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Appendix B Problems

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Table of Contents

Basic

Medium

Advanced

Basic

B-1. Basics of the Tensor Product \(S \otimes T\)

Hint

Use the distributive law \((a e_1 + b e_2) \otimes (c e_1 + d e_2) = ac\,e_1 \otimes e_1 + ad\,e_1 \otimes e_2 + bc\,e_2 \otimes e_1 + bd\,e_2 \otimes e_2\).

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B-2. Non-commutativity of Tensor Products

Hint

The tensor product is generally non-commutative: \(S \otimes T \neq T \otimes S\). Compare the coefficients of \(e_1 \otimes e_2\) and \(e_2 \otimes e_1\).

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B-3. Linear Combinations of Tensors

Hint

Add or subtract the coefficients of the same basis \(e_i \otimes e_j\). This is the same operation as linear combinations of ordinary vectors.

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B-4. Dimensions of Tensor Spaces

Hint

When \(V\) is \(n\)-dimensional, the dimension of \(T^r(V)\) is \(n^r\).

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B-5. Expansion of Einstein Summation

Hint

Write \(S = \sum_{i=1}^{3}\sum_{j=1}^{3} S^{ij}\,e_i \otimes e_j\) and count all combinations of \(i\) and \(j\).

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B-6. Violations of the Einstein Convention

  • (a) \(A^i B^j\,e_i \otimes e_j\)
  • (b) \(A^i B^i C^i\)
  • (c) \(S^{ij}\,T_{jk}\)
  • (d) \(A^i B_j\)
  • (e) \(S^{ij}\,e_i \otimes e_i\)
Hint

Rules of the convention: (1) When the same index appears once in an upper position and once in a lower position, summation is implied; (2) The same index must not appear three or more times in a single term; (3) Free indices (indices not summed over) must match across all terms.

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B-7. Reconstructing a Tensor from Components

Hint

Expand the contraction \(S = S^{ij}\,e_i \otimes e_j\) and substitute the values of the components.

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B-8. Tensor Product of Rank-1 and Rank-2 Tensors

Hint

\(A \otimes S\) is the tensor product of an element of \(T^1(V)\) and an element of \(T^2(V)\), so it is an element of \(T^{1+2}(V) = T^3(V)\). Expand using the distributive law.

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Medium

M-1. Condition for Decomposable Tensors

Hint

Set \(S = \alpha e_1 + \beta e_2\) and \(T = \gamma e_1 + \delta e_2\), expand \(S \otimes T\), and compare \(ad\) with \(bc\) where \(a = \alpha\gamma,\; b = \alpha\delta,\; c = \beta\gamma,\; d = \beta\delta\). Also show the converse direction (if \(ad = bc\), then the tensor is decomposable).

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M-2. Component Representation of Bilinear Maps

(a) For general vectors \(\vec{A} = A^1 e_1 + A^2 e_2\) and \(\vec{B} = B^1 e_1 + B^2 e_2\), express \(f(\vec{A}, \vec{B})\) in terms of the components \(A^i, B^j, f_{ij}\) using multilinearity.

(b) Find the value of \(f(\vec{A}, \vec{B})\) when \(\vec{A} = e_1 - 2e_2\) and \(\vec{B} = 3e_1 + e_2\).

(c) Verify using components that \(f(\vec{A}, \vec{B}) = f(\vec{B}, \vec{A})\) holds for arbitrary \(\vec{A}, \vec{B}\).

Hint

In (a), derive \(f(\vec{A}, \vec{B}) = A^i B^j f_{ij}\) similarly to equation (B.7). In (c), confirm that \(f_{ij} = f_{ji}\).

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M-3. Rank Addition via Tensor Product

Hint

Use the distributive law and scalar extraction properties (B.1)–(B.3) of the tensor product to expand \(S \otimes T\), and confirm that the basis takes the form \(e_{i_1} \otimes \cdots \otimes e_{i_r} \otimes e_{j_1} \otimes \cdots \otimes e_{j_m}\).

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M-4. \(T^0(V)\) and Scalars

Hint

\(T^0(V)\) is a 1-dimensional space whose only basis element is the "empty tensor product," and it can be identified with the real number field \(\mathbb{R}\). Apply (B.1) to \(\lambda \otimes S\).

