Tensor Dimension

Tensor Dimension

A tensor's dimension refers to the number of indices required to uniquely specify an element within the tensor. Mathematically, it can be expressed as:

Formal Definition:

For a tensor T, its dimension n is the number of independent indices $(i₁, i₂, ..., iₙ)$ needed to specify any component $Tᵢ₁ᵢ₂...ᵢₙ$

Mathematical Examples:

• Scalar (0D): $a$

• Vector (1D): $a_i$

• Matrix (2D): $a_{ij}$

• 3D Tensor: $a_{ijk}$

Real Mathematical Examples:

• CopyScalar: 5

• Vector: $v = [2, 3, 4]$

• Matrix: $\mathbf{M} = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$

• 3D Tensor: $\mathbf{T} = \begin{bmatrix} \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \end{bmatrix}$

Classification by Dimension

a. Scalar (0-dimension)

• Definition: No indices needed

• Notation: $s$ or $s₀$

• Examples:

  • Temperature: 25°C

  • Mass: 5 kg

  • Potential: $V = 10V$

b. Vector (1-dimension)

• Definition: One index needed

• Notation: $vᵢ$ or $v[i]$

• Examples:

  • Position vector: $r = [x, y, z]$

  • Force vector: $F = [Fx, Fy, Fz]$

  • Momentum: $p = [px, py, pz]$

c. Matrix (2-dimension)

• Definition: Two indices needed

• Notation: $Mᵢⱼ$ or $M[i,j]$

• Examples: Stress tensor: $\sigma_{ij} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{zx} & \sigma_{zy} & \sigma_{zz} \end{bmatrix}$ Inertia tensor: $I_{ij} = \begin{bmatrix} I_{xx} & I_{xy} & I_{xz} \\ I_{yx} & I_{yy} & I_{yz} \\ I_{zx} & I_{zy} & I_{zz} \end{bmatrix}$

d. 3-Dimensional Tensor

• Definition: Three indices needed

• Notation: $Tᵢⱼₖ$ or $T[i,j,k]$

• Examples:

  • Piezoelectric tensor: $d_{ᵢⱼₖ}$

  • Elastic stiffness tensor: $C_{ᵢⱼₖₗ}$

Mathematical Properties Based on Dimension

a. Transformation Rules

For a tensor $T$ of dimension $n$, under coordinate transformation $R$:

$T{\prime}{i_1 i_2 \ldots i_n} = R{i_1 j_1} R_{i_2 j_2} \ldots R_{i_n j_n} T_{j_1 j_2 \ldots j_n}$

b. Components Count

For a tensor in n-dimensional space:

• 0D (Scalar): $1$ component

• 1D (Vector): $n$ components

• 2D (Matrix): $n²$ components

• 3D Tensor: $n³$ components

Mathematical Operations by Dimension

a. Inner Products

• Vectors (1D): $\mathbf{a} \cdot \mathbf{b} = \sum_{i} a_i b_i$

• Matrices (2D): $\mathbf{A} : \mathbf{B} = \sum_{i,j} A_{ij} B_{ij}$

• 3D Tensors: $T_1 \cdot T_2 = \sum_{i,j,k} T_{1_{ijk}} T_{2_{ijk}}$

b. Outer Products

• Vectors to Matrix: $(a \otimes b)_{ij} = a_i b_j$

• Matrices to 4D Tensor: $(A \otimes B){ijkl} = A{ij} B_{kl}$

Implementation with PHP

Tensor Dimensions

<?php

// 0-dimensional tensor (Scalar)
$scalar = 5;

// 1-dimensional tensor (Vector)
$vector = [1, 2, 3];

// 2-dimensional tensor (Matrix)
$matrix = [
    [1, 2, 3],
    [4, 5, 6]
];

// 3-dimensional tensor
$tensor_3d = [
    [[1, 2], [3, 4]],
    [[5, 6], [7, 8]]
];

// Function to determine dimension
function getTensorDimension($tensor) {
    if (!is_array($tensor)) {
        return 0;  // Scalar
    }
    
    $dim = 1;
    $current = $tensor;
    
    while (is_array(current($current))) {
        $dim++;
        $current = current($current);
    }
    
    return $dim;
}

// Examples
echo "Scalar dimension: " . getTensorDimension($scalar) . "\n";  // 0
echo "Vector dimension: " . getTensorDimension($vector) . "\n";  // 1
echo "Matrix dimension: " . getTensorDimension($matrix) . "\n";  // 2
echo "3D Tensor dimension: " . getTensorDimension($tensor_3d) . "\n";  // 3