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\subsection{Volume, Area, Length}%
\label{sec:volume_area_length}

Given a $C^1$-diffeomorphism $φ: Ω → ℝ^d$
% Bild von Torus in deformierter Torus mit Seitenfläche
we want to compute volume and area of the image after deformation by $φ$.

\begin{definition}[right Cauchy-Green strain tensor] \label{def:right_cauchy_green_strain_tensor}
  (strain in german \enquote{Verzerrung})
  Let $Ω ⊂ ℝ^d$ be open, $φ : Ω → ℝ^d$ a $C^1$-diffeomorphism.
  The map $Ω \ni x ↦ Dφ(x)^t Dφ(x)$ is called the \define{right Cauchy-Green strain tensor}.
\end{definition} % end definition of right Cauchy-Green strain tensor
We recall the length of a curve $γ:[0,1] → ℝ^d$,
\begin{equation*}
  L(γ) := ∫_{0}^1 \abs{γ'(t)} \d t = ∫_0^1 √{γ'(t) · γ'(t)} \d t
\end{equation*}

\begin{lemma}[Length of a deformed curve] \label{thm:length_of_a_deformed_curve}
  Let $Ω ⊂ ℝ^d$ open,
  $φ: Ω → ℝ^d$ a $C^1$-diffeomorphism. Let $γ:[0,1] → Ω$ a $C^1$-curve.
  Then $φ \circ γ : [0,1] → ℝ^d$,
  $s ↦ φ(γ(s))$ is a $C^1$-curve and
  \begin{equation*}
    L(φ \circ γ) = ∫_0^1 √{γ'(t) · C(γ(t)) γ'(t)} \d t
  \end{equation*}
\end{lemma} % end lemma Length of a deformed curve

\begin{proof}
  Apply the chain rule and use the definition of $C$.
\end{proof}

\begin{theorem}[Volume of a deformed volume] \label{thm:volume_of_a_deformed_volume}
  Let $Ω ⊂ ℝ^d$ be open, $φ: Ω → ℝ^d$ a $C^1$-diffeomorphism.
  Then $f ∈ L^1(φ(Ω)) ⇔ f \circ φ √{\det C} = f \circ φ \abs{\det Dφ} ∈ L^1(Ω)$
\end{theorem} % end theorem Volume of a deformed volume
\begin{proof}
  See your favorite Analysis textbook.
  \begin{align*}
    ∫_{φ(Ω)} f (y) \d y =
    ∫_{Ω} f(φ(x)) \abs{\det Dφ(x)} \d x = 
    ∫_{Ω} f(φ(x)) √{C(x)} \d x 
  \end{align*}
\end{proof}

\begin{example}[Affine case] \label{ex:affine_case}
  If $φ(x) = Ax + c$ with $A ∈ ℝ^{d × d}$ and $c ∈ ℝ^d$, $A ∈ \GL(d)$.
  Then $D φ = A$, $C = A^tA$, $√ {\det C} = \abs{\det(A)}$.
  In the case of symmetric, positive definite matrix,
  it can be written as a diagonal matrix with eigenvalue entries.
  So $√(\det C)$ is the product of the square roots

  We call the cases
  \begin{align*}
    A &= λ_i (e_i \tens e_i) & \text{(dilation)} & √{\det C} = \abs{λ_1} … \abs{λ_d} \\
    % totikz: Bild: Kreis zu senkrechtem Oval: λ_2 < 1, λ_1 > 1
    A &∈ O(n) & \text{(rotation/ reflection)} & √{\det C} = 1\\
    % totikz: do: Bild: Verdrehtes lustiges Bild
    A &= \Id + λ(e_1 \tens e_2) & \text{(shear)} & √{\det C} = 1
    % totikz: Bild: Quadrat wird schief
  \end{align*}
\end{example} % end example Affine case

\begin{definition}[domain] \label{def:domain}
  \begin{enumerate}[label=\alph*)]
    \item We call $Ω ⊂ ℝ^d$ a \define{domain},
      if it is open, bounded and connected.
    \item A $C^1$-domain is a domain $Ω ⊂ ℝ^d$ such that
      $\rand Ω$ is a $(d-1)$-dimensional, orientable
      $C^1$-manifold and locally on one side of $Ω$.
      That means
      \begin{align*}
        ∀ x ∈ \rand Ω \ ∃ U ⊂ ℝ^d \text{ open},\,
        x &∈ U \and ∃f ∈ C_1(U) \text{ with } \\
        \nabla f &\neq 0 ∈ U \\
        \rand Ω ∩ U &= \set {f = 0} \\
        Ω ∩ U &= \set{f < 0} \\
              &\text{(Implicit characterisation of a manifold)}
      \end{align*}
    \item Let $Ω ⊂ ℝ^d$ be a domain with $C^1$-boundary.
      Let $x ∈ \rand Ω$.
      A vector $v ∈ ℝ^d$ is called a \define{tangent vector} to
      $\rand Ω$ at $x$,
      if there exists a $C^1$-curve
      $γ:(-1, 1) → ℝ^d$ with
      $γ((-1, 1)) ⊂ \rand Ω$, $γ(0) = x$, $γ'(0) = v$.
      $T_x(\rand Ω) = \set{v : v \text{ is tangent vecoter at x} }$
      is the \define{tangent space at $x$}.
    \item A vector $ν ∈ ℝ^d$ with $\abs{ν} = 1$ is called
      a \define{unit normal} to $\rand Ω$ at $x$,
      if $v · ν = 0$ for all $v ∈ T_x(\rand Ω)$
      and is called \define{outer unit normal} if
      $x + t ν \notin Ω$ for all $t > 0$ sufficiently small.
    \item A pair $(ϕ, V)$ is called a parametrization
      for $\rand Ω$ at $x$, if $V ⊂ ℝ^{d-1}$ open,
      $ϕ ∈ C^1(V ; ℝ^d)$ and
      \begin{itemize}
        \item $ϕ(v) ⊂ \rand Ω$
        \item $ϕ:V → ϕ(V)$ is a bijection with continous inverse
        \item $\rank(Dϕ) = d-1$.
          % todo: bild
      \end{itemize}
  \end{enumerate}
\end{definition} % end definition of [domain
\begin{lemma}\label{thm:local_parametrization}
  Let $Ω ⊂ ℝ^d$ be a $C^1$-domain,
  $x ∈ \rand Ω$. Then there there exists a parametrization
  $(ϕ, V)$ for $\rand Ω$ at $x$.
\end{lemma} % end lemma
\begin{proof}
  Use the Implicit function theorem.
\end{proof}
\begin{definition}[Surface integral] \label{def:surface_integral}
  \begin{align*}
    ∫_{\rand Ω} f(x) \d S(x) = ∫_{ϕ(V)} f(x) \d S(x)
    = ∫_V f(ϕ(x)) √{\det(Dφ^t Dφ} \d x
  \end{align*}
  where we assume $\supp f ⊂ ϕ(V)$.
  For $f$ having bigger support, use a partition of unity.

  Alternatively you can cover a surface by balls and
  reduce the radius of the balls towards $0$.
  This gives the \define{Hausdorff measure} $\mathcal H^{d-1}$.
  In all \enquote{nice} cases these definitions
  are equivalent.
\end{definition} % end definition of Surface integral