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M-5. Simple \(S \otimes T\) Expansion

Hint

Use the distributive law to expand \((e_1 + 2e_2) \otimes (3e_1 - e_2)\).

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M-6. Dimension of Tensor Spaces in 3-Dimensional Space

Hint

When \(V\) is \(n\)-dimensional, the dimension of \(T^r(V)\) is \(n^r\). Substitute \(n = 3\).

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M-7. Non-decomposability of Antisymmetric Tensors

Hint

Setting \(a = 0, b = 1, c = -1, d = 0\) gives \(ad - bc = 0 - (-1) = 1 \neq 0\). Use the result of Problem B.9.

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M-8. Evaluation of the Identity Bilinear Form

Hint

Substitute the components into \(f(\vec{A}, \vec{B}) = A^i B^j f_{ij}\). Note that \(f_{ij} = \delta_{ij}\) (the identity matrix).

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Advanced

A-1. Symmetric and Antisymmetric Decomposition

Define

\(W^{(S)} := \frac{1}{2}(W^{ij} + W^{ji})\,e_i \otimes e_j, \qquad W^{(A)} := \frac{1}{2}(W^{ij} - W^{ji})\,e_i \otimes e_j\)

(a) Show that any \(W \in T^2(V)\) can be uniquely decomposed as \(W = W^{(S)} + W^{(A)}\).

(b) Show that the set of all symmetric tensors \(\mathrm{Sym}^2(V) = \{W \in T^2(V) \mid W^{ij} = W^{ji}\}\) and the set of all antisymmetric tensors \(\mathrm{Alt}^2(V) = \{W \in T^2(V) \mid W^{ij} = -W^{ji}\}\) are each subspaces of \(T^2(V)\), and find the dimensions of \(\mathrm{Sym}^2(V)\) and \(\mathrm{Alt}^2(V)\) when \(V\) is \(n\)-dimensional.

(c) Given that the metric tensor \(g_{\mu\nu}\) is a symmetric tensor (\(g_{\mu\nu} = g_{\nu\mu}\)) and the electromagnetic field tensor (Faraday tensor) \(F_{\mu\nu}\) is an antisymmetric tensor (\(F_{\mu\nu} = -F_{\nu\mu}\)), find the number of independent components of \(g_{\mu\nu}\) and \(F_{\mu\nu}\) respectively in 4-dimensional spacetime (\(n = 4\)). Verify that their sum equals \(n^2\).

Hint

(b) The dimension of \(\mathrm{Sym}^2(V)\) is \(\frac{n(n+1)}{2}\), and the dimension of \(\mathrm{Alt}^2(V)\) is \(\frac{n(n-1)}{2}\). (c) Substitute \(n = 4\).

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A-2. Tensor Representation via Dual Spaces

(a) Write down the basis and dimension of the tensor product space \(V^* \otimes V^*\) of \(V^*\) and \(V^*\).

(b) Show that a \(\binom{0}{2}\) tensor \(f\) (a bilinear map that takes 2 vectors and returns a real number) can be expressed as an element of \(V^* \otimes V^*\) using the components \(f_{ij} = f(e_i, e_j)\):

\(f = f_{ij}\,e^i \otimes e^j\)

Demonstrate this by verifying that \(f(\vec{A}, \vec{B}) = A^k B^l f_{kl}\) is reproduced for arbitrary vectors \(\vec{A} = A^k e_k\), \(\vec{B} = B^l e_l\). Here, \((e^i \otimes e^j)(\vec{A}, \vec{B}) := e^i(\vec{A})\,e^j(\vec{B})\) is taken as the definition.

(c) Based on this result, discuss the fact that the space of \(\binom{0}{2}\) tensors is isomorphic to \(V^* \otimes V^*\), and that the space of \(\binom{0}{N}\) tensors is isomorphic to \(\underbrace{V^* \otimes \cdots \otimes V^*}_{N}\). Explain how the "connection between the two approaches" in Section B.11 of the main text generalizes.

Hint

(b) Substitute \((\vec{A}, \vec{B})\) into \(f_{ij}\,e^i \otimes e^j\) and use \(e^i(\vec{A}) = A^i\) (the definition of the dual basis). (c) In contrast to the contravariant tensor space \(T^r(V) = V \otimes \cdots \otimes V\), construct the argument for the covariant tensor space \(T_N(V) = V^* \otimes \cdots \otimes V^*\).

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