Chapter 3, A primer on tensor analysis in three-dimensional space Video Solutions, A Course on Plasticity Theory

Problem 1

Compute the $g_i, g_{\vec{\psi}}, g^j$, and $\mathrm{g}^i$ in spherical coordinates $\left\{\xi^i\right\}=\{r, \theta, \phi\}$, using
$$
y^1=r \cos \phi \cos \theta, \quad y^2=r \cos \phi \sin \theta, \quad y^3=r \sin \phi .
$$
At what points, if any, are the relations between the Cartesian and spherical coordinates non-invertible?

Carson Merrill

Numerade Educator

01:52

Problem 2

Establish the four representations
$$
u^i v^j g_{\bar{j}}=u^i v_i=u_j v_i g^{\vec{\theta}}=u_j v^j
$$
for the scalar product $\mathbf{u} \cdot \mathbf{v}$ of two 3-vectors.

Monica Miller

Numerade Educator

03:25

Problem 3

Establish the two representations
$$
\epsilon_{k i j} u^i v^j \mathbf{g}^k=\epsilon^{h i j} u_i v, g_k
$$
for the cross product $\mathbf{u} \times \mathbf{v}$ of two 3-vectors.

Jose Hannan

Numerade Educator

01:52

Problem 4

Find $\left\{\mathrm{g}_i\right\},\left\{g^i\right\}, g_{i j}$, and $g^{i j}$ in terms of $\left\{\xi^i\right\}=\{\xi, \eta, \phi\}$ for the following coordinate systems: (a) Parabolic coordinates: $y^1=\xi \eta \cos \phi, y^2=\xi \eta \sin \phi$, $y^3=\frac{1}{2}\left(\xi^2-\eta^2\right)$. (b) Elliptic-cylindrical coordinates: $y^1=\cosh \xi \cos \eta, y^2=\sinh \xi \sin \eta$, $y^3=\phi$

Linda Hand

Numerade Educator

03:08

Problem 5

Consider the complex-valued analytic function $g(z)$ of the complex variable $z=x+i y$, where $x\left(=y^1\right)$ and $y\left(=y^2\right)$ are Cartesian coordinates in the plane. Let $\left\{\xi^i\right\}=\left\{\phi, \psi, y^3\right\}$, where $\phi=\operatorname{Re} g$ and $\psi=\operatorname{Im} g$. Consider the example $g(z)=z^2$ and compute $\left\{\mathbf{g}_i\right\},\left\{\mathbf{g}^i\right\}, g_{i j}, g^{i j}$ in terms of the $y^i$. Sketch the coordinate surfaces, i.e., the surfaces on which each coordinate is constant.

Tanishq Gupta

Numerade Educator

02:05

Problem 6

Consider an oblique coordinate system $\left\{\xi^i\right\}$ for which $\mathrm{g}_1=\mathrm{i}_1$, $\mathrm{g}_2=\cos \phi \mathrm{i}_2+\sin \phi \mathrm{i}_1, \mathrm{~g}_3=\mathrm{i}_3$ with $\phi$ a constant angle.
(a) Find the $y^i$ as functions of the $\xi$. (Hint: construct the position vector and suppose the $\xi^i$ vanish at the origin.) Invert these relations.
(b) Obtain $g_{i j}, g^{i j}$ and find $\left\{\mathbf{g}^i\right\}$ in terms of $\left\{\mathbf{i}_i\right\}$.
(c) If $v_i^{(c)}$ are the Cartesian components of a vector, what are the co- and contravariant components of the vector in the $\left\{\xi^i\right\}$ system?

Surendra Kumar

Numerade Educator

01:02

Problem 7

Show that $\mathbf{A}^l=A_j^i \mathbf{g}^j \otimes \mathbf{g}_i$, and hence that $A_{j j}^i=A_j^i$ if $\mathbf{A}$ is symmetric. For symmetric A, show that $A_{i j}=A_{j i}$ and $A^{i j}=A^{i i}$. What if A is skew-symmetric, i.e., $\mathbf{A}^t=-\mathbf{A}$ ?

Hoan Nguyen

Numerade Educator

Problem 8

Compute the gammas for spherical coordinates.

Check back soon!

