Fundamental introduction for beginning student of absolute differential calculus and for those interested in applications of tensor calculus to mathematical physics and engineering. Topics include spaces and tensors; basic operations in Riemannian space, curvature of space, special types of space, relative tensors, ideas of volume, more.
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absolute derivative angle arbitrary Cartesian coordinates Cartesian tensor cell Christoffel symbols coefficients configuration-space Consider constant curvature contravariant vector coordinate system coordinates xr covariant derivative covariant tensor covariant vector curvature tensor curve curvilinear coordinates defined definition denote differential equations equations of motion Euclidean 3-space Exercise expression flat space fluid follows force formulae geometry given Green's theorem Hence homogeneous coordinates infinitesimal displacement integral Jacobian Kronecker delta line element linear connection Maxwell's equations metric form metric tensor notation obtain orthogonal parallel propagation parameter parametric lines particle permutation symbols physical components plane propagated parallelly prove rectangular Cartesian coordinates relative tensor Riemannian space rigid body satisfied second order set of quantities Show skew-symmetric sphere suffixes surface symmetric connection tensor calculus tensor character trajectory transformation Trmn unit vector values vanishes vector field vector Xr write written zero