Paper 3, Section II, B

Vector Calculus | Part IA, 2017

(a) The two sets of basis vectors ei\mathbf{e}_{i} and ei\mathbf{e}_{i}^{\prime} (where i=1,2,3i=1,2,3 ) are related by

ei=Rijej\mathbf{e}_{i}^{\prime}=R_{i j} \mathbf{e}_{j}

where RijR_{i j} are the entries of a rotation matrix. The components of a vector v\mathbf{v} with respect to the two bases are given by

v=viei=viei\mathbf{v}=v_{i} \mathbf{e}_{i}=v_{i}^{\prime} \mathbf{e}_{i}^{\prime}

Derive the relationship between viv_{i} and viv_{i}^{\prime}.

(b) Let T\mathbf{T} be a 3×33 \times 3 array defined in each (right-handed orthonormal) basis. Using part (a), state and prove the quotient theorem as applied to T\mathbf{T}.

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