- #1
PhysicsDude1
- 8
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Homework Statement
For the orthonormal coordinate system (X,Y) the metric is
\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
Calculate G' in 2 ways.
1) G'= M[itex]^{T}[/itex]*G*M
2) g[itex]\acute{}[/itex][itex]_{ij}[/itex] = [itex]\overline{a}\acute{}_{i}[/itex] . [itex]\overline{a}\acute{}_{j}[/itex]
Homework Equations
\begin{pmatrix} \overline{a}\acute{}_{1} \\ \overline{a}\acute{}_{2} \end{pmatrix}
= \begin{pmatrix} -cos(\phi).\overline{a}\acute{}_{1} -\overline{a}\acute{}_{2} \\
cos(\phi).\overline{a}\acute{}_{2}\end{pmatrix}
M= \begin{pmatrix} -cos(\phi) & 0 \\ -1 & cos(\phi) \end{pmatrix}
M[itex]^{T}[/itex] = \begin{pmatrix} -cos(\phi) & -1 \\ 0 & cos(\phi) \end{pmatrix}
G=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
=> G' = \begin{pmatrix} cos²(\phi) +1 & -cos(\phi) \\ -cos(\phi) & cos²(\phi) \end{pmatrix}
The Attempt at a Solution
So I'm having problems with the 2nd method i.e. g[itex]\acute{}[/itex][itex]_{ij}[/itex] = [itex]\overline{a}\acute{}_{i}[/itex] . [itex]\overline{a}\acute{}_{j}[/itex]
g[itex]\acute{}[/itex][itex]_{11}[/itex] = (-cos[itex](\phi)[/itex] . [itex]\overline{a}_{1}[/itex] -[itex]\overline{a}_{2}[/itex]) . (-cos[itex](\phi)[/itex] . [itex]\overline{a}_{1}[/itex] -[itex]\overline{a}_{2}[/itex]) = ?
What are the values for [itex]\overline{a}_{1}[/itex] and [itex]\overline{a}_{2}[/itex] ?
I think they're both 1 because they're both unit vectors of length 1 but I'm not sure.
Also, this is the first time ever I have used LaTeX so sorry if it's a bit sloppy.