The Relationship between the Fundamental Form and Coordinate Transformation
📂Geometry The Relationship between the Fundamental Form and Coordinate Transformation Overview Given the coordinate transformation f : V → U f : V \to U f : V → U g g g U U U g ‾ \overline{g} g V V V
Einstein notation is used.
For the metric g g g coordinate patch mapping x : U → R 3 \mathbf{x} : U \to \mathbb{R}^{3} x : U → R 3 g ‾ \overline{g} g y = x ∘ f : V → R 3 \mathbf{y} = \mathbf{x} \circ f : V \to \mathbb{R}^{3} y = x ∘ f : V → R 3 tangent vector X = X i x i = X ‾ α y α \mathbf{X} = X^{i}\mathbf{x}_{i} = \overline{X}^{\alpha} \mathbf{y}_{\alpha} X = X i x i = X α y α
X i = ∑ α X ‾ α ∂ u i ∂ v α g i j = ∑ α , β g ‾ α β ∂ v α ∂ u i ∂ v β ∂ u j g = g ‾ ( det [ ∂ v α ∂ u i ] ) 2 g k l = ∑ γ , δ g ‾ γ δ ∂ u k ∂ v γ ∂ u l ∂ u δ
\begin{align}
X^{i} &= \sum\limits_{\alpha} \overline{X}^{\alpha} \dfrac{\partial u^{i}}{\partial v^{\alpha}} \\
g_{ij} &= \sum\limits_{\alpha, \beta} \overline{g}_{\alpha \beta}\dfrac{\partial v^{\alpha}}{\partial u^{i}} \dfrac{\partial v^{\beta}}{\partial u^{j}} \\
g &= \overline{g} \left( \det \begin{bmatrix} \dfrac{\partial v^{\alpha}}{\partial u^{i}} \end{bmatrix} \right)^{2} \\
g^{kl} &= \sum\limits_{\gamma, \delta} \overline{g}^{\gamma \delta}\dfrac{\partial u^{k}}{\partial v^{\gamma}} \dfrac{\partial u^{l}}{\partial u^{\delta}} \\
\end{align}
X i g ij g g k l = α ∑ X α ∂ v α ∂ u i = α , β ∑ g α β ∂ u i ∂ v α ∂ u j ∂ v β = g ( det [ ∂ u i ∂ v α ] ) 2 = γ , δ ∑ g γ δ ∂ v γ ∂ u k ∂ u δ ∂ u l
X ‾ α = ∑ i X i ∂ v α ∂ u i g ‾ α β = ∑ i , j g i j ∂ u i ∂ v α ∂ u j ∂ v j g ‾ = g ( det [ ∂ u i ∂ v α ] ) 2 g ‾ γ δ = ∑ k , l g k l ∂ v γ ∂ u k ∂ v δ ∂ u l
\begin{align}
\overline{X}^{\alpha} &= \sum\limits_{i} X^{i} \dfrac{\partial v^{\alpha}}{\partial u^{i}} \\
\overline{g}_{\alpha \beta} &= \sum\limits_{i, j} g_{i j}\dfrac{\partial u^{i}}{\partial v^{\alpha}} \dfrac{\partial u^{j}}{\partial v^{j}} \\
\overline{g} &= g \left( \det
\begin{bmatrix}
\dfrac{\partial u^{i}}{\partial v^{\alpha}}
\end{bmatrix} \right)^{2} \\
\overline{g}^{\gamma \delta} &= \sum\limits_{k, l} g^{kl} \dfrac{\partial v^{\gamma}}{\partial u^{k}} \dfrac{\partial v^{\delta}}{\partial u^{l}}
\end{align}
