电脑技术：快速在word2016中输入公式并编号的简单教程（原创）
最近写论文，需要在word中录入公式，并且带公式编号。在网上看了一些方法，最后发现，word2016版本输入公式很方便了，现在就给大家分享一下。为了更加生动形象，我直接做一个视频教程，以供大家所需。
原创视频，转载请注明出处。
需要下载视频请访问网盘链接：https://pan.baidu.com/s/1o7VrAky 密码：cp9a
最近写论文，需要在word中录入公式，并且带公式编号。在网上看了一些方法，最后发现，word2016版本输入公式很方便了，现在就给大家分享一下。为了更加生动形象，我直接做一个视频教程，以供大家所需。
原创视频，转载请注明出处。
需要下载视频请访问网盘链接：https://pan.baidu.com/s/1o7VrAky 密码：cp9a
转载自http://mathworld.wolfram.com/MetricTensor.html
Roughly speaking, the metric tensor is a function which tells how to compute the distance between any two points in a given space. Its components can be viewed as multiplication factors which must be placed in front of the differential displacements in a generalized Pythagorean theorem:
(1)

In Euclidean space, where is the Kronecker delta (which is 0 for and 1 for ), reproducing the usual form of the Pythagorean theorem
(2)

In this way, the metric tensor can be thought of as a tool by which geometrical characteristics of a space can be “arithmetized” by way of introducing a sort of generalized coordinate system (Borisenko and Tarapov 1979).
In the above simplification, the space in question is most often a smooth manifold , whereby a metric tensor is essentially a geometrical object taking two vector inputs and calculating either the squared length of a single vector or a scalar product of two different vectors (Misner et al. 1978). In this analogy, the inputs in question are most commonly tangent vectors lying in the tangent space for some point, a fact which facilitates the more common definition of metric tensor as an assignment of differentiable inner products to the collection of all tangent spaces of a differentiable manifold (O’Neill 1967). For this reason, some literature defines a metric tensor on a differentiable manifold to be nothing more than a symmetric nondegenerate bilinear form (Dodson and Poston 1991).
An equivalent definition can be stated using the language of tensor fields and indices thereon. Along these lines, some literature defines a metric tensor to be a symmetric tensor field on a smooth manifold so that, for all , is nondegenerate and for some nonnegative integer (Sachs and Wu 1977). Here, is called the index of and the expression refers to the index of the respective quadratic form. This definition seems to occur less commonly than those stated above.
Metric tensors have a number of synonyms across the literature. In particular, metric tensors are sometimes called fundamental tensors (Fleisch 2012) or geometric structures (O’Neill 1967). Manifolds endowed with metric tensors are sometimes called geometric manifolds (O’Neill 1967), while a pair consisting of a real vector space and a metric tensor is called a metric vector space (Dodson and Poston 1991). Symbolically, metric tensors are most often denoted by or , although the notations (O’Neill 1967), (Fleisch 2012), and (Dodson and Poston 1991) are also sometimes used.
When defined as a differentiable inner product of every tangent space of a differentiable manifold , the inner product associated to a metric tensor is most often assumed to be symmetric, nondegenerate, and bilinear, i.e., it is most often assumed to take two vectors as arguments and to produce a real number such that
(3)

(4)

(5)

(6)

Note, however, that the inner product need not be positive definite, i.e., the condition
(7)

with equality if and only if need not always be satisfied. When the metric tensor is positive definite, it is called a Riemannian metric or, more precisely, a weak Riemannian metric; otherwise, it is called nonRiemannian, (weak) pseudoRiemannian, or semiRiemannian, though the latter two terms are sometimes used differently in different contexts. The simplest example of a Riemannian metric is the Euclidean metric discussed above; the simplest example of a nonRiemannian metric is the Minkowski metric of special relativity, the fourdimensional version of the more general metric of signature which induces the standard Lorentzian Inner Product on dimensional Lorentzian space. In some literature, the condition of nondegeneracy is varied to include either weak or strong nondegeneracy (Marsden et al. 2002); one may also consider metric tensors whose associated quadratic forms fail to be symmetric, though this is far less common.
In coordinate notation (with respect to a chosen basis), the metric tensor and its inverse satisfy a number of fundamental identities, e.g.
(8)

