Abstract: According to celebrated Hurwitz theorem, there exists
four division algebras consisting of R (real numbers), C (complex
numbers), H (quaternions) and O (octonions). Keeping in view
the utility of octonion variable we have tried to extend the three
dimensional vector analysis to seven dimensional one. Starting with
the scalar and vector product in seven dimensions, we have redefined
the gradient, divergence and curl in seven dimension. It is shown
that the identity n(n - 1)(n - 3)(n - 7) = 0 is satisfied only
for 0, 1, 3 and 7 dimensional vectors. We have tried to write all
the vector inequalities and formulas in terms of seven dimensions
and it is shown that same formulas loose their meaning in seven
dimensions due to non-associativity of octonions. The vector formulas
are retained only if we put certain restrictions on octonions and split
octonions.