X α g α β g g γ δ = i ∑ X i ∂ u i ∂ v α = i , j ∑ g ij ∂ v α ∂ u i ∂ v j ∂ u j = g ( det [ ∂ v α ∂ u i ] ) 2 = k , l ∑ g k l ∂ u k ∂ v γ ∂ u l ∂ v δ
Explanation ( 1 ) ( 4 ) (1) ~ (4) ( 1 ) ( 4 ) U U U V V V ( 1 ) , ( 4 ) (1), (4) ( 1 ) , ( 4 ) Jacobian of f : V → U f : V \to U f : V → U ( 2 ) , ( 3 ) (2), (3) ( 2 ) , ( 3 ) g = f − 1 = U → V g = f^{-1} = U \to V g = f − 1 = U → V ( 1 ) , ( 4 ) (1), (4) ( 1 ) , ( 4 ) contravariant transformations . When J J J Jacobian of f f f
[ X 1 X 2 ] = [ ∂ u 1 ∂ v 1 ∂ u 1 ∂ v 2 ∂ u 2 ∂ v 1 ∂ u 2 ∂ v 2 ] [ X ‾ 1 X ‾ 2 ] = J [ X ‾ 1 X ‾ 2 ]
\begin{equation}
\begin{bmatrix}
X^{1} \\ X^{2}
\end{bmatrix}
= \begin{bmatrix}
\dfrac{\partial u^{1}}{\partial v^{1}} & \dfrac{\partial u^{1}}{\partial v^{2}} \\[1em]
\dfrac{\partial u^{2}}{\partial v^{1}} & \dfrac{\partial u^{2}}{\partial v^{2}}
\end{bmatrix}
\begin{bmatrix}
\overline{X}^{1} \\ \overline{X}^{2}
\end{bmatrix}
= J \begin{bmatrix}
\overline{X}^{1} \\ \overline{X}^{2}
\end{bmatrix}
\tag{1}
\end{equation}
[ X 1 X 2 ] = ∂ v 1 ∂ u 1 ∂ v 1 ∂ u 2 ∂ v 2 ∂ u 1 ∂ v 2 ∂ u 2 [ X 1 X 2 ] = J [ X 1 X 2 ] ( 1 )
[ g 11 g 12 g 21 g 22 ] = [ ∂ u 1 ∂ v 1 ∂ u 1 ∂ v 2 ∂ u 2 ∂ v 1 ∂ u 2 ∂ v 2 ] [ g ‾ 11 g ‾ 12 g ‾ 21 g ‾ 22 ] [ ∂ u 1 ∂ v 1 ∂ u 2 ∂ v 1 ∂ u 1 ∂ v 2 ∂ u 2 ∂ v 2 ] = J [ g ‾ 11 g ‾ 12 g ‾ 21 g ‾ 22 ] J t
\begin{equation}
\begin{align*}
\begin{bmatrix}
g^{11} & g^{12} \\[1em] g^{21} & g^{22}
\end{bmatrix}
&= \begin{bmatrix}
\dfrac{\partial u^{1}}{\partial v^{1}} & \dfrac{\partial u^{1}}{\partial v^{2}} \\[1em]
\dfrac{\partial u^{2}}{\partial v^{1}} & \dfrac{\partial u^{2}}{\partial v^{2}}
\end{bmatrix}
\begin{bmatrix}
\overline{g}^{11} & \overline{g}^{12} \\[1em]
\overline{g}^{21} & \overline{g}^{22}
\end{bmatrix}
\begin{bmatrix}
\dfrac{\partial u^{1}}{\partial v^{1}} & \dfrac{\partial u^{2}}{\partial v^{1}} \\[1em]
\dfrac{\partial u^{1}}{\partial v^{2}} & \dfrac{\partial u^{2}}{\partial v^{2}}
\end{bmatrix} \\
&= J
\begin{bmatrix}
\overline{g}^{11} & \overline{g}^{12} \\[1em]
\overline{g}^{21} & \overline{g}^{22}
\end{bmatrix}
J^{t}
\end{align*}
\tag{4}
\end{equation}
g 11 g 21 g 12 g 22 = ∂ v 1 ∂ u 1 ∂ v 1 ∂ u 2 ∂ v 2 ∂ u 1 ∂ v 2 ∂ u 2 g 11 g 21 g 12 g 22 ∂ v 1 ∂ u 1 ∂ v 2 ∂ u 1 ∂ v 1 ∂ u 2 ∂ v 2 ∂ u 2 = J g 11 g 21 g 12 g 22 J t ( 4 )
Transformations like ( 2 ) (2) ( 2 ) covariant transformations .