(9)

and
(10)

where is the matrix of metric coefficients. One example of identity (0) comes from special relativity where is the matrix of metric coefficients for the Minkowski metric of signature , i.e.
(11)

Generally speaking, identities (3), (2), and (1) can be succinctly written as
(12)

where
(13)


(14)

What’s more,
(15)

gives
(16)

and hence yields a quantitative relationship between a metric tensor and its inverse.
In the event that the metric is positive definite, the metric discriminants are positive. For a metric in twospace, this fact can be expressed quantitatively by the inequality
(17)

The orthogonality of contravariant and covariant metrics stipulated by
(18)

for gives linear equations relating the quantities and . Therefore, if metrics are known, the others can be determined, a fact summarized by saying that the existence of metric tensors gives a geometrical way of changing from contravariant tensors to covariant ones and vice versa (Dodson and Poston 1991).
In twospace,
(19)


(20)


(21)

Therefore, if is symmetric,
(22)


(23)

In any symmetric space (e.g., in Euclidean space),
(24)

and so
(25)

The angle between two parametric curves is given by
(26)

so
(27)

and
(28)

In arbitrary (finite) dimension, the line element can be written
(29)

where Einstein summation has been used. In three dimensions, this yields
(30)

and so it follows that the metric tensor in threespace can be written as
(31)

Moreover, because for when working with respect to orthogonal coordinate systems, the line elementfor threespace becomes
(32)


(33)

