Tensor CalculusCourier Corporation, 01.01.1978 - 324 Seiten "This book is an excellent classroom text, since it is clearly written, contains numerous problems and exercises, and at the end of each chapter has a summary of the significant results of the chapter." — Quarterly of Applied Mathematics. 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, and more. |
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absolute derivative arbitrary C₁ Cartesian coordinates Cartesian tensor cell Christoffel symbols coefficients configuration-space Consider constant curvature contravariant vector coordinate system covariant derivative covariant tensor covariant vector curvature tensor curve curvilinear coordinates defined definition denote differential equations ds² dx¹ dx² equations of motion Euclidean 3-space Exercise expression flat space fluid follows formulae geometry given Green's theorem Hence homogeneous coordinates infinitesimal displacement integral Jacobian Kronecker delta line element M-cell Maxwell's equations metric form metric tensor normal notation obtain orthogonal parallel propagation parameter parametric lines particle permutation symbols physical components plane Prove rectangular Cartesian coordinates relative tensor Riemannian space rigid body satisfied second order set of quantities Show skew-symmetric sphere suffixes surface tensor calculus tensor character theorem trajectory transformation unit vector V₁ V₂ values vanishes vector field write zero δι მე
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Seite 166 - Moon's motion is therefore reduced to the determination of the motion of a particle of mass M, under the action of a true force-function MF, where M E+M . in which x ', y', z' are the known functions of the time obtained from purely elliptic motion.
Seite 161 - The equation (2.2.49) expresses the fact that the rate of change of angular momentum is equal to the moment of the external forces about the origin.
Seite 13 - A set of quantities (Y""') (here 16) are said to be components of a contravariant tensor of the second order if they transform according to the equation Y...
Seite 241 - ... valid in all coordinate systems. In this case the relative tensors on two sides of equations must be of the same weight. A little reflection will convince the reader that : (a) Relative tensors of the same type and weight may be added, and the sum is a relative tensor of the same type and weight. (6) Relative tensors may be multiplied, the weight of the product being the sum of the weights of tensors entering in the product. (c) The operation of contraction on a relative tensor yields a relative...
Seite 10 - Then the n quantities are said to be the components of a contra-variant vector if they transform according to the same rule as do the differentials of the co-ordinates. Thus by (2.5), the Vr transform to V'r where F...
Seite 37 - In fact, a better definition is to say that a geodesic is a curve whose length has a stationary value with respect to arbitrary small variations of the curve, the end points being held fixed.
Seite 39 - It follows from the Fundamental Lemma of the Calculus of Variations that dt l To prove the lemma, note that Eq.