10.1 Tensor Algebra and Index Notation

Tensors are powerful mathematical tools that generalize scalars, vectors, and matrices. They're essential in physics for describing complex phenomena like stress, strain, and spacetime curvature. Their ability to represent multidimensional relationships makes them indispensable in various fields.

Tensor operations, including addition, multiplication, and contraction, allow physicists to manipulate these objects effectively. Understanding tensor algebra is crucial for tackling advanced topics in mathematical physics, from continuum mechanics to general relativity and beyond.

Tensor Fundamentals

Tensors and their rank

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• Generalize scalars, vectors, and matrices to higher-dimensional objects
  • Scalars represent single values with no directional information (rank-0 tensors)
  • Vectors represent quantities with magnitude and direction (rank-1 tensors)
  • Matrices represent linear transformations between vectors (rank-2 tensors)
• Determined by the number of indices needed to specify their components
  • Each index corresponds to a dimension in the tensor's space
  • Example: a rank-3 tensor $T_{ijk}$ has three indices and operates in a 3D space
• Represent physical quantities and their relationships in mathematical physics
  • Stress tensor describes internal forces acting on a material ($\sigma_{ij}$)
  • Strain tensor describes deformation of a material ($\varepsilon_{ij}$)
  • Moment of inertia tensor describes an object's resistance to rotational acceleration ($I_{ij}$)
  • Curvature tensor describes the curvature of spacetime in general relativity ($R_{ijkl}$)
• Coordinate-independent, allowing description of physical laws in a general manner
  • Tensors maintain their properties under coordinate transformations
  • Enables formulation of physical theories in a coordinate-free way (general relativity)

Tensor Manipulation and Operations

Index notation for tensors

• Compact representation of tensor components and operations using subscripts and superscripts
  • Subscripts denote covariant components ($a_i$) that transform contravariantly with coordinate changes
  • Superscripts denote contravariant components ($a^i$) that transform covariantly with coordinate changes
• Einstein summation convention simplifies tensor expressions by implying summation over repeated indices
  • Repeated indices in a term, one upper and one lower, are summed over their range
  • $a_ib^i = a_1b^1 + a_2b^2 + \cdots + a_nb^n$ (sum over $i$ from 1 to $n$)
  • Reduces notational clutter and emphasizes the coordinate-independent nature of tensors
• Tensor contraction operation reduces the rank of a tensor by summing over a pair of indices
  • Covariant and contravariant indices are paired and summed, resulting in a lower-rank tensor
  • $A_{ij}B^{jk} = C_i^k$ contracts over index $j$, resulting in a rank-2 tensor $C_i^k$

Operations with tensors

• Component-wise addition and subtraction for tensors of the same rank and dimensions
  • $(A_{ij} + B_{ij})_{ij} = A_{ij} + B_{ij}$ adds corresponding components of rank-2 tensors
  • Tensors must have matching indices and dimensions for these operations
• Various types of tensor multiplication to create new tensors
  • Outer product ($\otimes$) increases rank by combining tensors into a higher-rank tensor
    • $a_i \otimes b_j = C_{ij}$ combines rank-1 tensors into a rank-2 tensor
  • Inner product (contraction) reduces rank by summing over paired indices
    • $A_{ij}B^{j} = c_i$ contracts rank-2 and rank-1 tensors into a rank-1 tensor
  • Tensor product combines tensors of different spaces into a higher-rank tensor
    • $(A \otimes B)_{ijkl} = A_{ij}B_{kl}$ combines rank-2 tensors into a rank-4 tensor

Tensor algebra in physics

• Stress tensor ($\sigma_{ij}$) describes internal forces acting on a material
  • Rank-2 tensor with components representing stress in $j$-direction on surface normal to $i$-direction
  • $\sigma_{11}, \sigma_{22}, \sigma_{33}$ are normal stresses; $\sigma_{12}, \sigma_{13}, \sigma_{23}$ are shear stresses
• Strain tensor ($\varepsilon_{ij}$) describes deformation of a material
  • Rank-2 tensor derived from displacement vector $u_i$: $\varepsilon_{ij} = \frac{1}{2}(\partial_i u_j + \partial_j u_i)$
  • Relates change in length and angles to original configuration
• Hooke's law connects stress and strain tensors in linear elastic materials
  1. $\sigma_{ij} = C_{ijkl}\varepsilon_{kl}$, a tensor equation involving double contraction
  2. $C_{ijkl}$ is the rank-4 elastic stiffness tensor containing material properties
  3. Tensor equation represents a system of linear equations relating stress and strain components
• Tensor algebra is a powerful tool for deriving and solving equations in various fields
  • Continuum mechanics: stress, strain, and constitutive relations
  • General relativity: curvature tensor, Einstein field equations, and geodesic equations
  • Fluid dynamics: velocity gradient tensor, strain rate tensor, and Navier-Stokes equations
  • Electromagnetism: electromagnetic field tensor, Maxwell's equations in covariant form