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the algebra of vectors
Linear dependence or independence of a set of number ntuples
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algebraic arrows associated assumed axis basis calculus Cartesian coordinate system Cartesian vector Chapter Christoffel symbols coefficients cofactor completes the proof Compute Consider constant contravariant vector corresponding covariant and contravariant covariant vector cross product defined dependent determinant differential domain dt dt dX dX dX1 dX2 elements Euclidean space Example fact fundamental metric tensor geometric interpretation given gradient indicated introduced invariant line integral linear linearly independent magnitude mathematical matrix metric tensor multiplication n-tuples notation obtain orthogonal Cartesian group pair parametric equations partial derivatives perpendicular physical plane Problem properties reader real numbers rectangular Cartesian coordinate rectangular Cartesian system relation representation represented respect result Riemannian space rotation satisfy the transformation scalar field Section Show sin2 skew symmetric space curve Suppose surface tangent vector theory three-space transformation equations transformation group transformation law triple vector concept vector field zero