Lectures on Tensor Calculus and Differential Geometry |
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Page 55
... vector space Y is parallel to a flat associated with a vector space 3 , 2 and 3 being subspaces of X , when either ... coordinate system with origin O. The vector x is the coordinate vector of P with respect to 0 . This coordinate vector ...
... vector space Y is parallel to a flat associated with a vector space 3 , 2 and 3 being subspaces of X , when either ... coordinate system with origin O. The vector x is the coordinate vector of P with respect to 0 . This coordinate vector ...
Page 56
... vector x of P is represented by 1 , ... ) X = X × x * . ( 5.1-7 ) The numbers * are called the coordinates of the point P with respect to the given origin and the given basis . Such a combination will be referred to as a coordinate ...
... vector x of P is represented by 1 , ... ) X = X × x * . ( 5.1-7 ) The numbers * are called the coordinates of the point P with respect to the given origin and the given basis . Such a combination will be referred to as a coordinate ...
Page 86
... vector . It follows that d ̧y is orthogonal to d ̧x and this con- cludes the proof of the theorem . By the centre of curvature at a point s of a curve x ( s ) we mean a point whose coordinate vector is x ( s ) + u ( s ) p 1 ( 6.3-12 ) ...
... vector . It follows that d ̧y is orthogonal to d ̧x and this con- cludes the proof of the theorem . By the centre of curvature at a point s of a curve x ( s ) we mean a point whose coordinate vector is x ( s ) + u ( s ) p 1 ( 6.3-12 ) ...
Contents
LINEAR VECTOR SPACES | 1 |
METRIC VECTOR SPACES | 14 |
BILINEAR AND QUADRATIC FORMS | 26 |
Copyright | |
15 other sections not shown
Common terms and phrases
angle arc length basis bivector called Christoffel symbols conditions of integrability congruence consider constant Riemannian curvature coordinate vector corresponding cos² covariant components covariant differentiation curve x(s denote differential equations dimension eigenvalues eigenvector expression finite dimensional follows geodesic curve given curve given point h h+1 h=1 h h Hence hyperplane hypersphere hypersurface identically line of curvature linear operator linearly independent metric space metric tensor metric vector space multilinear form multiplying both members n-dimensional normal number space obtained orthogonal orthonormal frame osculating parameter point space principal curvatures principal directions prove replace respect Riccian Riemannian curvature scalar invariant second fundamental tensor subspace symmetric taking account tangent space tangent vector tensor of valency theorem transformation Transvecting uniquely determined unit vector values vanishes vector function vector space zero vector βλ βλμ βμ Καβλμ νμ Σ σ Χλμ