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\begin{align}
  & \frac{d{{s}^{2}}{{|}_{{{{\bar{x}}}^{0}}+\delta {{{\bar{x}}}^{0}}=\text{const}}}}{d{{s}^{2}}{{|}_{{{{\bar{x}}}^{0}}=\text{const}}}}=1+\delta {{{\bar{x}}}^{0}}h\left( {{{\bar{x}}}^{0}} \right)+O\left( {{\left( \delta {{{\bar{x}}}^{0}} \right)}^{2}} \right) \\
 & \Rightarrow {{g}_{ik}}\left( {{{\bar{x}}}^{0}},{{x}^{k}} \right)={{S}^{2}}\left( {{{\bar{x}}}^{0}} \right){{\gamma }_{ij}}\left( {{x}^{k}} \right) \\
 & \Rightarrow d{{s}^{2}}={{\left( d{{{\bar{x}}}^{0}} \right)}^{2}}+{{S}^{2}}\left( {{{\bar{x}}}^{0}} \right){{\gamma }_{ij}}\left( {{x}^{k}} \right) \\
\end{align}

TeX (checked): 

{\begin{aligned}&{\frac {d{{s}^{2}}{{|}_{{{\bar {x}}^{0}}+\delta {{\bar {x}}^{0}}={\text{const}}}}}{d{{s}^{2}}{{|}_{{{\bar {x}}^{0}}={\text{const}}}}}}=1+\delta {{\bar {x}}^{0}}h\left({{\bar {x}}^{0}}\right)+O\left({{\left(\delta {{\bar {x}}^{0}}\right)}^{2}}\right)\\&\Rightarrow {{g}_{ik}}\left({{\bar {x}}^{0}},{{x}^{k}}\right)={{S}^{2}}\left({{\bar {x}}^{0}}\right){{\gamma }_{ij}}\left({{x}^{k}}\right)\\&\Rightarrow d{{s}^{2}}={{\left(d{{\bar {x}}^{0}}\right)}^{2}}+{{S}^{2}}\left({{\bar {x}}^{0}}\right){{\gamma }_{ij}}\left({{x}^{k}}\right)\\\end{aligned}}

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ds2|x¯0+δx¯0=constds2|x¯0=const=1+δx¯0h(x¯0)+O((δx¯0)2)⇒gik(x¯0,xk)=S2(x¯0)γij(xk)⇒ds2=(dx¯0)2+S2(x¯0)γij(xk)
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data-mjx-texclass="ORD"><mover><mi>x</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msup><mo>,</mo><msup><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><msup><mi>S</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>x</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><msub><mi>&#x03B3;</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>&#x21D2;</mo><mi>d</mi><msup><mi>s</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>d</mi><msup><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>x</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo>+</mo><msup><mi>S</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>x</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><msub><mi>&#x03B3;</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>

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Identifiers

  • d
  • s
  • x¯
  • δ
  • x¯
  • d
  • s
  • x¯
  • δ
  • x¯
  • h
  • x¯
  • O
  • δ
  • x¯
  • gik
  • x¯
  • x
  • k
  • S
  • x¯
  • γij
  • x
  • k
  • d
  • s
  • d
  • x¯
  • S
  • x¯
  • γij
  • x
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