Math Problems Vault

Chapter IX

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Problem IX.1.1

Let $\mathcal{B}$ be the linear space of bilinear functions defined on an $n$ -dimensional linear space $\mathcal{L}$ , and let $\mathrm{A}$ be a linear transformation of the space $\mathcal{B}$ . Prove that $\mathrm{A}$ is a tensor of type $(2,2)$ in the space $\mathcal{L}$ .

Problem IX.1.2

(a)(b)

(a)

How many components does a tensor of valence three have in a four-dimensional space?

(b)

How many terms does the expression of any one of its components in a new basis through the components in the old basis contain?

Problem IX.1.3

A tensor of type $(0, n)$ in an $n$ -dimensional linear space has in a basis $\mathbf{e}$ the components $\varepsilon_{i_{1} \ldots i_{n}}=0$ , if among the values $i_{1}, \ldots , i_{n}$ there are equal ones, and

\varepsilon _{i_{1} \ldots i_{n}}=(-1)^{N\left(i_{1}, \ldots , i_{n}\right)}

otherwise. Find the components of this tensor in the basis $\mathbf{e}^{\prime }=\mathbf{e} S$ .

Problem IX.1.4

A linear function $\mathrm{f}$ is given in a basis $\mathbf{e}$ by a row $\mathbf{f}$ , and a vector $a$ by a column $\mathbf{a}$ . Find the matrix of the tensor $a \otimes \mathrm{f}$ . What geometric meaning does this tensor have?

Problem IX.1.5

How many different tensors can be formed by means of contraction from a tensor of type $(2,2)$ ?

Problem IX.1.6

Prove that the tensor from problem 3 is antisymmetric in any subset of the set of indices.

Problem IX.1.7

Prove that for any tensor of type $(1,1)$ the equalities

a_{j}^{(i} a_{l}^{k)}=a_{(j}^{i} a_{l)}^{k}, \quad a_{j}^{[i} a_{l}^{k]}=a_{[j}^{i} a_{l]}^{k} .

hold.

Section IX.2§ 2[3/100%]

Problem IX.2.1

In a basis $\mathbf{e}$ the metric tensor is given by a matrix $\Gamma$ , and the tensor $a_{i j}$ by a matrix $A$ :

\Gamma =\left\| \begin{array}{ll} 1 & 1 \\ 1 & 2 \end{array}\right\| , \quad A=\left\| \begin{array}{ll} 3 & 4 \\ 5 & 7 \end{array}\right\| .

Find the matrices $B$ , $C$ and $D$ of the tensors $a_{. j}^{k}$ , $a_{j}^{k}$ . and $a^{j k}$ .

Problem IX.2.2

One may notice that in problem 1 the determinants of the matrices, $A, B, C$ and $D$ coincide, and the traces of $B$ and $C$ are the same. Explain this.

Problem IX.2.3

Simplify the expression

\left(\delta _{r}^{m} a_{i j}^{r} g^{j k}+\delta _{i}^{j} a_{l j}^{m} g^{l k}\right) g_{k s}+a_{[j k]}^{m} g^{j k} g_{i s} .

Section IX.3§ 3[4/100%]

Problem IX.3.1

A function of two vectors in a three-dimensional Euclidean vector space assigns to any two vectors $\mathbf{x}$ and $\mathbf{y}$ the scalar triple product $(\mathbf{a}, \mathbf{x}, \mathbf{y})$ , where $\mathbf{a}$ is a fixed vector. Prove that this function is a 2-form. Express its matrix in a given basis $\mathbf{e}$ through the coordinates of the vector $\mathbf{a}$ .

Problem IX.3.2

Let $x_{1}, \ldots , x_{n}$ be linearly independent vectors in an $n$ -dimensional linear space $\mathcal{L}$ . Prove that the bivectors $x_{k} \wedge x_{l}$ for all $k<l$ form a basis in the space of bivectors of the space $\mathcal{L}$ .

Problem IX.3.3

(a)(b)(c)

Let $x_{1}, \ldots , x_{p}$ be a basis of a subspace $\mathcal{L}^{\prime } \subset \mathcal{L}$ . Let us call $w=x_{1} \wedge \ldots \wedge x_{p}$ the direction $p$ -vector of the subspace.

Prove that

(a)

a vector $y$ lies in $\mathcal{L}^{\prime }$ if and only if $y \wedge w=0$ .

(b)

Any two direction $p$ -vectors of the subspace $\mathcal{L}^{\prime }$ differ from one another by a numerical factor.

(c)

If the space is Euclidean, this factor equals the ratio of the volumes of the oriented parallelepipeds built on the corresponding bases.

Problem IX.3.4

Prove that in a three-dimensional space every bivector is simple.