[ g 11 g 12 g 21 g 22 ] = [ ∂ v 1 ∂ u 1 ∂ v 2 ∂ u 1 ∂ v 1 ∂ u 2 ∂ v 2 ∂ u 2 ] [ g ‾ 11 g ‾ 12 g ‾ 21 g ‾ 22 ] [ ∂ v 1 ∂ u 1 ∂ v 1 ∂ u 2 ∂ v 2 ∂ u 1 ∂ v 2 ∂ u 2 ] = ( J − 1 ) t [ g ‾ 11 g ‾ 12 g ‾ 21 g ‾ 22 ] J − 1
\begin{equation}
\begin{align*}
\begin{bmatrix}
g_{11} & g_{12} \\[1em] g_{21} & g_{22}
\end{bmatrix}
&= \begin{bmatrix}
\dfrac{\partial v^{1}}{\partial u^{1}} & \dfrac{\partial v^{2}}{\partial u^{1}} \\[1em]
\dfrac{\partial v^{1}}{\partial u^{2}} & \dfrac{\partial v^{2}}{\partial u^{2}}
\end{bmatrix}
\begin{bmatrix}
\overline{g}_{11} & \overline{g}_{12} \\[1em]
\overline{g}_{21} & \overline{g}_{22}
\end{bmatrix}
\begin{bmatrix}
\dfrac{\partial v^{1}}{\partial u^{1}} & \dfrac{\partial v^{1}}{\partial u^{2}} \\[1em]
\dfrac{\partial v^{2}}{\partial u^{1}} & \dfrac{\partial v^{2}}{\partial u^{2}}
\end{bmatrix} \\
&= (J^{-1})^{t}
\begin{bmatrix}
\overline{g}_{11} & \overline{g}_{12} \\[1em]
\overline{g}_{21} & \overline{g}_{22}
\end{bmatrix}
J^{-1}
\end{align*}
\tag{2}
\end{equation}
g 11 g 21 g 12 g 22 = ∂ u 1 ∂ v 1 ∂ u 2 ∂ v 1 ∂ u 1 ∂ v 2 ∂ u 2 ∂ v 2 g 11 g 21 g 12 g 22 ∂ u 1 ∂ v 1 ∂ u 1 ∂ v 2 ∂ u 2 ∂ v 1 ∂ u 2 ∂ v 2 = ( J − 1 ) t g 11 g 21 g 12 g 22 J − 1 ( 2 )
( 3 ) (3) ( 3 ) not a transformation because it does not include matrix multiplication. If ( 5 ) , ( 6 ) , ( 8 ) (5), (6), (8) ( 5 ) , ( 6 ) , ( 8 )
[ X ‾ 1 X ‾ 2 ] = [ ∂ v 1 ∂ u 1 ∂ v 1 ∂ u 2 ∂ v 2 ∂ u 1 ∂ v 2 ∂ u 2 ] [ X ‾ 1 X ‾ 2 ] = J − 1 [ X ‾ 1 X ‾ 2 ]
\begin{equation}
\begin{bmatrix}
\overline{X}^{1} \\ \overline{X}^{2}
\end{bmatrix}
= \begin{bmatrix}
\dfrac{\partial v^{1}}{\partial u^{1}} & \dfrac{\partial v^{1}}{\partial u^{2}} \\[1em]
\dfrac{\partial v^{2}}{\partial u^{1}} & \dfrac{\partial v^{2}}{\partial u^{2}}
\end{bmatrix}
\begin{bmatrix}
\overline{X}^{1} \\ \overline{X}^{2}
\end{bmatrix}
= J^{-1} \begin{bmatrix}
\overline{X}^{1} \\ \overline{X}^{2}
\end{bmatrix}
\tag{5}
\end{equation}
[ X 1 X 2 ] = ∂ u 1 ∂ v 1 ∂ u 1 ∂ v 2 ∂ u 2 ∂ v 1 ∂ u 2 ∂ v 2 [ X 1 X 2 ] = J − 1 [ X 1 X 2 ] ( 5 )
[ g ‾ 11 g ‾ 12 g ‾ 21 g ‾ 22 ] = [ ∂ u 1 ∂ v 1 ∂ u 2 ∂ v 1 ∂ u 1 ∂ v 2 ∂ u 2 ∂ v 2 ] [ g 11 g 12 g 21 g 22 ] [ ∂ u 1 ∂ v 1 ∂ u 1 ∂ v 2 ∂ u 2 ∂ v 1 ∂ u 2 ∂ v 2 ] = J t [ g 11 g 12 g 21 g 22 ] J
