Integral calculus

1. Divergence Theorem

Divergence Theorem

Set an open set \(\Omega \in \mathbf{R}^d\), the bound \(\partial \Omega = \Gamma\), \(\phi \in H^1(\Omega)\) and the outlet unit normal vector \(\mathbf{n}\) :

\[\int_{\Omega}{\nabla \cdot \phi} = \int_{\Gamma}{\phi \cdot \mathbf{n}}\]

2. Formula of Green

Formula of Green (First Form)

Set an open set \(\Omega \in \mathbb{R}^d\), \(u, \, v \in H^1(\Omega)\) and the outlet unit normal vector \(\mathbf{n} = \begin{pmatrix} n_i \end{pmatrix}_{1 \leq i \leq d}\), so, for \(0 \leq i \leq d\) :

\[ \int_{\Omega}{ \partial_{x_i} u \, v } = - \int_{\Omega}{ u \, \partial_{x_i} v } + \int_{\Gamma}{ u \, v \, n_i }\]

Formula of Green (Second Form)

Set an open set \(\Omega \in \mathbb{R}^d\), \(u \in H^2(\Omega)\), \(v \in H^1(\Omega)\) and the outlet unit normal vector \(\mathbf{n}\), so :

\[ \int_{\Omega}{ \Delta u \cdot v } = \int_{\Gamma}{ \frac{\partial u}{\partial \mathbf{n}} \cdot v } - \int_{\Omega}{ \nabla u \cdot \nabla v }\]

With : \(\frac{\partial u}{\partial \mathbf{n}} = \nabla u \cdot \mathbf{n}\)

3. Formula of variable change in integrals

Thus a set \(\Omega\) with bound \(\Gamma = \partial \Omega\), \(\mathbf{n}\) the outlet unit vector of \(\Gamma\) on \(\Omega\), \(f \in H^2(\Omega)\) and \(\mathbf{F} \in H^2(\Omega)^d\) (\(d \in \mathbb{N}^*\))

3.1. In cylindrical coordinates

Thus \(\phi\) the function of changement of variable to cylindric to cartesian and \((\mathbf{e_r}, \mathbf{e_{\theta}}, \mathbf{e_z})\).

Thus \(\Omega^{cyl} = \phi^{-1}(\Omega)\), \(\Gamma^{cyl} = \phi^{-1}(\Gamma)\), \(\mathbf{n}^{cyl} = \phi^{-1}(\mathbf{n})\), \(f^{cyl} = f \circ \phi\) and \(\mathbf{F}^{cyl} = \mathbf{F} \circ \phi\).

We have :

Formula of changement in cylindrical coordinates

Integral of function on a domain :

\[ \int_{\Omega} \ f(x,y,z) \ dxdydz\ = \int_{\Omega_{cyl}} f_{cyl}(r,\theta,z) r \ dr d\theta dz\]

Integral of jacobian of function on a domain :

\[ \int_{\Omega} \nabla f(x,y,z) \ dxdydz = \int_{\Omega_{cyl}}{ \left( \frac{\partial f_{cyl}}{\partial r} \mathbf{e_r} + \frac{1}{r} \frac{\partial f_{cyl}}{\partial \theta} \mathbf{e_{\theta}} + \frac{\partial f_{cyl}}{\partial z} \mathbf{e_z} \right) \ r \ drd\theta dz}\]

Integral of function on bound of a domain :

\[ \int_{\partial \Omega}{ f(x) dx } = \int{ \int_{T^{cyl}}{ f\begin{pmatrix} r(s,t) \\ \theta(s,t) \\ z(s,t) \end{pmatrix} } \, \left\Vert \begin{pmatrix} r cos\theta & sin \theta & 0 \\ -r sin \theta & cos \theta & 0 \\ 0 & 0 & r \end{pmatrix} \frac{\partial \begin{pmatrix} r \\ \theta \\ z \end{pmatrix}}{\partial s}(s,t) \times \frac{\partial \begin{pmatrix} r \\ \theta \\ z \end{pmatrix}}{\partial t}(s,t) \right\Vert \, ds dt }\]

With \((s,t) \in T^{cyl} \subset \mathbb{R}^2 \rightarrow \begin{pmatrix} r(s,t) \\ \theta(s,t) \\ z(s,t) \end{pmatrix} \in \mathbb{R} \times (0,2\pi) \times \mathbb{R}\) a parametrization of \(\partial \Omega\).

