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Chapter 16: Problem 1

Let \(x^{1}=x, x^{2}=y, x^{3}=z\) be coorduates on the manifold \(\mathbb{R}^{3}\). Write out the components \(\alpha_{u}\) and \((\mathrm{d} \alpha)_{n k}\), etc. for each of the following 2 -forms: $$ \begin{aligned} &\alpha=\mathrm{d} y \wedge \mathrm{d} z+\mathrm{d} x \wedge \mathrm{d} y \\ &\beta=x \mathrm{~d} z \wedge \mathrm{d} y+y \mathrm{~d} x \wedge \mathrm{d} z+z \mathrm{\phi} y \wedge \mathrm{d} x \\ &\gamma=\mathrm{d}\left(r^{2}(x \mathrm{~d} x+y \mathrm{~d} y+z \mathrm{~d} z)\right) \text { where } r^{2}=x^{2}+y^{2}+z^{2}. \end{aligned} $$

Short Answer

Expert verified
The non-zero components for the forms are: \(\alpha_{23} = \alpha_{31} = 1\), \(\alpha_{12} = -1\); \(\beta_{32} = x\), \(\beta_{13} = y\), \(\beta_{21} = z\); \(\gamma_{32} = \gamma_{13} = \gamma_{21} = 2x^2 + 2y^2 + 2z^2\).

Step by step solution

01

Find the components of \(\alpha\)

In this differential form, \(\alpha = \mathrm{d} y \wedge \mathrm{d} z + \mathrm{d} x \wedge \mathrm{d} y\). The non-zero components of \(\alpha\) are directly given from the 2-form itself, i.e., \(\alpha_{23} = \alpha_{31} = 1\), and \(\alpha_{12} = -1\) (this comes from reversing the order of the wedge product).
02

Find the components of \(\beta\)

In this differential form, \(\beta = x\, \mathrm{d} z \wedge \mathrm{d} y + y\, \mathrm{d} x \wedge \mathrm{d} z + z\, \mathrm{d} y \wedge \mathrm{d} x\). We have \(\beta_{32} = x\), \(\beta_{13} = y\), and \(\beta_{21} = z\).
03

Find the components of \(\gamma\)

In this differential form, \(\gamma = \mathrm{d}(r^{2}(x\, \mathrm{d} x + y\, \mathrm{d} y + z\, \mathrm{d} z)\) where \(r^{2}=x^{2}+y^{2}+z^{2}\). Notice that \(\gamma\) is actually the exterior derivative of another 1-form, which is \(r^{2}(x\, \mathrm{d} x + y\, \mathrm{d} y + z\, \mathrm{d} z)\). Expand the exterior derivative, and we get \(\gamma_{32} = 2x^2 + 2y^2 + 2z^2\), \(\gamma_{13} = 2x^2 + 2y^2 + 2z^2\), and \(\gamma_{21} = 2x^2 + 2y^2 + 2z^2\).

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Most popular questions from this chapter

Let \(\omega=y z \mathrm{~d} x+x z+z^{2} \mathrm{~d} z\). Show that the Pfaffian system \(\omega=0\) has integral surfaces \(g=z^{3} \mathrm{e}^{x y}=\) const, and express \(\omega\) in the form \(f \mathrm{~d} g\)

Given an \(r \times r\) matrix of 1 -forms \(\Omega\), show that the equation $$ d A=\Omega A-A \Omega $$ 15 soluble for an \(r \times r\) matrix of functions \(A\) only if $$ \Theta A=A \Theta $$ where \(\Theta=d \Omega-\Omega \wedge \Omega\) 1f the equation has a solution for arbitrary initial values \(A=A_{0}\) at any pornt \(p \in M\), show that there exists a 2 -form \(\alpha\) such that \(\Theta=\alpha\\}\) and \(d \alpha=0\).

On the manifold \(\mathbb{R}^{n}\) compute the exteror derivative \(\mathrm{d}\) of the differential form $$ \alpha=\sum_{t=1}^{n}(-1)^{i-1} x^{i} \mathrm{~d} x^{\prime} \wedge \cdots \wedge d x^{l-1} \wedge d x^{\prime+1} \wedge \cdots \wedge d x^{n}. $$ Do the same for \(\beta=r^{-n} \alpha\) where \(r^{2}=\left(x^{1}\right)^{2}+\cdots+\left(x^{n}\right)^{2}\).

For a reversible process \(\sigma: T \rightarrow K\), using absolute temperature \(T\) as the parameter, set $$ \sigma^{*} \theta=c \mathrm{~d} T $$ where \(c\) is known as the specife heat for the process. For a perfect gas show that for a process at constant volume, \(V=\) const., the specific heat is given by $$ c_{V}=\left(\frac{\partial U}{\partial T}\right)_{V} $$ For a process at constant pressure show that $$ c_{p}=c_{V}+R $$ while for an adiabatic process, \(\sigma^{\circ} \theta=0\). $$ p V^{r}=\text { const. where } \gamma=\frac{c_{p}}{c_{V}}. $$

Let \(\varphi: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}\) be the map $$ (x, y) \rightarrow(u, v, w) \text { where } u=\sin (r v), r=x+y, w=2 $$ For the 1 -form \(\omega=w_{1} \mathrm{~d} u+w_{2} \mathrm{~d} v+w_{3} \mathrm{dw}\) on \(\Omega^{3}\) evaluate \(\varphi^{\circ} \omega\). For any function \(f: \mathbb{R}^{3} \rightarrow \mathbb{R}\) verify Theorem 16.2, that \(\mathrm{d}\left(\varphi^{*} f\right)=\varphi^{*} \mathrm{~d} f\).
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