\begin{equation}
\begin{align*}
\begin{bmatrix}
\overline{g}_{11} & \overline{g}_{12} \\[1em]
\overline{g}_{21} & \overline{g}_{22}
\end{bmatrix}
&= \begin{bmatrix}
\dfrac{\partial u^{1}}{\partial v^{1}} & \dfrac{\partial u^{2}}{\partial v^{1}} \\[1em]
\dfrac{\partial u^{1}}{\partial v^{2}} & \dfrac{\partial u^{2}}{\partial v^{2}}
\end{bmatrix}
\begin{bmatrix}
g_{11} & g_{12} \\[1em] g_{21} & g_{22}
\end{bmatrix}
\begin{bmatrix}
\dfrac{\partial u^{1}}{\partial v^{1}} & \dfrac{\partial u^{1}}{\partial v^{2}} \\[1em]
\dfrac{\partial u^{2}}{\partial v^{1}} & \dfrac{\partial u^{2}}{\partial v^{2}}
\end{bmatrix} \\
&= J^{t}
\begin{bmatrix}
g_{11} & g_{12} \\[1em] g_{21} & g_{22}
\end{bmatrix}
J
\end{align*}
\tag{6}
\end{equation}
g 11 g 21 g 12 g 22 = ∂ v 1 ∂ u 1 ∂ v 2 ∂ u 1 ∂ v 1 ∂ u 2 ∂ v 2 ∂ u 2 g 11 g 21 g 12 g 22 ∂ v 1 ∂ u 1 ∂ v 1 ∂ u 2 ∂ v 2 ∂ u 1 ∂ v 2 ∂ u 2 = J t g 11 g 21 g 12 g 22 J ( 6 )
[ g ‾ 11 g ‾ 12 g ‾ 21 g ‾ 22 ] = [ ∂ v 1 ∂ u 1 ∂ v 1 ∂ u 2 ∂ v 2 ∂ u 1 ∂ v 2 ∂ u 2 ] [ g 11 g 12 g 21 g 22 ] [ ∂ v 1 ∂ u 1 ∂ v 2 ∂ u 1 ∂ v 1 ∂ u 2 ∂ v 2 ∂ u 2 ] = J − 1 [ g 11 g 12 g 21 g 22 ] ( J − 1 ) t
\begin{equation}
\begin{align*}
\begin{bmatrix}
\overline{g}^{11} & \overline{g}^{12} \\[1em]
\overline{g}^{21} & \overline{g}^{22}
\end{bmatrix}
&= \begin{bmatrix}
\dfrac{\partial v^{1}}{\partial u^{1}} & \dfrac{\partial v^{1}}{\partial u^{2}} \\[1em]
\dfrac{\partial v^{2}}{\partial u^{1}} & \dfrac{\partial v^{2}}{\partial u^{2}}
\end{bmatrix}
\begin{bmatrix}
g^{11} & g^{12} \\[1em] g^{21} & g^{22}
\end{bmatrix}
\begin{bmatrix}
\dfrac{\partial v^{1}}{\partial u^{1}} & \dfrac{\partial v^{2}}{\partial u^{1}} \\[1em]
\dfrac{\partial v^{1}}{\partial u^{2}} & \dfrac{\partial v^{2}}{\partial u^{2}}
\end{bmatrix} \\
&= J^{-1}
\begin{bmatrix}
g^{11} & g^{12} \\[1em] g^{21} & g^{22}
\end{bmatrix}
(J^{-1})^{t}
\end{align*}
\tag{8}
\end{equation}
g 11 g 21 g 12 g 22 = ∂ u 1 ∂ v 1 ∂ u 1 ∂ v 2 ∂ u 2 ∂ v 1 ∂ u 2 ∂ v 2 g 11 g 21 g 12 g 22 ∂ u 1 ∂ v 1 ∂ u 2 ∂ v 1 ∂ u 1 ∂ v 2 ∂ u 2 ∂ v 2 = J − 1 g 11 g 21 g 12 g 22 ( J − 1 ) t ( 8 )
Derivation Let it be said as follows.
