Vector Analysis

1. Basic

1.1. Vector fields

Definition of Vector Field

Vector field is an application :

\[\begin{eqnarray*} \mathbf{u} : && \Omega \in \mathbb{R}^d \longrightarrow \mathbb{R}^d \\ && \mathbf{x} = \begin{pmatrix} x_1 \\ \vdots \\ x_d \end{pmatrix} \longmapsto \mathbf{u}(x) = \begin{pmatrix} u_1(x) \\ \vdots \\ u_d(x) \end{pmatrix} \end{eqnarray*}\]

A vector field is usually continuous and infinitely differentiable.

1.2. Tensor of order two

Definition of Tensor of order two

A tensor of order two is an bilinear application on \(\mathbb{R}^3\) :

\[ T : \mathbb{R}^3 \times \left( \mathbb{R}^3 \right)^* \rightarrow \mathbf{R} \\ \mathbf{u} , \mathbf{v} \rightarrow T(\mathbf{u},\mathbf{v})\]

With \(\left( \mathbb{R}^3 \right)^*\) the dual of \(\mathbb{R}^3\).

A tensor of order two on \(\mathbb{R}^3\) can be represented by a matrix \(3 \times 3\) :

\[ T = \begin{pmatrix} {T^i}_j \end{pmatrix}_{1 \leq i,j \leq 3}\]

With \({T^i}_j = T(e_i,e^j)\), \(\left( e_1, \, e_2, \, e_3 \right)\) a basis of \(\mathbb{R}^3\) and \(\left( e^1, \, e^2, \, e^3 \right)\) a basis of \(\left( \mathbb{R}^3 \right)^*\). With this notation, we have :

\[ T(\mathbf{u},\mathbf{v^*}) = \mathbf{u} \times T \times \mathbf{v^*}\]

  • Representation in cartesian coordinates \(\left( e_x, \, e_y, \, e_z \right)\)

\[ T = \begin{pmatrix} {T^x}_x & {T^x}_y & {T^x}_z \\ {T^y}_x & {T^y}_y & {T^y}_z \\ {T^z}_x & {T^z}_y & {T^z}_z \end{pmatrix}\]

  • Representation in cylindric coordinates \(\left( e_r, \, e_{\theta}, \, e_z \right)\)

\[ T = \begin{pmatrix} {T^r}_r & {T^r}_{\theta} & {T^r}_z \\ {T^{\theta}}_r & {T^{\theta}}_{\theta} & {T^{\theta}}_z \\ {T^z}_r & {T^z}_{\theta} & {T^z}_z \end{pmatrix}\]

1.3. Scalar Product

Scalar Product of two vector

The cross product is an application \((. \cdot . ) : \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}\) :

\[\begin{pmatrix} u_1 \\ \vdots \\ u_3 \end{pmatrix} \cdot \begin{pmatrix} v_1 \\ \vdots \\ v_3 \end{pmatrix} = \sum_{i=0}^d{u_i \, v_i}\]

Some propertie of scalar product :

  • \(\mathbf{u} \cdot (\mathbf{v} \cdot \mathbf{w}) = \mathbf{w} \cdot (\mathbf{u} \cdot \mathbf{v}) = \mathbf{v} \cdot (\mathbf{w} \cdot \mathbf{u})\)

1.4. Cross Product

Cross Product of two vector

The cross product is an application \(\cdot \times \cdot : \mathbb{R}^3 \times \mathbb{R}^3 \rightarrow \mathbb{R}^3\) :

\[\begin{pmatrix} u_1 \\ u_2 \\ u_3 \end{pmatrix} \times \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix} = \begin{pmatrix} u_2 v_3 - u_3 v_2 \\ u_1 v_3 - u_3 v_1 \\ u_1 v_2 - u_2 v_1 \end{pmatrix}\]

Some properties of cross product :

  • bilinear

  • \(\mathbf{u} \times \mathbf{v} = - \mathbf{v} \times \mathbf{u}\)

  • \(\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = \mathbf{w} \cdot (\mathbf{u} \times \mathbf{v}) = \mathbf{v} \cdot (\mathbf{w} \times \mathbf{u})\)

2. Analysis

2.1. Gradient

Gradient of Vector Field
\[ \nabla \mathbf{u} = \left( \partial_{x_j} u_i \right)_{i,j} = \begin{pmatrix} \partial_{x_1} u_1 & \cdots & \partial_{x_d} u_1 \\ \partial_{x_1} u_2 & \cdots & \partial_{x_d} u_2 \\ \vdots & \ddots & \vdots \\ \partial_{x_1} u_d & \cdots & \partial_{x_d} u_d \end{pmatrix}\]

\(\nabla\) can see as :

\[\nabla = \left(\frac{\partial}{\partial x_j} \right)_{i=1,...,J}\]

In \(\mathbb{R}^3\) (\(\mathbf{u} : \mathbb{R}^3 \rightarrow \mathbb{R}^3\)) :

