From 299c06ff52cb81a1c3c58f84636c624f3ee45ef4 Mon Sep 17 00:00:00 2001 From: Luc Date: Thu, 25 Jan 2024 11:50:43 +0100 Subject: [PATCH] Finished divergence in vector analysis. --- .../electromagnetism/optics/polarisation.md | 2 +- .../curvilinear-coordinates.md | 2 +- .../vector-analysis/divergence.md | 131 ++++++++++++++++++ 3 files changed, 133 insertions(+), 2 deletions(-) diff --git a/docs/en/physics/electromagnetism/optics/polarisation.md b/docs/en/physics/electromagnetism/optics/polarisation.md index 9ebb53e..a62d0b0 100644 --- a/docs/en/physics/electromagnetism/optics/polarisation.md +++ b/docs/en/physics/electromagnetism/optics/polarisation.md @@ -94,7 +94,7 @@ Using the properties of birefringence, wave plates (retarders) can be created. T ## Jones formalism of polarisation -Jones formalism of polarisation with vectors and matrices cna make it easier to calculate the effect of optical elements such as linear polarizers and wave plates. +Jones formalism of polarisation with vectors and matrices can make it easier to calculate the effects of optical elements such as linear polarizers and wave plates. > *Definition*: for an electromagnetic wave $\mathbf{E}: \mathbb{R}^2 \to \mathbb{R}^3$ with wavenumber $k \in \mathbb{R}$ and angular frequency $\omega \in \mathbb{R}$ propagating in the positive $z$-direction given by > diff --git a/docs/en/physics/mathematical-physics/vector-analysis/curvilinear-coordinates.md b/docs/en/physics/mathematical-physics/vector-analysis/curvilinear-coordinates.md index 910f2d4..7194a63 100644 --- a/docs/en/physics/mathematical-physics/vector-analysis/curvilinear-coordinates.md +++ b/docs/en/physics/mathematical-physics/vector-analysis/curvilinear-coordinates.md @@ -245,7 +245,7 @@ $$ \|\mathbf{u}\| = \sqrt{u^{(i)} u_{(i)}}. $$ -We will discuss as an example the representations of the cartesian, cylindrical and spherical coordinate systems viewed from a cartesian perspective. This means that the coordinate maps are based on the cartesian interpretation of then. Every other interpretation could have been used, but our brains have a preference for cartesian it seems.z +We will discuss as an example the representations of the cartesian, cylindrical and spherical coordinate systems viewed from a cartesian perspective. This means that the coordinate maps are based on the cartesian interpretation of them. Every other interpretation could have been used, but our brains have a preference for cartesian it seems. Let $\mathbf{x}: \mathbb{R}^3 \to \mathbb{R}^3$ map a cartesian coordinate system given by diff --git a/docs/en/physics/mathematical-physics/vector-analysis/divergence.md b/docs/en/physics/mathematical-physics/vector-analysis/divergence.md index e69de29..18cc6ea 100644 --- a/docs/en/physics/mathematical-physics/vector-analysis/divergence.md +++ b/docs/en/physics/mathematical-physics/vector-analysis/divergence.md @@ -0,0 +1,131 @@ +# The divergence of a vector field + +## Flux densities + +Considering a medium with a mass density $\rho: \mathbb{R}^4 \to \mathbb{R}$ and a velocity field $\mathbf{v}: \mathbb{R}^4 \to \mathbb{R}^3$ consisting of a orientable finite sized surface element $d\mathbf{A} \in \mathbb{R}^3$. + +> *Definition*: a surface must be orientable for the surface integral to exist. It must be able to move along the surface continuously without ending up on the "other side". + +We then have a volume $dV \in \mathbb{R}$ defined by the parallelepiped formed by $dV = \langle d\mathbf{x}, d\mathbf{A} \rangle$ with the vector $d\mathbf{x} = \mathbf{v} dt$, for a time interval $dt \in \mathbb{R}$. The mass flux $d\Phi$ per unit of time through the surface element $d\mathbf{A}$ may then be given by + +$$ + d \Phi = \rho \langle \mathbf{v}, d\mathbf{A} \rangle. +$$ + +The mass flux $\Phi: \mathbb{R} \to \mathbb{R}$ through a orientable finite sized surface $A \subseteq \mathbb{R}^3$ is then given by + +$$ + \Phi(t) = \int_A \Big\langle \rho(\mathbf{x}, t) \mathbf{v}(\mathbf{x}, t), d\mathbf{A} \Big\rangle, +$$ + +for all $t \in \mathbb{R}$. + +> *Definition*: let $\mathbf{\Gamma}: \mathbb{R}^4 \to \mathbb{R}^3$ be the (mass) flux density given by +> +> $$ +> \mathbf{\Gamma}(\mathbf{x},t) := \rho(\mathbf{x},t) \mathbf{v}(\mathbf{x},t), +> $$ +> +> for all $(\mathbf{x},t) \in \mathbb{R}^4$. + +The (mass) flux density is a vector-valued function of position and time that