g i j = ⟨ x i , x j ⟩
g_{ij} = \left\langle \mathbf{x}_{i}, \mathbf{x}_{j} \right\rangle
g ij = ⟨ x i , x j ⟩
Regarding coordinate patch mapping y = x ∘ f : V → R 3 \mathbf{y} = \mathbf{x} \circ f : V \to \mathbb{R}^{3} y = x ∘ f : V → R 3
y α = ∂ y ∂ v α , g ‾ α β = ⟨ y α , y β ⟩
\mathbf{y}_{\alpha} = \dfrac{\partial \mathbf{y}}{\partial v^{\alpha}},\quad \overline{g}_{\alpha \beta} = \left\langle \mathbf{y}_{\alpha}, \mathbf{y}_{\beta} \right\rangle
y α = ∂ v α ∂ y , g α β = ⟨ y α , y β ⟩
g ‾ = det [ g ‾ α β ] , [ g ‾ γ β ] = [ g ‾ α β ] − 1
\overline{g} = \det \begin{bmatrix} \overline{g}_{\alpha \beta} \end{bmatrix},\quad \begin{bmatrix} \overline{g}^{\gamma \beta} \end{bmatrix} = \begin{bmatrix} \overline{g}_{\alpha \beta} \end{bmatrix}^{-1}
g = det [ g α β ] , [ g γ β ] = [ g α β ] − 1
By the chain rule , the following is obtained.
x i = ∂ y ∂ v 1 ∂ v 1 ∂ u i + ∂ y ∂ v 2 ∂ v 2 ∂ u i = ∑ α ∂ y ∂ v α ∂ v α ∂ u i = y α ∂ v α ∂ u i
\mathbf{x}_{i} = \dfrac{\partial \mathbf{y}}{\partial v^{1}} \dfrac{\partial v^{1}}{\partial u^{i}} + \dfrac{\partial \mathbf{y}}{\partial v^{2}} \dfrac{\partial v^{2}}{\partial u^{i}} = \sum \limits_{\alpha} \dfrac{\partial \mathbf{y}}{\partial v^{\alpha}} \dfrac{\partial v^{\alpha}}{\partial u^{i}} = \mathbf{y}_{\alpha} \dfrac{\partial v^{\alpha}}{\partial u^{i}}
x i = ∂ v 1 ∂ y ∂ u i ∂ v 1 + ∂ v 2 ∂ y ∂ u i ∂ v 2 = α ∑ ∂ v α ∂ y ∂ u i ∂ v α = y α ∂ u i ∂ v α
y α = ∂ x ∂ u 1 ∂ u 1 ∂ v α + ∂ x ∂ u 2 ∂ u 2 ∂ v α = ∑ i ∂ x ∂ u i ∂ u i ∂ v α = x i ∂ u i ∂ v α
\mathbf{y}_{\alpha} = \dfrac{\partial \mathbf{x}}{\partial u^{1}} \dfrac{\partial u^{1}}{\partial v^{\alpha}} + \dfrac{\partial \mathbf{x}}{\partial u^{2}} \dfrac{\partial u^{2}}{\partial v^{\alpha}} = \sum \limits_{i} \dfrac{\partial \mathbf{x}}{\partial u^{i}} \dfrac{\partial u^{i}}{\partial v^{\alpha}} = \mathbf{x}_{i} \dfrac{\partial u^{i}}{\partial v^{\alpha}}
y α = ∂ u 1 ∂ x ∂ v α ∂ u 1 + ∂ u 2 ∂ x ∂ v α ∂ u 2 = i ∑ ∂ u i ∂ x ∂ v α ∂ u i = x i ∂ v α ∂ u i
Therefore, g i j g_{ij} g ij
g i j = ⟨ x i , x j ⟩ = ⟨ y α , y β ⟩ ∂ v α ∂ u i ∂ v β ∂ u j = g ‾ α β ∂ v α ∂ u i ∂ v β ∂ u j
g_{ij} = \left\langle \mathbf{x}_{i}, \mathbf{x}_{j} \right\rangle = \left\langle \mathbf{y}_{\alpha}, \mathbf{y}_{\beta} \right\rangle \dfrac{\partial v^{\alpha}}{\partial u^{i}} \dfrac{\partial v^{\beta}}{\partial u^{j}} = \overline{g}_{\alpha \beta}\dfrac{\partial v^{\alpha}}{\partial u^{i}} \dfrac{\partial v^{\beta}}{\partial u^{j}}
g ij = ⟨ x i , x j ⟩ = ⟨ y α , y β ⟩ ∂ u i ∂ v α ∂ u j ∂ v β = g α β ∂ u i ∂ v α ∂ u j ∂ v β
When represented as matrix multiplication, it looks like below.
