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author committer Naoki Pross 2021-10-10 18:47:32 +0200 Naoki Pross 2021-10-10 18:47:32 +0200 469f1c8c733857c2957c98815bfdcff7a5453329 (patch) 54384c3a3e886e7c87bc58c31dd899b946822698 Vector analysis and electrodynamics recap (diff) ElMag-469f1c8c733857c2957c98815bfdcff7a5453329.tar.gzElMag-469f1c8c733857c2957c98815bfdcff7a5453329.zip
Partial derivatives
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1 files changed, 41 insertions, 6 deletions
 diff --git a/ElMag.tex b/ElMag.texindex 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*}