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authorNaoki Pross <np@0hm.ch>2021-10-10 18:47:32 +0200
committerNaoki Pross <np@0hm.ch>2021-10-10 18:47:32 +0200
commit469f1c8c733857c2957c98815bfdcff7a5453329 (patch)
tree54384c3a3e886e7c87bc58c31dd899b946822698
parentVector analysis and electrodynamics recap (diff)
downloadElMag-469f1c8c733857c2957c98815bfdcff7a5453329.tar.gz
ElMag-469f1c8c733857c2957c98815bfdcff7a5453329.zip
Partial derivatives
-rw-r--r--ElMag.tex47
1 files changed, 41 insertions, 6 deletions
diff --git a/ElMag.tex b/ElMag.tex
index 940f00b..8c21412 100644
--- a/ElMag.tex
+++ b/ElMag.tex
@@ -88,6 +88,35 @@
\section{Vector Analysis Recap}
+\begin{definition}[Partial derivative]
+ A vector valued function \(f: \mathbb{R}^m\to\mathbb{R}\), with
+ \(\vec{v}\in\mathbb{R}^m\), has a partial derivative with respect to \(v_i\)
+ defined as
+ \[
+ \partial_{v_i} f(\vec{v})
+ % = f_{v_i}(\vec{v})
+ = \frac{\partial f}{\partial v_i}
+ = \lim_{h\to 0} \frac{f(\vec{v} + h\vec{e}_i) - f(\vec{v})}{h}
+ \]
+\end{definition}
+
+\begin{theorem}[Integration of partial derivatives]
+ Let \(f: \mathbb{R}^m\to\mathbb{R}\) be a partially differentiable function
+ of many \(x_i\). When \(x_i\) is \emph{indipendent} with respect to all other
+ \(x_j\) \((0 < j \leq m, j \neq i)\) then
+ \[
+ \int \partial_{x_i} f \,d x_i = f + C,
+ \]
+ where \(C\) is a function of \(x_1, \ldots, x_m\) but not of \(x_i\).
+\end{theorem}
+
+To illustrate the previous theorem, in a simpler case with \(f(x,y)\), we get
+\[
+ \int \partial_x f(x,y) \,dx = f(x, y) + C(y).
+\]
+Beware that this is valid only if \(x\) and \(y\) are indipendent.
+If there is a relation \(x(y)\) or \(y(x)\) the above does not hold.
+
\begin{definition}[Gradient vector]
The \emph{gradient} of a function \(f(\vec{x}), \vec{x}\in\mathbb{R}^m\) is a
column vector containing the partial derivatives
@@ -103,7 +132,8 @@
\end{definition}
\begin{theorem}[Gradient in curvilinear coordinates]
- Let \(f: \mathbb{R}^3 \to \mathbb{R}\) be a scalar field. In cylindrical coordinates \((r,\phi,z)\)
+ Let \(f: \mathbb{R}^3 \to \mathbb{R}\) be a scalar field. In cylindrical
+ coordinates \((r,\phi,z)\)
\[
\grad f = \uvec{r}\,\partial_r f
+ \uvec{\phi}\,\frac{1}{r}\partial_\phi f
@@ -127,7 +157,8 @@
\end{definition}
\begin{theorem}[Divergence in curvilinear coordinates]
- Let \(\vec{F}: \mathbb{R}^3 \to \mathbb{R}^3\) be a field. In cylindrical coordinates \((r,\phi,z)\)
+ Let \(\vec{F}: \mathbb{R}^3 \to \mathbb{R}^3\) be a field. In cylindrical
+ coordinates \((r,\phi,z)\)
\[
\div \vec{F} = \frac{1}{r} \partial_r (r F_r)
+ \frac{1}{r}\partial_\phi F_\phi
@@ -170,7 +201,8 @@
\end{definition}
\begin{definition}[Curl in curvilinear coordinates]
- Let \(\vec{F}: \mathbb{R}^3 \to \mathbb{R}^3\) be a field. In cylindrical coordinates \((r,\phi,z)\)
+ Let \(\vec{F}: \mathbb{R}^3 \to \mathbb{R}^3\) be a field. In cylindrical
+ coordinates \((r,\phi,z)\)
\begin{align*}
\curl \vec{F} =
&\left(\frac{1}{r} \partial_\phi F_z - \partial_z F_\phi \right) \uvec{r} \\
@@ -212,7 +244,8 @@
\end{definition}
\begin{theorem}[Laplacian in curvilinear coordinates]
- Let \(f: \mathbb{R}^3 \to \mathbb{R}\) be a scalar field. In cylindrical coordinates \((r,\phi,z)\)
+ Let \(f: \mathbb{R}^3 \to \mathbb{R}\) be a scalar field. In cylindrical
+ coordinates \((r,\phi,z)\)
\[
\laplacian f = \frac{1}{r} \partial_r (r \partial_r f)
+ \frac{1}{r^2} \partial_\phi^2 f
@@ -294,11 +327,13 @@ Maxwell's equations in matter in their integral form are
\oint_{\partial V} \vec{B} \dotp d\vec{s} &= 0.
\end{align}
\end{subequations}
-Where \(\vec{J}\) and \(\rho\) are the \emph{free current density} and \emph{free charge density} respectively.
+Where \(\vec{J}\) and \(\rho\) are the \emph{free current density} and
+\emph{free charge density} respectively.
\subsection{Isotropic linear materials and boundary conditions}
-Inside of so called isotropic linear materials the fields and flux (or current) densities are proportional, i.e.
+Inside of so called isotropic linear materials the fields and flux (or current)
+densities are proportional, i.e.
\begin{align*}
\vec{D} &= \varepsilon \vec{E}, & \vec{J} &= \sigma \vec{E}, & \vec{B} &= \mu \vec{H}.
\end{align*}