[ g 11 g 12 g 21 g 22 ] = [ ∂ v 1 ∂ u 1 ∂ v 2 ∂ u 1 ∂ v 1 ∂ u 2 ∂ v 2 ∂ u 2 ] [ g ‾ 11 g ‾ 12 g ‾ 21 g ‾ 22 ] [ ∂ v 1 ∂ u 1 ∂ v 1 ∂ u 2 ∂ v 2 ∂ u 1 ∂ v 2 ∂ u 2 ]
\begin{bmatrix}
g_{11} & g_{12} \\[1em] g_{21} & g_{22}
\end{bmatrix}
= \begin{bmatrix}
\dfrac{\partial v^{1}}{\partial u^{1}} & \dfrac{\partial v^{2}}{\partial u^{1}} \\[1em]
\dfrac{\partial v^{1}}{\partial u^{2}} & \dfrac{\partial v^{2}}{\partial u^{2}}
\end{bmatrix}
\begin{bmatrix}
\overline{g}_{11} & \overline{g}_{12} \\[1em]
\overline{g}_{21} & \overline{g}_{22}
\end{bmatrix}
\begin{bmatrix}
\dfrac{\partial v^{1}}{\partial u^{1}} & \dfrac{\partial v^{1}}{\partial u^{2}} \\[1em]
\dfrac{\partial v^{2}}{\partial u^{1}} & \dfrac{\partial v^{2}}{\partial u^{2}}
\end{bmatrix}
g 11 g 21 g 12 g 22 = ∂ u 1 ∂ v 1 ∂ u 2 ∂ v 1 ∂ u 1 ∂ v 2 ∂ u 2 ∂ v 2 g 11 g 21 g 12 g 22 ∂ u 1 ∂ v 1 ∂ u 1 ∂ v 2 ∂ u 2 ∂ v 1 ∂ u 2 ∂ v 2
Assuming J J J Jacobian of f : V → U f : V \to U f : V → U
J = [ ∂ u 1 ∂ v 1 ∂ u 1 ∂ v 2 ∂ u 2 ∂ v 1 ∂ u 2 ∂ v 2 ] and J − 1 = [ ∂ v 1 ∂ u 1 ∂ v 1 ∂ u 2 ∂ v 2 ∂ u 1 ∂ v 2 ∂ u 2 ]
J = \begin{bmatrix}
\dfrac{\partial u^{1}}{\partial v^{1}} & \dfrac{\partial u^{1}}{\partial v^{2}} \\[1em]
\dfrac{\partial u^{2}}{\partial v^{1}} & \dfrac{\partial u^{2}}{\partial v^{2}}
\end{bmatrix}
\quad \text{and} \quad
J^{-1} = \begin{bmatrix}
\dfrac{\partial v^{1}}{\partial u^{1}} & \dfrac{\partial v^{1}}{\partial u^{2}} \\[1em]
\dfrac{\partial v^{2}}{\partial u^{1}} & \dfrac{\partial v^{2}}{\partial u^{2}}
\end{bmatrix}
J = ∂ v 1 ∂ u 1 ∂ v 1 ∂ u 2 ∂ v 2 ∂ u 1 ∂ v 2 ∂ u 2 and J − 1 = ∂ u 1 ∂ v 1 ∂ u 1 ∂ v 2 ∂ u 2 ∂ v 1 ∂ u 2 ∂ v 2
Then,
[ g i j ] = ( J − 1 ) t [ g ‾ α β ] J − 1
\begin{bmatrix}
g_{ij}
\end{bmatrix}
= (J^{-1})^{t}
\begin{bmatrix}
\overline{g}_{\alpha \beta}
\end{bmatrix}
J^{-1}
[ g ij ] = ( J − 1 ) t [ g α β ] J − 1
g = det [ g i j ] = det ( ( J − 1 ) t [ g ‾ α β ] J − 1 ) = det ( J − 1 ) t det [ g ‾ α β ] det J − 1 = det [ g ‾ α β ] ( det J − 1 ) 2 = g ‾ ( det [ ∂ v α ∂ u i ] ) 2
\begin{align*}
