15.1 Introduction to tensors and index notation

Tensors are powerful tools in physics and math, extending the ideas of scalars and vectors. They help describe complex systems and properties that change with coordinates. This intro to tensors covers their basics, notation, and key concepts.

Index notation makes working with tensors easier, using subscripts and superscripts to show components. We'll learn about tensor ranks, coordinate transformations, and important symbols like the Kronecker delta and Levi-Civita symbol.

Tensor Basics

Definition and Notation

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• Tensor generalizes the concept of scalar, vector, and matrix to higher dimensions
• Defined as a mathematical object that transforms according to certain rules under a change of coordinates
• Index notation represents tensors using indices (subscripts and superscripts) to indicate their components
  • Example: $A_{ij}$ represents a second-rank tensor with components indexed by $i$ and $j$
• Rank of a tensor refers to the number of indices required to specify its components
  • Scalar is a tensor of rank 0 (no indices)
  • Vector is a tensor of rank 1 (one index, e.g., $v_i$)
  • Matrix is a tensor of rank 2 (two indices, e.g., $A_{ij}$)
  • Higher-rank tensors have more indices (e.g., $T_{ijk}$ is a third-rank tensor)

Coordinate Transformations

• Tensors are defined independently of the choice of coordinate system
• Components of a tensor change when the coordinate system is transformed
• Transformation rules ensure that the tensor itself remains unchanged
  • Example: Components of a vector $v_i$ transform as $v'_i = \frac{\partial x'_i}{\partial x_j} v_j$ under a coordinate transformation from $x_i$ to $x'_i$
• Tensors of higher rank have more complex transformation rules involving multiple partial derivatives

Tensor Components

Contravariant and Covariant Components

• Contravariant components are denoted by superscript indices (e.g., $A^i$)
  • Transform inversely to the basis vectors under a coordinate transformation
  • Example: Components of a contravariant vector $v^i$ transform as $v'^i = \frac{\partial x'^i}{\partial x_j} v^j$
• Covariant components are denoted by subscript indices (e.g., $A_i$)
  • Transform in the same way as the basis vectors under a coordinate transformation
  • Example: Components of a covariant vector $v_i$ transform as $v'_i = \frac{\partial x_j}{\partial x'^i} v_j$
• Mixed tensors have both contravariant and covariant components (e.g., $T^i_j$)

Metric Tensor

• Metric tensor $g_{ij}$ is a symmetric, second-rank tensor that defines the inner product and distance in a space
• Used to raise or lower indices of tensor components
  • Raising an index: $A^i = g^{ij} A_j$, where $g^{ij}$ is the inverse of the metric tensor
  • Lowering an index: $A_i = g_{ij} A^j$
• In Euclidean space, the metric tensor is the Kronecker delta $\delta_{ij}$ (identity matrix)

Tensor Operations and Symbols

Einstein Summation Convention

• Einstein summation convention simplifies tensor expressions by implying summation over repeated indices
• When an index appears twice in a term (once as a superscript and once as a subscript), summation over that index is implied
  • Example: $A_i B^i = \sum_{i} A_i B^i$ (sum over the repeated index $i$)
• Greatly simplifies tensor equations by eliminating the need for explicit summation symbols

Kronecker Delta

• Kronecker delta $\delta_{ij}$ is a second-rank tensor defined as:
  • $\delta_{ij} = 1$ if $i = j$
  • $\delta_{ij} = 0$ if $i \neq j$
• Acts as the identity tensor, analogous to the identity matrix
• Used for raising or lowering indices in Euclidean space
  • Example: $A^i = \delta^{ij} A_j$ (raising an index in Euclidean space)

Levi-Civita Symbol

• Levi-Civita symbol $\epsilon_{ijk}$ is a third-rank tensor that represents the permutation of indices
• Defined as:
  • $\epsilon_{ijk} = +1$ if $(i, j, k)$ is an even permutation of $(1, 2, 3)$
  • $\epsilon_{ijk} = -1$ if $(i, j, k)$ is an odd permutation of $(1, 2, 3)$
  • $\epsilon_{ijk} = 0$ if any two indices are equal
• Used to express cross products and determinants in tensor notation
  • Example: Cross product of two vectors $\vec{a}$ and $\vec{b}$ can be written as $(a \times b)^i = \epsilon^{ijk} a_j b_k$