Problem 9

Prove this.
Then,
$$
\text { c } \cdot\left[\operatorname{div} \mathbf{A}-\left(\mathrm{A}_{, i}\right) \mathrm{g}^i\right]=0
$$
and as this purports to hold for all fixed c, we may take it to coincide with the square bracket, fixed at the point in question, to conclude that the norm of the bracket, and hence the bracket itself, vanishes. Thus,
$$
\operatorname{div} \mathbf{A}=\left(\mathbf{A}_{, j}\right) \mathbf{g}^j
$$
This exceedingly useful formula conceals fairly complicated component expressions. For example, starting with the representation (3.39), differentiating everything in sight and applying the product rule to the tensor products, we get
$$
\mathbf{A}_j=A_{i j}^{i k} \mathbf{g}_i \otimes \mathbf{g}_k
$$
where
$$
A_j^{i k}=A_{j j}^{i k}+A^{i l} \Gamma_{l j}^k+A^{i k} \Gamma_{l j}^i
$$
is one among four kinds of covariant derivative. Another kind follows by using the first of Eqs. (3.40): Thus,
$$
\mathbf{A}_{, j}=A_{i k j j} \mathbf{g}^i \otimes \mathbf{g}^k
$$
where
$$
A_{i k j j}=A_{i k_j j}-A_{i k} \Gamma_{i j}^l-A_i \Gamma_{k j}^l
$$

Check back soon!

07:06

Problem 10

Show that for two differentiable vector fields $\mathbf{u}$ and $\mathbf{v}$,
$$
(\mathbf{u} \otimes \mathbf{v})_{, i}=\mathbf{u}_{, i} \otimes \mathbf{v}+\mathbf{u} \otimes \mathbf{v}_{, i}
$$

Carlos Pinilla

Numerade Educator

Problem 11

Verify (3.82) and (3.84), and write out expressions for the remaining two kinds of covariant derivative, namely $A_{\cdot k, j}^i$ and $A_{k, j}^i$

Check back soon!

00:55

Problem 12

Write out the Laplacian $\Delta f(\xi, \eta, \phi)$ explicitly in terms of $f$ and its partial derivatives with respect to $\xi, \eta, \phi$ for the coordinate systems of Problem 3.4.

Hast Aggarwal

Numerade Educator

01:23

Problem 13

In cylindrical polar coordinates $\{r, \theta, z\}$ the position function is given by $\mathbf{y}=r \mathrm{e}_r(\theta)+z \mathbf{i}_3$, where $r$ is the radius from the $z$-axis, $z$ measures distance along this axis, $\theta$ is the azimuthal angle, and $\mathbf{e}_r(\theta)=\cos \theta \mathbf{i}_1+\sin \theta \mathbf{i}_2$. In domains with coneshaped boundaries, it may be more convenient to use a conical-polar coordinate system. For this purpose we introduce coordinates $\{\bar{r}, \theta, \bar{z}\}$, where $r=\bar{r} \cos \phi$,
The Levi-Civita connection
53
$\bar{z}=z-\bar{r} \sin \phi$, and $\phi$ is the constant cone angle. The position function may then be written $\mathrm{y}=\overline{\mathrm{e}}_r(\theta)+\bar{z}_3$, where $\overline{\mathbf{e}}_r(\theta)=\cos \phi \mathrm{e}_r(\theta)+\sin \phi \mathrm{i}_3$ (draw a figure). Let $\left\{\xi^i\right\}=\{\bar{r}, \theta, \bar{z}\}$ and write out the divergence $A_{i j}^{\bar{i}}$ explicitly in terms of $A^{i j}$ and their partial derivatives.

James Kiss

Numerade Educator

Problem 14

Show that the Laplacian of a function $f(r, \theta, \phi)$ in spherical coordinates is
$$
\Delta f=\frac{1}{r^2}\left[\left(r^2 f_r\right)_r+\left(\sec ^2 \phi\right) f_{\theta \theta}+\sec \phi\left(f_\phi \cos \phi\right) \phi\right]
$$
where the subscripts stand for partial derivatives.

Check back soon!

Problem 15

Position in a helical coordinate system $\left\{\xi^i\right\}=\{r, \theta, s\}$ is described by
$$
\mathbf{y}\left(\xi^i\right)=\mathbf{r}(s)+r \cos \theta \mathbf{n}(s)+r \sin \theta \mathbf{b}(s)
$$
where $\mathbf{r}(s)$ is the equation of the center-line of a helical tube, $s$ is arclength along the center-line, $\mathbf{n}(s)$ is the principal normal to the center-line, and $\mathbf{b}(s)$ is the binormal. The coordinates $r$ and $\theta$, respectively, are the radius from the center-line and the counterclockwise azimuth measured from the principal normal in a plane cross section of the tube. The curvature of the center-line is $x(s)=\left|\mathbf{t}^{\prime}(s)\right|$, where $\mathbf{t}=\mathbf{r}^{\prime}(s)$ is the unit tangent to the center-line. Wherever $x$ is non-zero, we stipulate that $\mathbf{n}=\boldsymbol{x}^{-1} \mathbf{t}^{\prime}(s)$ and $\mathbf{b}=\mathbf{t} \times \mathbf{n}$ so that $\{\mathbf{t}, \mathbf{n}, \mathbf{b}\}$ is an orthonormal basis at every point of the curve.