g = \det \begin{bmatrix}
g_{ij}
\end{bmatrix}
= \det \Big( (J^{-1})^{t}
\begin{bmatrix}
\overline{g}_{\alpha \beta}
\end{bmatrix}
J^{-1} \Big) &= \det (J^{-1})^{t}
\det \begin{bmatrix}
\overline{g}_{\alpha \beta}
\end{bmatrix}
\det J^{-1} \\
&= \det \begin{bmatrix}
\overline{g}_{\alpha \beta}
\end{bmatrix} (\det J^{-1})^{2} \\
&= \overline{g} \left( \det
\begin{bmatrix}
\dfrac{\partial v^{\alpha}}{\partial u^{i}}
\end{bmatrix} \right)^{2}
\end{align*}
g = det [ g ij ] = det ( ( J − 1 ) t [ g α β ] J − 1 ) = det ( J − 1 ) t det [ g α β ] det J − 1 = det [ g α β ] ( det J − 1 ) 2 = g ( det [ ∂ u i ∂ v α ] ) 2
Also, the inverse matrix is,
[ g k l ] = [ g i j ] − 1 = ( ( J − 1 ) t [ g ‾ α β ] J − 1 ) − 1 = J [ g ‾ α β ] − 1 J t = J [ g ‾ γ δ ] J t
\begin{bmatrix}
g^{kl}
\end{bmatrix}
= \begin{bmatrix}
g_{ij}
\end{bmatrix}^{-1}
= \Big( (J^{-1})^{t}
\begin{bmatrix}
\overline{g}_{\alpha \beta}
\end{bmatrix}
J^{-1} \Big)^{-1}
= J \begin{bmatrix}
\overline{g}_{\alpha \beta}
\end{bmatrix}^{-1} J^{t}
= J \begin{bmatrix}
\overline{g}^{\gamma \delta}
\end{bmatrix} J^{t}
[ g k l ] = [ g ij ] − 1 = ( ( J − 1 ) t [ g α β ] J − 1 ) − 1 = J [ g α β ] − 1 J t = J [ g γ δ ] J t
⟹ g k l = g ‾ γ δ ∂ u k ∂ v γ ∂ u l ∂ u δ
\implies g^{kl} = \overline{g}^{\gamma \delta}\dfrac{\partial u^{k}}{\partial v^{\gamma}} \dfrac{\partial u^{l}}{\partial u^{\delta}}
⟹ g k l = g γ δ ∂ v γ ∂ u k ∂ u δ ∂ u l
If the tangent vector is considered as X = X i x i = X ‾ α y α \mathbf{X} = X^{i}\mathbf{x}_{i} = \overline{X}^{\alpha} \mathbf{y}_{\alpha} X = X i x i = X α y α
X i x i = X ‾ α y α = X ‾ α ∂ u i ∂ v α x i ⟹ X i = X ‾ α ∂ u i ∂ v α
X^{i}\mathbf{x}_{i} = \overline{X}^{\alpha} \mathbf{y}_{\alpha} = \overline{X}^{\alpha} \dfrac{\partial u^{i}}{\partial v^{\alpha}} \mathbf{x}_{i} \implies X^{i} = \overline{X}^{\alpha} \dfrac{\partial u^{i}}{\partial v^{\alpha}}
X i x i = X α y α = X α ∂ v α ∂ u i x i ⟹ X i = X α ∂ v α ∂ u i
By considering U U U V V V
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