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A X B angular velocity arc length axial vector axis B X C calculate Chapter column commutative components consider constant vector coordinate axes coordinate system coordinate transformation cos(x curl curl curl grad curvilinear coordinates denote derivative describe differential operators direction cosines discussion of vector divergence theorem dx2 dx3 equal equation equivalent evaluated example expression field vector fluid Gauss given grad)B gradient operator improper rotation infinitesimal rotation inverse irrotational irrotational field line integral magnitude multiplied normal notation obtain origin orthogonal transformation orthogonality relations outward flow particle permutation physical plane polar coordinates point Q proof proper rotations prove quantity radius vector represented result right-hand rotation matrices scalar function scalar product Show specified spherical coordinates Stokes summation surface element surface integral symbol tensor transformation matrix transformation properties true vector unit vectors VECTOR ANALYSIS vector field vector function vector operators vector product volume write x-axis