  • In Cartesian coordinates :

\[ \nabla \, \mathbf{u} = \begin{pmatrix} \frac{\partial u_x}{\partial x} & \frac{\partial u_x}{\partial y} & \frac{\partial u_x}{\partial z} \\ \frac{\partial u_y}{\partial x} & \frac{\partial u_y}{\partial y} & \frac{\partial u_y}{\partial z} \\ \frac{\partial u_z}{\partial x} & \frac{\partial u_z}{\partial y} & \frac{\partial u_z}{\partial z} \end{pmatrix}\]

  • In Cylindric coordinates :

\[ \nabla \, \mathbf{u} = \begin{pmatrix} \frac{\partial u_r}{\partial r} & \frac{1}{r} \left( \frac{\partial u_r}{\partial \theta} - u_{\theta} \right) & \frac{\partial u_r}{\partial z} \\ \frac{\partial u_{\theta}}{\partial r} & \frac{1}{r} \left( \frac{\partial u_{\theta}}{\partial \theta} - u_r \right) & \frac{\partial u_{\theta}}{\partial z} \\ \frac{\partial u_z}{\partial r} & \frac{1}{r} \frac{\partial u_z}{\partial \theta} & \frac{\partial u_z}{\partial z} \end{pmatrix}\]

2.2. Divergence

2.2.1. Vector Field

Divergence of Vectorial Field in Cartesian Coordinates

Thus a vector field \(u\) on \(\mathbb{R}^3\) :

\[ \nabla \cdot \mathbf{u} = div(u) = \sum_{i=1}^{n}{\partial_{x_i} u_i} \label{div} \tag{Divergence}\]

In cartesian coordinates, in inferior dimensions :

  • \(d=1\) : \(\nabla \cdot \mathbf{u} = \frac{\partial u}{\partial x}\)

  • \(d=2\) : \(\nabla \cdot \mathbf{u}(x,y) = \frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y}\) (with \(u = \begin{pmatrix} u_x \\ u_y \end{pmatrix}\))

  • \(d=3\) : \(\nabla \cdot \mathbf{u}(x,y,z) = \frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} + \frac{\partial u_z}{\partial z}\) (with \(u = \begin{pmatrix} u_x \\ u_y \\ u_z \end{pmatrix}\))

In cylindric coordinates in three dimensions \((r,\theta,z)\) (\(d=3\)) :

Divergence of Vectorial Field in Cylindric Coordinates

Thus a vector field \(\mathbf{u} = \begin{pmatrix} u_r \\ u_{\theta} \\ u_z \end{pmatrix}_{cyl}\) on \(\mathbb{R}^3\) :

\[ \nabla \cdot \mathbf{u} = \frac{1}{r} \frac{\partial (ru_r)}{\partial r} + \frac{1}{r} \frac{\partial u_{\theta}}{\partial \theta} + \frac{\partial u_z}{\partial z}\]

2.2.2. Tensor of Order Two

Divergence of Tensor of order two in Cartesian Coordinates

Thus \(T\) a tensor of order two of \(\mathbb{R}^3\)

\[ \nabla \cdot T = \begin{pmatrix} \frac{\partial {T^x}_x}{\partial x} + \frac{\partial {T^x}_y}{\partial y} + \frac{\partial {T^x}_z}{\partial z} \\ \frac{\partial {T^y}_x}{\partial x} + \frac{\partial {T^y}_y}{\partial y} + \frac{\partial {T^y}_z}{\partial z} \\ \frac{\partial {T^z}_x}{\partial x} + \frac{\partial {T^z}_y}{\partial y} + \frac{\partial {T^z}_z}{\partial z} \end{pmatrix}\]
Divergence of Tensor of order two in Cylindric Coordinates

Thus \(T\) a tensor of order two of \(\mathbb{R}^3\)

\[ \nabla \cdot T = \begin{pmatrix} \frac{\partial {T^r}_r}{\partial r} + \frac{1}{r} \frac{\partial {T^r}_{\theta}}{\partial \theta} + \frac{\partial {T^r}_z}{\partial z} + \frac{{T^r}_r - {T^{\theta}_{\theta}}}{r} \\ \frac{\partial {T^{\theta}}_r}{\partial r} + \frac{1}{r} \frac{\partial {T^{\theta}}_{\theta}}{\partial \theta} + \frac{\partial {T^{\theta}}_z}{\partial z} + \frac{{T^r}_{\theta} + {T^{\theta}_r}}{r} \\ \frac{\partial {T^z}_r}{\partial r} + \frac{1}{r} \frac{\partial {T^z}_{\theta}}{\partial \theta} + \frac{\partial {T^z}_z}{\partial z} + \frac{{T^z}_r}{r} \end{pmatrix}\]

2.3. Laplacian

2.3.1. Scalar

Laplacian of Scalar in Cartesian Coordinates

Thus \(f: \mathbb{R}^3 \rightarrow \mathbb{R}\) :

\[ \Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}\]
Laplacian of Scalar in Cylindric Coordinates