The Serret-Frenet equations for smooth space curves are (see any elementary geometry or dynamics text).
$$
\mathbf{n}^{\prime}(s)=\tau \mathbf{b}-x \mathbf{t} \text { and } \mathbf{b}^{\prime}(s)=-\tau \mathbf{n},
$$
where $\tau(s)$ is the torsion of the center-line (not to be confused with the torsion, i.e., the skew part of the connection, referred to in the text).

Let $\left\{\xi^i\right\}=\{r, \theta, s\}$ and obtain $\left\{\mathbf{g}_i\right\},\left\{\mathrm{g}^i\right\}, g_{i j}, g^{i j}$ in terms of $\mathbf{t}, \mathbf{n}, \mathbf{b}, \kappa, \tau$ and the coordinates. Show that this coordinate system is non-orthogonal; i.e., the matrix $\left(g_{i j}\right)$ is not diagonal.

Helices of uniform radius and pitch are distinguished by the property that the curvature and torsion are constants, independent of arclength. Obtain an explicit expression for the Laplacian $\Delta f(r, \theta)$ in this special case, for a function $f$ that is independent of arclength. This problem is of obvious importance in diverse applications including, for example, heat conduction in helical wires and fluid flow in helical tubes.

Check back soon!

01:01

Problem 16

Work out the transformation formula
$$
\bar{A}_{\cdot i j}^k=\frac{\partial \bar{\xi}^k}{\partial \xi^l} \frac{\partial \xi^m}{\partial \bar{\xi}^i} \frac{\partial \xi^n}{\partial \bar{\xi}^j} A_{\cdot m n}^l
$$
for components $A_{\cdot i j}^k$ of a third-order tensor.
(a) Thus, show that the connection coefficients $\Gamma_{i j}^k$ are not the components of a tensor, firstly by proceeding from the formula $(3.56)$ for the connection coefficients, and secondly by proceeding directly from the Levi-Civita form (3.103) without using (3.56). In fact this result follows from the tensorial property of the covariant derivatives of a tensor, whether or not the connection has either of the forms (3.56) or (3.103). See the book by Lovelock and Rund or Szekeres' book for fuller discussions.
(b) Suppose the connection is asymmetric, with a non-zero torsion $T_{\cdot i j}^k=\Gamma_{[i]}^k$. Show that the $T_{\cdot i j}^k$ are the components of a tensor, provided that the coordinates $\bar{\xi}^i$ are twice-differentiable functions of the $\xi^i$.
(c) Using (3.16), argue that $\epsilon_{i j k ;}=0$ and $\epsilon_{i l}^{i j k}=0$ without invoking the formulae for the covariant derivatives of third-order tensor components. Use a similar argument to explain why the metric and reciprocal metric are covariantly constant without resorting to Ricci's lemma.

Raj Bala

Numerade Educator

View

Problem 17

Combine (3.130) and (3.131) to obtain $R_{-m b b^k}^{u^m}=0$.

Alison Rodriguez

Numerade Educator

Problem 18

Show that (3.62) is true, with the gammas given by (3.56), if and only if $R_{-m j j}^h=0$.

Check back soon!

01:01

Problem 19

Consider a vector field with covariant components $v_i$, and let $A_{i j}=v_{i j j}$. Suppose the $v_i$ are twice differentiable, so that $v_{i, j k}=v_{i, k j}$.
Torsion and the Weitzenböck connection
59
(a) Show that, in a general affine space with both torsion and curvature,
$$
A_{i j ; k]}+T_{\cdot j k}^l A_{i l}=\frac{1}{2} R_{i j k}^m v_m .
$$
(b) Note that the left-hand side of this equation is tensorial; i.e., it represents the covariant components of a third-order tensor. Use this fact to establish the transformation formula giving $\bar{R}_{\cdot i j k}^m$ in terms of the $R_{-\mathrm{Aqr}}^n$ and hence conclude that $R_{\cdot i j k}^{m^{\prime}}$ are indeed the components of a fourth-order tensor. (You could proceed from (3.136) to reach the same conclusion, but the present method is considerably simpler.)
(c) It follows from the above formula that, unlike partial differentiation, covariant differentiation generally does not commute, i.e., $v_{i, j k} \neq v_{i, k j}$, where $v_{i, j k}$ is shorthand for $\left(v_{i, j}\right)_{; k}$. Exceptionally, $v_{i, j k}=v_{i, k j}$ if the space is Euclidean $\left(T_{, j k}^i=0\right.$ and $\left.R_{r i j k}^m=0\right)$. Prove this fact directly, using only the tensorial property of $v_{\dot{k} j k}$ and the fact that it is possible to parametrize Euclidean space in terms of Cartesian coordinates.

Raj Bala

Numerade Educator

Problem 20

Assuming the differentials (3.60) to be exact, apply Stokes' theorem to arrive at an integral constraint involving the curvature tensor.

Check back soon!