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@@ -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*} |