expresses the rate of transport of a quantity per unit of time of area perpendicular to its direction. + +The mass flux $\Phi$ through $A$ may then be given by + +$$ + \Phi(t) = \int_A \Big\langle \mathbf{\Gamma}(\mathbf{x},t), d\mathbf{A} \Big\rangle, +$$ + +for all $t \in \mathbb{R}$. + +## Definition of the divergence + +> *Definition*: the divergence of a flux density $\mathbf{\Gamma}: \mathbb{R}^4 \to \mathbb{R}^3$ is given by +> +> $$ +>\nabla \cdot \mathbf{\Gamma}(\mathbf{x}, t) = \lim_{V \to 0} \frac{1}{V} \int_A \Big\langle \mathbf{\Gamma}(\mathbf{x}, t), d\mathbf{A} \Big\rangle, +> $$ +> +> for all $(\mathbf{x}, t) \in \mathbb{R}^4$ for a volume $V \subset \mathbb{R}^3$ with closed orientable boundary surface $A \subset V$. + +Note that this "dot product" between the nabla operator and the flux density $\mathbf{\Gamma}$ does not imply anything and is only there for notational sake. An alternative to this notation is using $\text{div } \mathbf{\Gamma}$ to denote the divergence. + +The definition of hte divergence can be interpreted with the species mass balance for a medium with a particle density $n: \mathbb{R}^4 \to \mathbb{R}$ and a velocity field $\mathbf{v}: \mathbb{R}^4 \to \mathbb{R}^3$. Furthermore we have that the particles are produced at a rate $S: \mathbb{R}^4 \to \mathbb{R}^3$. + +We then have the particle flux density $\mathbf{\Gamma}: \mathbb{R}^4 \to \mathbb{R}^3$ given by + +$$ + \mathbf{\Gamma}(\mathbf{x},t) = n(\mathbf{x},t) \mathbf{v}(\mathbf{x},t), +$$ + +for all $(\mathbf{x},t) \in \mathbb{R}^4$. + +For a volume $V \subseteq \mathbb{R}^3$ with a closed orientable boundary surface $A \subseteq \mathbb{R}^3$ we have that the amount of particles inside this volume for a specific time is given by + +$$ + \int_V n(\mathbf{x}, t) dV, +$$ + +for all $t \in \mathbb{R}$. We have that the particle flux through $A$ is given by + +$$ + \int_A \Big\langle \mathbf{\Gamma}(\mathbf{x},t), d\mathbf{A} \Big\rangle, +$$ + +for all $t \in \mathbb{R}$ and we have that the particle production rate in this volume $V$ is given by + +$$ + \int_V S(\mathbf{x}, t)dV, +$$ + +for all $t \in \mathbb{R}$. We conclude that the sum of the particle flux through $A$ and the time derivative of the particles inside the volume $V$ must be equal to the production rate inside this volume $V$. Therefore we have + +$$ + d_t \int_V n(\mathbf{x}, t) dV + \int_A \Big\langle \mathbf{\Gamma}(\mathbf{x},t), d\mathbf{A} \Big\rangle = \int_V S(\mathbf{x}, t)dV, +$$ + +for all $t \in \mathbb{R}$. + +Assuming the system is stationary the time derivative of the particles inside the volume $V$ must vanish. The divergence is then defined to be the total production for a position $\mathbf{x} \in V$. + +## Divergence in curvilinear coordinates + +> *Theorem*: the divergence of a flux density $\mathbf{\Gamma}: \mathbb{R}^4 \to \mathbb{R}^3$ for a curvilinear coordinate system is given by +> +> $$ +> \nabla \cdot \mathbf{\Gamma}(\mathbf{x}, t) = \frac{1}{\sqrt{g(\mathbf{x})}} \partial_i \Big(\Gamma^i(\mathbf{x},t) \sqrt{g(\mathbf{x})} \Big) +> $$ +> +> for all $\mathbf{x} \in \mathbb{R}^3$ and $i \in \{1, 2, 3\}$. + +??? note "*Proof*:" + + Will be added later. + +We may also give the divergence for ortho-curvilinear coordinate systems. + +> *Corollary*: the divergence of a flux density $\mathbf{\Gamma}: \mathbb{R}^3 \to \mathbb{R}^3$ for a ortho-curvilinear is given by +> +> $$ +> \nabla \cdot \mathbf{\Gamma}(\mathbf{x}, t) = \frac{1}{h_1h_2h_3} \partial_i \Big(\Gamma^i(\mathbf{x},t)\frac{1}{h_i} h_1 h_2 h_3 \Big) +> $$ +> +> for all $\mathbf{x} \in \mathbb{R}^3$ and $i \in \{1, 2, 3\}$. + +??? note "*Proof*:" + + Will be added later. + +It has been found that the volume integral over the divergence of a vector field is equal to the integral of the vector field itself over the surface that bounds the volume. It is known as the divergence theorem given below.s + +> *Theorem*: for a volume $V \subset \mathbb{R}^3$ with a closed and orientable boundary surface $A \subset V$ with a continuously differentiable flux density $\mathbf{\Gamma}: \mathbb{R}^4 \to \mathbb{R}^3$ we have that +> +> $$ +> \int_A \Big\langle \mathbf{\Gamma}(\mathbf{x}, t), d\mathbf{A} \Big\rangle = \int_V \nabla \cdot \mathbf{\Gamma}(\mathbf{x}, t) dV, +> $$ +> +> for all $t \in \mathbb{R}$. + +??? note "*Proof*:" + + Will be added later. \ No newline at end of file