Thus \(f: \mathbb{R}^3 \rightarrow \mathbb{R}\) :

\[ \Delta f = \frac{1}{r} \, \frac{\partial}{\partial r} \left( r \frac{\partial f}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 f}{\partial \theta^2} + \frac{\partial^2 f}{\partial z^2}\]

2.3.2. Vector Field

Laplacian of Vector Field in Cartesian Coordinates

Thus \(\mathbf{u}: \mathbb{R}^3 \rightarrow \mathbb{R}^3\) :

\[ \Delta \mathbf{u} = \begin{pmatrix} \Delta u_x \\ \Delta u_y \\ \Delta u_z \end{pmatrix} = \begin{pmatrix} \frac{\partial^2 u_x}{\partial x^2} + \frac{\partial^2 u_x}{\partial y^2} + \frac{\partial^2 u_x}{\partial z^2} \\ \frac{\partial^2 u_y}{\partial x^2} + \frac{\partial^2 u_y}{\partial y^2} + \frac{\partial^2 u_y}{\partial z^2} \\ \frac{\partial^2 u_z}{\partial x^2} + \frac{\partial^2 u_z}{\partial y^2} + \frac{\partial^2 u_z}{\partial z^2} \end{pmatrix}\]
Laplacian of Vector Field in Cylindric Coordinates

Thus \(\mathbf{u}: \mathbb{R}^3 \rightarrow \mathbb{R}^3\) :

\[ \Delta \mathbf{u} = \begin{pmatrix} \Delta u_x \\ \Delta u_y \\ \Delta u_z \end{pmatrix}_{cyl} = \begin{pmatrix} \frac{\partial^2 u_r}{\partial r^2} + \frac{1}{r^2} \frac{\partial^2 u_r}{\partial \theta^2} + \frac{\partial^2 u_r}{\partial z^2} + \frac{1}{r} \frac{\partial u_r}{\partial r} - \frac{2}{r^2} \frac{\partial u_{\theta}}{\partial r} - \frac{u_r}{r^2} \\ \frac{\partial^2 u_{\theta}}{\partial r^2} + \frac{1}{r^2} \frac{\partial^2 u_{\theta}}{\partial \theta^2} + \frac{\partial^2 u_{\theta}}{\partial z^2} + \frac{1}{r} \frac{\partial u_{\theta}}{\partial r} + \frac{2}{r^2} \frac{\partial u_r}{\partial \theta} - \frac{u_{\theta}}{r^2} \\ \frac{\partial^2 u_z}{\partial r^2} + \frac{1}{r^2} \frac{\partial^2 u_z}{\partial \theta^2} + \frac{\partial^2 u_z}{\partial z^2} + \frac{1}{r} \frac{\partial u_z}{\partial u_z} \end{pmatrix}_{cyl}\]

2.4. Curl

Curl of Vectorial Field in Cartesian Coordinates

Thus vector field \(u = \begin{pmatrix} u_x \\ u_y \\ u_z \end{pmatrix}\) in 3 dimension (\(d=3\)) :

\[\nabla \times u = curl(u) = \begin{pmatrix} \frac{\partial u_z}{\partial y} - \frac{\partial u_y}{\partial z} \\ \frac{\partial u_x}{\partial z} - \frac{\partial u_z}{\partial x} \\ \frac{\partial u_y}{\partial x} - \frac{\partial u_x}{\partial y} \end{pmatrix}\]

In cylindric coordinates, the formulate become :

Curl of Vectorial Field in Cylindric Coordinates

Thus vector field \(u = \begin{pmatrix} u_r \\ u_{\theta} \\ u_z \end{pmatrix}\) in 3 dimension (\(d=3\)) :

\[\nabla \times u = \begin{pmatrix} \frac{1}{r} \frac{\partial u_z}{\partial \theta} - \frac{\partial u_{\theta}}{\partial z} \\ \frac{\partial u_r}{\partial z} - \frac{\partial u_z}{\partial r} \\ \frac{1}{r} \left( \frac{\partial (r u_{\theta})}{\partial r} - \frac{\partial u_r}{\partial \theta} \right) \end{pmatrix}_{cyl}\]

2.5. Relation

Some relations exist between \(grad\), \(div\) and \(curl\) :

  • \(\nabla \cdot (\mathbf{u} \times \mathbf{v}) = \mathbf{v} \cdot (\nabla \times \mathbf{u}) - \mathbf{u} \cdot (\nabla \times \mathbf{v})\)

  • \(\mathbf{u} \cdot (\mathbf{v} \cdot \mathbf{w}) = \mathbf{w} \cdot (\mathbf{u} \cdot \mathbf{v}) = \mathbf{v} \cdot (\mathbf{w} \cdot \mathbf{u})\)

  • \(\nabla \cdot (\mathbf{u} \cdot \mathbf{v}) = \mathbf{v} \cdot \nabla \mathbf{u} + \mathbf{u} \cdot \nabla \cdot \mathbf{v}\)

3. Documentation