-3n-7=0 is satisfied only for 0, 1, 3 and 7 dimensional vectors. Wehave tried to write all the vector inequalities and formulas in terms of sevendimensions and it is shown that same formulas loose their meaning in sevendimensions due to non-associativity of octonions. The vector formulas areretained only if we put certain restrictions on octonions and split octonions.

Author: Bhupendra C. S. Chauhan, P. S. Bisht, O. P. S. Negi

Source: https://arxiv.org/

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