where are called the scale factors. Many of these notions can be generalized to higher dimensions and to more general contexts.
REFERENCES:
Borisenko, A. I. and Tarapov, I. E. Vector and Tensor Analysis with Applications. New York: Dover Publications, Inc., 1979.
Dodson, C. T. J. and Poston, T. Tensor Geometry: The Geometric Viewpoint and its Uses, 2nd Edition. New York: SpringerVerlag, 1991.
Fleisch, D. A Student’s Guide to Vectors and Tensors. New York: Cambridge University Press, 2012.
Marsden, J. E.; Ratiu, T.; and Abraham, R. Manifolds, Tensor Analysis, and Applications, 3rd Edition. SpringerVerlag Publishing Company, 2002.
Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. “The Metric Tensor.” §2.4 in Gravitation. San Francisco, CA: W. H. Freeman, pp. 5153, 1973.
O’Neill, B. Elementary Differential Geometry, 2nd Edition. Burlington, MA: Academic Press, 2006.
Ratcliffe, J. G. Foundations of Hyperbolic Manifolds. New York: Springer, 2006.
Sachs, R. K. and Wu, H. General Relativity for Mathematicians. New York: SpringerVerlag, 1977.
Snygg, J. A New Approach to Differential Geometry using Clifford’s Geometric Algebra. New York: Springer Science+Business Media, 2012.
Referenced on WolframAlpha: Metric Tensor
CITE THIS AS:
Stover, Christopher and Weisstein, Eric W. “Metric Tensor.” From MathWorld–A Wolfram Web Resource.http://mathworld.wolfram.com/MetricTensor.html
张量计算比较繁琐，尤其是广义相对论和黎曼几何结合起来的计算更是冗繁。好在现在已经出现了很多计算张量的工具包。吴老师使用Maple的 GRTenser 计算，我打算看看针对 Mathematica 的 EDC and RGTC。
EDC and RGTC，即 Riemannian Geometry & Tensor Calculus @ Mathematica，链接：http://www.inp.demokritos.gr/~sbonano/RGTC/
Download RGTC (Version 3.8.9 – May 2013)
 Download all files – compressed: .sit format (100 KB), .zip format (135 KB)
 Uncompressed files (~1000 KB): RGTC.nb — OperatorPLT.nb — NPsymbolPLT.nb — EDCRGTCcode.m. (Only the combined matrixEDC and RGTC code in package format is included — it must be placed in an appropriate directory).
 Note: RGTC cannot be used for calculations with abstract tensors (manipulation of tensor expressions with abstract indices). It only operates on explicit tensors (nested lists of components which are functions of the coordinates). For abstract calculations try the package xTensor.
Additional Examples can be found here.
英文版维基百科的介绍如下（来自 Tensor software https://en.wikipedia.org/wiki/Tensor_software），红色字体是和广义相对论计算有关的工具，我专门标注了出来。
Standalone software
 SPLATT^{[1]} is an open source software package for highperformance sparse tensor factorization. SPLATT ships a standalone executable, C/C++ library, and Octave/MATLABAPI.
 Cadabra^{[2]} is a computer algebra system (CAS) designed specifically for the solution of problems encountered in field theory. It has extensive functionality for tensor polynomial simplification including multiterm symmetries, fermions and anticommuting variables, Clifford algebras and Fierz transformations, implicit coordinate dependence, multiple index types and many more. The input format is a subset of TeX. Both a commandline and a graphical interface are available.
 Tela^{[3]} is a software package similar to Matlab and (GNU) Octave, but designed specifically for tensors.
Software for use with Mathematica
 Tensor^{[4]} is a tensor package written for the Mathematica system. It provides many functions relevant for General Relativity calculations in general RiemannCartan geometries.
 Ricci^{[5]} is a system for Mathematica 2.x and later for doing basic tensor analysis, available for free.
 TTC^{[6]} Tools of Tensor Calculus is a Mathematica package for doing tensor and exterior calculus on differentiable manifolds.
 EDC and RGTC,^{[7]} “Exterior Differential Calculus” and “Riemannian Geometry & Tensor Calculus,” are free Mathematica packages for tensor calculus especially designed but not only for general relativity.
 Tensorial^{[8]} “Tensorial 4.0” is a general purpose tensor calculus package for Mathematica.
 xAct:^{[9]} Efficient Tensor Computer Algebra for Mathematica. xAct is a collection of packages for fast manipulation of tensor expressions.
 GREAT^{[10]} is a free package for Mathematica that computes the Christoffel connection and the basic tensors of General Relativity from a given metric tensor.
 Atlas 2 for Mathematica^{[11]} is a powerful Mathematica toolbox which allows to do a wide range of modern differential geometry calculations
 GRTensorM^{[12]} is a computer algebra package for performing calculations in the general area of differential geometry.
 MathGR^{[13]} is a package to manipulate tensor and GR calculations with either abstract or explicit indices, simplify tensors with permutational symmetries, decompose tensors from abstract indices to partially or completely explicit indices and convert partial derivatives into total derivatives.
 TensoriaCalc^{[14]} is a tensor calculus package written for Mathematica 9 and higher, aimed at providing userfriendly functionality and a smooth consistency with the Mathematica language itself. As of January 2015, given a metric and the coordinates used, TensoriaCalc can compute Christoffel symbols, the Riemann curvature tensor, and Ricci tensor/scalar; it allows for userdefined tensors and is able to perform basic operations such as taking the covariant derivatives of tensors. TensoriaCalc is continuously under development due to time constraints faced by its inventor/developer.
Software for use with Maple
