Tensor | Wikipedia audio article
Channel: Subhajit Sahu
Date: 2018-12-25

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Tensor


00:03:44 1 Definition
00:05:47 1.1 As multidimensional arrays
00:06:07 1.2 As multilinear maps
00:27:07 1.3 Using tensor products
00:30:58 1.4 Tensors in infinite dimensions
00:33:56 1.5 Tensor fields
00:34:59 2 Examples
00:38:02 3 Notation
00:39:21 3.1 Ricci calculus
00:39:37 3.2 Einstein summation convention
00:40:01 3.3 Penrose graphical notation
00:40:31 3.4 Abstract index notation
00:40:55 3.5 Component-free notation
00:41:20 4 Operations
00:41:40 4.1 Tensor product
00:42:16 4.2 Contraction
00:45:46 4.3 Raising or lowering an index
00:49:54 5 Applications
00:50:58 5.1 Continuum mechanics
00:51:07 5.2 Other examples from physics
00:52:38 5.3 Applications of tensors of order &gt; 2
00:53:35 6 Generalizations
00:56:35 6.1 Holors
00:56:44 6.2 Tensor products of vector spaces
00:57:59 6.3 Tensors in infinite dimensions
00:58:45 6.4 Tensor densities
00:59:25 6.5 Geometric objects
01:05:52 6.6 Spinors
01:06:32 7 History
01:07:46 8 See also
01:10:46 8.1 Foundational
01:10:55 8.2 Applications
01:11:16 9 Notes



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SUMMARY
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In mathematics, a tensor is an arbitrarily complex geometric object that maps in a (multi-)linear manner geometric vectors, scalars, and other tensors to a resulting tensor. Thereby, vectors and scalars themselves, often used already in elementary physics and engineering applications, are considered as the simplest tensors. Additionally, vectors from the dual space of the vector space, which supplies the geometric vectors, are also included as tensors. Geometric in this context is chiefly meant to emphasize independence of any selection of a coordinate system.
Elementary examples of such relations include the dot product, mapping two vectors to a tensor, which is a simple scalar. A more complex example is the Cauchy stress tensor T, which takes a directional unit vector v as input and maps it to the stress vector T(v), which is the force (per unit area) exerted by material on the negative side of the plane orthogonal to v against the material on the positive side of the plane, thus expressing a relationship between these two vectors, shown in the figure (right). The cross product, where two vectors are mapped to a third one, is strictly speaking not a tensor, because it changes its sign under those transformations that change the orientation of the coordinate system. The totally anti-symmetric symbol 
  
    
      
        
          ε
          
            i
            j
            k
          
        
      
    
    {\displaystyle \varepsilon _{ijk}}
   allows nevertheless a convenient handling of the cross product in equally oriented three dimensional coordinate systems.
Assuming a basis of a real vector space, e.g., a coordinate frame in the ambient space, a tensor can be represented as an organized multidimensional array of numerical values with respect to this specific basis. Changing the basis transforms the values in the array in a characteristic way that allows to define tensors as objects adhering to this transformational behavior. For example, there are invariants of tensors that must be preserved under any change of the basis, thereby making only certain multidimensional arrays of numbers a tensor. Compare this to the array representing 
  
    
      
        
          ε
          
            i
            j
            k
          
        
      
    
    {\displaystyle \varepsilon _{ijk}}
   not being a tensor, for the sign change under transformations changing the orientation.
Because the components of vectors and their duals transform differently under the change of their dual bases, there is a covariant and/or contravariant transformation law that relates the arrays, which represent the tensor with respect to one basis and that with respect to the other one. The numbers of, respectively, vectors: n (contravaria ...