Tensor Analysis, and Applications, 2nd ed. Hemisphere moment of inertia tensor of solid ![]() These can be achieved through multiplication by a so-called metricĬircle area moment of inertia tensor of enclosed laminaĬonical frustum moment of inertia tensor of solid Lowering and index raising as special cases. Of tensor indices to produce identities or to simplify expressions is known as index gymnastics, which includes index Kronecker delta) or by tensor operators (suchĪs the covariant derivative). Tensors may be operated on by other tensors (such as metric tensors, the permutation tensor, or the , which would be written in tensor notation. Objects that transform like zeroth-rank tensors are called scalars, those that transform like first-rank tensors are called vectors,Īnd those that transform like second-rank tensors are called matrices. Space, and such tensors are known as Cartesian While the distinction between covariant and contravariant indices must be made for general tensors, the two are equivalent for tensors in three-dimensional Euclidean Note that the positions of the slots in which contravariant and covariant indicesĪre placed are significant so, for example, is distinct from. In addition, a tensor with rank may be of mixed type, consisting of so-called "contravariant" (upper) indices The notation for a tensor is similar to that of a matrix (i.e., ), except that a tensor, ,, etc., may have an arbitrary number of indices. ![]() Tensors provide a natural and concise mathematical framework for formulating and solving problems in areas of physics such as elasticity, fluid mechanics, and general relativity. Have exactly two indices) to an arbitrary number of indices. (that have exactly one index), and matrices (that Tensors are generalizations of scalars (that have no indices), vectors (with the notable exception of the contracted Kroneckerĭelta). However, the dimension of the space is largely irrelevant in most tensor equations ![]() Of a tensor ranges over the number of dimensions of space. If you have studied the book thoroughly, you will be prepared to start working on the physics of gravitation as described by general relativity.Tensor in -dimensional space is a mathematicalĬomponents and obeys certain transformation rules. This book serves as a good basis for mastering tasks. For exercises, there are enough examples in the relevant textbooks that can be used to deepen a topic. For this purpose, many examples and detailed extra introductions have been made. And it is thus the stage on which the physical processes and procedures show themselves.Ĭalculation tasks have been deliberately omitted. Space is a component of the representation of the physical real. Therefore, the emphasis in this textbook has been placed on understanding space in its geometric configuration. General relativity is in its deeper sense a geometric theory. Great importance is always attached to the clarity of the explanations and derivations of the topics. This textbook starts with basic topics such as vector space and vectors (chapter 1), dual space and covectors (chapter 2), tensors (chapter 3), etc. Some knowledge of linear algebra and analysis are required. This book is intended for physics students who want to prepare for lectures on general relativity. For knowledge is limited to all we now know and understand, while imagination embraces the entire world, and all there ever will be to know and understand.” Albert Einstein Start learning the foundations of General Relativity today… “Imagination is more important than knowledge.
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