 GRTensorII^{[15]} is a computer algebra package for performing calculations in the general area of differential geometry.
 Atlas 2 for Maple^{[16]} is a modern differential geometry for Maple.
 DifferentialGeometry^{[17]} is a package which performs fundamental operations of calculus on manifolds, differential geometry, tensor calculus, General Relativity, Lie algebras, Lie groups, transformation groups, jet spaces, and the variational calculus. It is included with Maple.
Software for use with Matlab
 Tensorlab^{[18]} is a MATLAB toolbox for multilinear algebra and structured data fusion.
 Tensor Toolbox^{[19]} Multilinear algebra MATLAB software.
 MPCA and MPCA+LDA^{[20]}] Multilinear subspace learning software: Multilinear principal component analysis.
 UMPCA^{[21]}Multilinear subspace learning software: Uncorrelated multilinear principal component analysis.
 UMLDA^{[22]}Multilinear subspace learning software: Uncorrelated multilinear discriminant analysis.
Software for use with Maxima
Maxima^{[23]} is a free open source general purpose computer algebra system which includes several packages for tensor algebra calculations in its core distribution. It is particularly useful for calculations with abstract tensors, i.e., when one wishes to do calculations without defining all components of the tensor explicitly. It comes with three tensor packages:^{[24]}
 itensor for abstract (indicial) tensor manipulation,
 ctensor for componentdefined tensors, and
 atensor for algebraic tensor manipulation.
Software for use with R
 Tensor^{[25]} is an R package for basic tensor operations.
 rTensor^{[26]} provides several tensor decomposition approaches.
 tensorBF^{[27]} is an R package for Bayesian Tensor decomposition.
 MTF^{[28]} Bayesian MultiTensor Factorization for data fusion and Bayesian versions of Tensor PCA and Tensor CCA. Software: MTF
Libraries
 Redberry^{[29]} is an open source computer algebra system designed for symbolic tensor manipulation. Redberry provides common tools for expression manipulation, generalized on tensorial objects, as well as tensorspecific features: indices symmetries, LaTeXstyle input, natural dummy indices handling, multiple index types etc. The HEP package includes tools for Feynman diagrams calculation: Dirac and SU(N) algebra, LeviCivita simplifications, tools for calculation of oneloop counterterms etc. Redberry is written in Java and provides extensive Groovybased programming language.
 libxm^{[30]} is a lightweight distributedparallel tensor library written in C.
 FTensor^{[31]} is a high performance tensor library written in C++.
 TL^{[32]} is a multithreaded tensor library implemented in C++ used in Dynare++. The library allows for folded/unfolded, dense/sparse tensor representations, general ranks (symmetries). The library implements Faa Di Bruno formula and is adaptive to available memory. Dynare++ is a standalone package solving higher order Taylor approximations to equilibria of nonlinear stochastic models with rational expectations.
 vmmlib^{[33]} is a C++ linear algebra library that supports 3way tensors, emphasizing computation and manipulation of several tensor decompositions.
 Spartns^{[34]} is a Sparse Tensor framework for Common Lisp.
 FAstMat^{[35]} is a threadsafe general tensor algebra library written in C++ and specially designed for FEM/FVM/BEM/FDM element/edge wise computations.
 Cyclops Tensor Framework ^{[36]} is a distributed memory library for efficient decomposition of tensors of arbitrary type and parallel MPI+OpenMP execution of tensor contractions/functions.
 TiledArray^{[37]} is a scalable, blocksparse tensor library that is designed to aid in rapid composition of highperformance algebraic tensor equation. It is designed to scale from a single multicore computer to a massivelyparallel, distributedmemory system.
 libtensor ^{[38]} is a set of performance linear tensor algebra routines for large tensors found in postHartreeFock methods in quantum chemistry.
 ITensor ^{[39]} features automatic contraction of matching tensor indices. It is written in C++ and has higherlevel features for quantum physics algorithms based on tensor networks.
 Fastor ^{[40]} is a high performance C++ tensor algebra library that supports tensors of any arbitrary dimensions and all their possible contraction and permutation thereof. It employs compiletime graph search optimisations to find the optimal contraction sequence between arbitrary number of tensors in a network. It has high level domain specific features for solving nonlinear multiphysics problem using FEM.
 Xerus ^{[41]} is a C++ tensor algebra library for tensors of arbitrary dimensions and tensor decomposition into general tensor networks (focusing on matrix product states). It offers Einstein notation like syntax and optimizes the contraction order of any network of tensors at runtime so that dimensions need not be fixed at compiletime.
我对广义相对论的很多计算并不是很清楚，基本上也没怎么计算过度规、张量、四维电磁势等等东西，只是在现成的度规下开始做黑洞解，然后算一些温度、熵，深一些就做一些泰勒展开，或者用一下留数定理等，对张量的完整计算并不熟悉。
吴老师使用Maple的GRTenser张量包来计算黑洞相关的解，效果非常好，只可惜我没用过Maple，也不是很懂这个软件。对于Mathematica用的稍微多一点的我，在网上找到了一些书，这些书基本上是外文图书，但是内容都很不错。突然感叹，外国人虽少，但对某一点是真专注，再小众也有人做的非常深非常好，佩服。
这是在亚马逊搜索到的图书：查看链接
我下载了几本电子书，感觉还可以，放到这里可以下载：
MATHEMATICA全书（第4版）.pdf
Mathmatica for theoretical physics I.pdf
Mathmatica for theoretical physics II.pdf
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