Tensors, Differential Forms, and Variational Principles
Incisive, self-contained account of tensor analysis and the calculus of exterior differential forms, interaction between the concept of invariance and the calculus of variations. Emphasis is on analytical techniques, with large number of problems, from routine manipulative exercises to technically difficult assignments.
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Affine tensor algebra in Euclidean geometry
Tensor analysis on manifolds
9 other sections not shown
_ dxJ 1-forms absolute differential according accordingly affine connection affine tensor algebra applied arbitrary assumed calculus of variations Christoffel symbols class C2 clearly condition connection coefficients constant constitute the components construct contravariant vector coordinate transformation corresponding covariant derivative covariant vector curvature tensor curve curvilinear coordinate d2xJ defined denote differentiable manifold differentiable manifold Xn differential equations dxh dxk dxJ dx elements equivalent Euler-Lagrange equations expressed follows functions fundamental integral geodesic field geometry given hypersurface implies independent indices invariant Kronecker delta Lagrangian latter left-hand side Lie derivative linear metric tensor notation obtain parameter problem properties quantities relation relative tensor Remark respect Riemannian space right-hand side satisfied scalar density Section skew-symmetric subspace Cm substitute symmetric tangent space tensor density tensor field tensor of rank tensor of type theorem theory Tn(P transformation law vanishes identically vector field virtue yields