Integral of dot between vectorial field and normal on bound of a domain :

\[ \int_{\partial \Omega}{ \mathbf{F}(x) \cdot \mathbf{n}(x) dx} = ???\]

3.2. In axisymetric condition

We suppose that \(\Omega\) is symmetric by \(Oz\) axis, \(f^{cyl}\) and \(\mathbf{F}^{cyl}\) are independant by \(\theta\).

We note \(\Omega^{axis}\) (respectively \(\Gamma^{axis}\) and \(\mathbf{n}^{axis}\)) the representation of \(\Omega\) (respectively \(\Gamma\) and \(\mathbf{n}\)) in axisymmetric coordinates). And we note \(f^{axis} \in H^2(\Omega^{axis})\) and \(\mathbf{F}^{axis} \in H^2(\Omega^{axis})^d\) sucht that \(f^{axis}(r,z) = f^{cyl}(r,\theta,z)\) and \(\mathbf{F}^{axis}(r,z) = \mathbf{F}^{cyl}(r,\theta,z)\).

We have :

Formula of changement in axisymmetrical coordinates

Integral of function on a domain :

\[ \int_{\Omega} \ f(x,y,z) \ dxdydz\ = 2\pi \int_{\Omega_{axis}} f_{axis}(r,z) r \ dr dz\]

Integral of jacobian of function on a domain :

\[ \int_{\Omega} \nabla f(x,y,z) \ dxdydz = \int_{\Omega_{axis}}{ \left( \frac{\partial f_{cyl}}{\partial r} \mathbf{e_r} + \frac{\partial f_{cyl}}{\partial z} \mathbf{e_z} \right) \ r \ dr dz}\]

Integral of function on bound of domain :
- On \(O_\theta O_z\) plan :
  
  \[ \int_{\partial \Omega}{ f(x) dx } = 2\pi \int_{\tau \in T^{axis}}{ f\begin{pmatrix} r(\tau) \\ z(\tau) \end{pmatrix} \sqrt{ \left(\frac{\partial z}{\partial \tau}\right)^2 + \left(\frac{\partial r}{\partial \tau}\right)^2 } \, r \, d\tau }\]
  
  with parametrization \(\tau \in T^{axis} \rightarrow \begin{pmatrix} r(\tau) \\ z(\tau) \end{pmatrix}\), it’s a parametrization of a curve, but in fact this curve represents a surface to axisymmetric to three dimensions.
- On \(O_r O_z\) plan (it isn’t a really axysimmetric surface but its value is constant by \(\theta\)) :
  
  \[ \int_{O_r O_z}{ f(x) dx } = \int_{(r,z) \in O_r O_z}{ f\begin{pmatrix} r \\z \end{pmatrix} dr dz }\]
Integral of dot between vectorial field and normal on bound of domain :
- On \(O_\theta O_z\) plan :
  
  \[ \int_{\partial \Omega}{ \mathbf{F}(x) \cdot \mathbf{n}(x) dx} = ???\]
  
  with parametrization \(\tau \in T^{axis} \rightarrow \begin{pmatrix} r(\tau) \\ z(\tau) \end{pmatrix}\), it’s a parametrization of a curve, but in fact this curve represents a surface to axisymmetric to three dimensions.
- On \(O_r O_z\) plan (it isn’t a really axysimmetric surface but its value is constant by \(\theta\)) :
  
  \[ \int_{\partial \Omega}{ \mathbf{F}(x) \cdot \mathbf{n}(x) dx} = ???\]