Emulsions are locally embedded.
📂Geometry Emulsions are locally embedded. 3173) An immersion is locally embedded. Theorem Proof Assuming that ϕ \phi ϕ d ϕ p d\phi{p} d ϕ p ϕ ∣ V \phi|_{V} ϕ ∣ V ( ϕ ∣ V ) − 1 (\phi|_{V})^{-1} ( ϕ ∣ V ) − 1 R m \mathbb{R}^{m} R m [ ∂ y 1 ∂ x 1 ∂ y 1 ∂ x 2 … ∂ y 1 ∂ x n ∂ y 2 ∂ x 1 ∂ y 2 ∂ x 2 … ∂ y 2 ∂ x n ⋮ ⋮ ⋱ ⋮ ∂ y m ∂ x 1 ∂ y m ∂ x 2 … ∂ y m ∂ x n ]
\begin{bmatrix}
\dfrac{\partial y_{1}}{\partial x_{1}} & \dfrac{\partial y_{1}}{\partial x_{2}} & \dots & \dfrac{\partial y_{1}}{\partial x_{n}}
\\[1em] \dfrac{\partial y_{2}}{\partial x_{1}} & \dfrac{\partial y_{2}}{\partial x_{2}} & \dots & \dfrac{\partial y_{2}}{\partial x_{n}}
\\[1ex] \vdots & \vdots & \ddots & \vdots
\\[1ex] \dfrac{\partial y_{m}}{\partial x_{1}} & \dfrac{\partial y_{m}}{\partial x_{2}} & \dots & \dfrac{\partial y_{m}}{\partial x_{n}}
\end{bmatrix}
∂ x 1 ∂ y 1 ∂ x 1 ∂ y 2 ⋮ ∂ x 1 ∂ y m ∂ x 2 ∂ y 1 ∂ x 2 ∂ y 2 ⋮ ∂ x 2 ∂ y m … … ⋱ … ∂ x n ∂ y 1 ∂ x n ∂ y 2 ⋮ ∂ x n ∂ y m
To apply the Inverse Function Theorem , consider the following mapping.
φ : ( U 1 × R m − n = k ) ⊂ R m → R m
\varphi : (U_{1} \times \mathbb{R}^{m-n=k})\subset \mathbb{R}^{m} \to \mathbb{R}^{m}
φ : ( U 1 × R m − n = k ) ⊂ R m → R m
φ ( x 1 , … , x n , t 1 , … , t k ) = ( y 1 ( x 1 , … , x n ) , … , y n ( x 1 , … , x n ) , y n + 1 ( x 1 , … , x n ) + t 1 , … , y n + k ( x 1 , … , x n ) + t k )
\begin{align*}
&\ \varphi ( x_{1}, \dots, x_{n}, t_{1}, \dots, t_{k} )
\\ =&\ \Big( y_{1}(x_{1},\dots,x_{n}), \dots, y_{n}(x_{1},\dots,x_{n}), y_{n+1}(x_{1},\dots,x_{n}) + t_{1}, \dots, y_{n+k}(x_{1},\dots,x_{n}) + t_{k} \Big)
\end{align*}
= φ ( x 1 , … , x n , t 1 , … , t k ) ( y 1 ( x 1 , … , x n ) , … , y n ( x 1 , … , x n ) , y n + 1 ( x 1 , … , x n ) + t 1 , … , y n + k ( x 1 , … , x n ) + t k )
det ( d φ q ) = ∂ ( y 1 , … , y n , y n + 1 , … , y n + k ) ∂ ( x 1 , … , x n , t 1 , … , t k ) = [ ∂ y 1 ∂ x 1 ∂ y 1 ∂ x 2 … ∂ y 1 ∂ x n ∂ y 2 ∂ x 1 ∂ y 2 ∂ x 2 … ∂ y 2 ∂ x n ⋮ ⋮ ⋱ ⋮ ∂ y m ∂ x 1 ∂ y m ∂ x 2 … ∂ y m ∂ x n ] =
\begin{align*}
\det( d \varphi _{q}) =&\ \dfrac{\partial (y_{1}, \dots, y_{n}, y_{n+1}, \dots, y_{n+k})}{\partial (x_{1}, \dots, x_{n}, t_{1}, \dots, t_{k})}
\\ =&\ \begin{bmatrix}
\dfrac{\partial y_{1}}{\partial x_{1}} & \dfrac{\partial y_{1}}{\partial x_{2}} & \dots & \dfrac{\partial y_{1}}{\partial x_{n}}
\\[1em] \dfrac{\partial y_{2}}{\partial x_{1}} & \dfrac{\partial y_{2}}{\partial x_{2}} & \dots & \dfrac{\partial y_{2}}{\partial x_{n}}
\\[1ex] \vdots & \vdots & \ddots & \vdots
\\[1ex] \dfrac{\partial y_{m}}{\partial x_{1}} & \dfrac{\partial y_{m}}{\partial x_{2}} & \dots & \dfrac{\partial y_{m}}{\partial x_{n}}
\end{bmatrix}
\\ =&\
\end{align*}
det ( d φ q ) = = = ∂ ( x 1 , … , x n , t 1 , … , t k ) ∂ ( y 1 , … , y n , y n + 1 , … , y n + k ) ∂ x 1 ∂ y 1 ∂ x 1 ∂ y 2 ⋮ ∂ x 1 ∂ y m ∂ x 2 ∂ y 1 ∂ x 2 ∂ y 2 ⋮ ∂ x 2 ∂ y m … … ⋱ … ∂ x n ∂ y 1 ∂ x n ∂ y 2 ⋮ ∂ x n ∂ y m
