201902210108 Exercise 5.3.2

Calculate all eight partial derivatives $\displaystyle{\frac{\partial x'^\mu}{\partial x^\nu}}$ and $\displaystyle{\frac{\partial x^\mu}{\partial x'^\nu}}$ .

Roughwork.

Equation (5.23):

$p(x,y)=x$ , $q(x,y)=y-cx^2$ ;

Equation (5.24):

$x(p,q)=p$ , $y(p,q)=cp^2+q$ .

Then, the eight partial derivatives are

$\displaystyle{\frac{\partial p}{\partial x}}=1$ , $\displaystyle{\frac{\partial p}{\partial y}}=0$ , $\displaystyle{\frac{\partial q}{\partial x}}=-2cx$ , $\displaystyle{\frac{\partial q}{\partial y}}=1$ ,

$\displaystyle{\frac{\partial x}{\partial p}}=1$ , $\displaystyle{\frac{\partial x}{\partial q}}=0$ , $\displaystyle{\frac{\partial y}{\partial p}}=2cp$ , $\displaystyle{\frac{\partial y}{\partial q}}=1$ .

Equation (5.10):

$\mathrm{d}s^2=g_{pq}\,\mathrm{d}p\,\mathrm{d}q$ .

Equation (5.11):

$g'_{\mu\nu}=\displaystyle{\frac{\partial x^\alpha}{\partial x'^\mu}}\displaystyle{\frac{\partial x^\beta}{\partial x'^\nu}}\, g_{\alpha\beta}$

The metric tensor for the cartesian coordinates is given by equation (5.25):

$g_{\alpha\beta}=\begin{bmatrix} 1&0\\0&1 \end{bmatrix}$

Using Equations (5.11) and (5.25):

$\begin{aligned} g_{pp} & =\displaystyle{\frac{\partial x^\alpha}{\partial p}}\displaystyle{\frac{\partial x^\beta}{\partial p}}g_{\alpha\beta} \\ & = \displaystyle{\frac{\partial x}{\partial p}}\displaystyle{\frac{\partial x}{\partial p}}g_{xx}+\displaystyle{\frac{\partial x}{\partial p}}\displaystyle{\frac{\partial y}{\partial p}}g_{xy}+\displaystyle{\frac{\partial y}{\partial p}}\displaystyle{\frac{\partial x}{\partial p}}g_{yx}+\displaystyle{\frac{\partial y}{\partial p}}\displaystyle{\frac{\partial y}{\partial p}}g_{yy}\\ & = (1)(1)(1) + (1)(2cp)(0) + (2cp)(1)(0) + (2cp)(2cp)(1) \\ &= 1+4c^2p^2 \end{aligned}$

$\begin{aligned} g_{pq} & =\displaystyle{\frac{\partial x^\alpha}{\partial p}}\displaystyle{\frac{\partial x^\beta}{\partial q}}g_{\alpha\beta} \\ & = \displaystyle{\frac{\partial x}{\partial p}}\displaystyle{\frac{\partial x}{\partial q}}g_{xx}+\displaystyle{\frac{\partial x}{\partial p}}\displaystyle{\frac{\partial y}{\partial q}}g_{xy}+\displaystyle{\frac{\partial y}{\partial p}}\displaystyle{\frac{\partial x}{\partial q}}g_{yx}+\displaystyle{\frac{\partial y}{\partial p}}\displaystyle{\frac{\partial y}{\partial q}}g_{yy}\\ & = (1)(0)(1) + (1)(1)(0) + (2cp)(0)(0) + (2cp)(1)(1) \\ &= 2cp \end{aligned}$

$\begin{aligned} g_{qp} & =\displaystyle{\frac{\partial x^\alpha}{\partial q}}\displaystyle{\frac{\partial x^\beta}{\partial p}}g_{\alpha\beta} \\ & = \displaystyle{\frac{\partial x}{\partial q}}\displaystyle{\frac{\partial x}{\partial p}}g_{xx}+\displaystyle{\frac{\partial x}{\partial q}}\displaystyle{\frac{\partial y}{\partial p}}g_{xy}+\displaystyle{\frac{\partial y}{\partial q}}\displaystyle{\frac{\partial x}{\partial p}}g_{yx}+\displaystyle{\frac{\partial y}{\partial q}}\displaystyle{\frac{\partial y}{\partial p}}g_{yy}\\ & = (0)(1)(1) + (0)(2cp)(0) + (2cp)(0)(0) + (1)(2cp)(1) \\ &= 2cp \end{aligned}$

$\begin{aligned} g_{qq} & =\displaystyle{\frac{\partial x^\alpha}{\partial q}}\displaystyle{\frac{\partial x^\beta}{\partial q}}g_{\alpha\beta} \\ & = \displaystyle{\frac{\partial x}{\partial q}}\displaystyle{\frac{\partial x}{\partial q}}g_{xx}+\displaystyle{\frac{\partial x}{\partial q}}\displaystyle{\frac{\partial y}{\partial q}}g_{xy}+\displaystyle{\frac{\partial y}{\partial q}}\displaystyle{\frac{\partial x}{\partial q}}g_{yx}+\displaystyle{\frac{\partial y}{\partial q}}\displaystyle{\frac{\partial y}{\partial q}}g_{yy}\\ & = (0)(0)(1) + (0)(1)(0) + (1)(0)(0) + (1)(1)(1) \\ &= 1 \end{aligned}$

The metric tensor for $p$ – $q$ coordinate system is given by Equation (5.26):

$g'_{\mu\nu}=\begin{bmatrix} 1+4c^2p^2 & 2cp \\ 2cp & 1 \\ \end{bmatrix}$ .

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