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zs~M48)Uh5gw?#qAvef#O`lF(%8*^hwoN7a9PX7{PfKSI_4ZWv|_9)jSP@rVkB)$T&R6?2QL1$4ZJjZ0_;b@V>q(JFhS`Rh?= z7GBjJc!K^Y-HGUkmmE{5=sYEFhtRW<9M5)wvo8e#5q^}P4%@MpkNLIb9 z53K-NJaV^V%c&?G;k|$;@LUzYCj90zNjRd^IsQ@pN!Bpkr8o!u$?%aONoA38gmy=3 zD=aW5M8UOBk~K5dq2}t5Q`j5pal_cH7%I4So#>~Wf1RH^sFFYMJPyYU`l#~(=q>mq z;`hc+zG5tpR}2(i_UaVAr&z-_-)Lt5hhx){Z;XGs*O+G|r>>{)Cm)+5d_d3i zAA+I3*=_)japwlhK9$cB*GJ%I-3v?0J|)jTQQfLt|GMS6+2!96Aw+S1_}8xmzA54b zT%AM`a$|JQ!Ikg9=fnw~g!gcxeL5jd4;aA@5$D8`XouBvBatz~ADfbo7|~K1!5*1V z9vRVK?y}OO=LO(gwkVi7+2uen=7%b-j6MMk3wfOEM zrZ7mUsO;=fabjD&O2?L*v-NVlHjiWz-# 0\) centred at a -point \(\symbf{p} \in \R^n\) is denoted \(B_r\left( \symbf{p} -\right)\), and is defined as +\subsection{Topological Properties} +\begin{definition} + An open ball of radius \(r > 0\) centred at a + point \(\symbf{p} \in \R^n\) is denoted \(B_r\left( \symbf{p} + \right)\), and is defined as + \begin{equation*} + B_r\left( \symbf{p} \right) = \left\{ \symbf{x} \in \R^n : \norm{\symbf{x} - \symbf{p}} < r \right\}. + \end{equation*} + This region includes all points less than a distance \(r\) from the + vector \(\symbf{p}\), where the distance is typically defined by the + \(L_2\)-norm: + \begin{equation*} + \norm{\symbf{x} - \symbf{p}}_2 = \left( \sum_{i=1}^n \left( x_i - p_i \right)^2 \right)^{1/2}. + \end{equation*} +\end{definition} +\begin{definition}[Open Set] + A set \(S \subset \R^n\) is open if for every point \(\symbf{x} \in S\), + there exists \(\delta > 0\), such that the open ball + \(B_{\delta}\left( \symbf{x} \right) \in S\). +\end{definition} +\begin{definition}[Closed Set] + A set \(S \subset \R^n\) is closed if its complement + \(\R^n \setminus S\) is open. +\end{definition} +\begin{definition}[Connected Set] + A set \(S \subset \R^n\) is connected if it cannot be represented as + the union of two or more disjoint non-empty open subsets. +\end{definition} +\begin{definition}[Path Connected Set] + A set \(S \subset \R^n\) is path connected if for every pair of + points \(\symbf{x}, \symbf{y} \in S\), there exists a continuous + path \(\symbf{r}\left( t \right)\) from \(\symbf{x}\) to \(\symbf{y}\). +\end{definition} +\begin{definition}[Simply Connected Set] + A set \(S \subset \R^n\) is simply connected if it is path-connected + and if every closed path in \(S\) can be continuously contracted to + a point in \(S\). +\end{definition} +\subsection{Parametrisations of Curves} +\begin{definition}[Path] + A path is a continuous function + \begin{equation*} + \symbf{r} : \interval{a}{b} \subset \R \to \R^n + \end{equation*} + where \(t \mapsto \symbf{r}\left( \symbf{x}\left( t \right) \right)\) + is the parameter of the path. +\end{definition} +\begin{definition}[Curve] + A curve \(\mathscr{C}\) is the set of points in \(\R^n\) + corresponding to the range of a path \(\symbf{r}\). + \begin{equation*} + \mathscr{C} = \left\{ \symbf{r}\left( t \right) : t \in \rinterval{a}{b} \right\} + \end{equation*} +\end{definition} +A curve \(\mathscr{C}\) is \textbf{parametrised} by \(\symbf{r}\left( t \right)\), +and a path \(\symbf{r}\left( t \right)\) is a \textbf{parametrisation} +of \(\mathscr{C}\). +\begin{definition}[Closed Path] + A path is closed if it starts and ends at the same point: + \begin{equation*} + \symbf{r}\left( a \right) = \symbf{r}\left( b \right). + \end{equation*} +\end{definition} +\begin{definition}[Simple Path] + A path is simple if the map \(t \mapsto \symbf{r}\left( t \right)\) + is injective. That is, the path does not intersect itself. +\end{definition} +\begin{definition}[Regular Path] + A path is regular (or smooth) if has nonzero continuous first derivatives: + \begin{equation*} + \symbf{r}\left( t \right) \in \C^1 \quad \text{and} \quad + \symbf{r}'\left( t \right) \neq \symbf{0} + \end{equation*} + for all \(t\). This restriction ensures that the path does not have + any cusps and allows a unit tangent vector to be defined. +\end{definition} +\begin{definition}[Piecewise Regular Path] + A path is piecewise regular if it can be divided into a finite + number of regular paths. +\end{definition} +\begin{definition}[Path Concatenation] + The path concatenation of two paths \(\symbf{r}_1 : \interval{a}{b} \subset \R \to \R^n\) + and \(\symbf{r}_2 : \interval{b}{c} \subset \R \to \R^n\) is defined as + \begin{equation*} + \symbf{r}\left( t \right) = \left( \symbf{r}_1 \vee \symbf{r}_2 \right)\left( t \right) = + \begin{cases} + \symbf{r}_1\left( t \right) & t \in \interval{a}{b} \\ + \symbf{r}_2\left( t \right) & t \in \interval{b}{c} + \end{cases} + \end{equation*} + where \(a < b < c\). +\end{definition} +\begin{definition}[Travelling Direction] + The travelling direction of a path \(\symbf{r}\left( t \right)\) is + the direction of increasing \(t\). + A regular curve can be oriented by choosing one of the two + travelling directions. +\end{definition} +\subsubsection{Remarks} +\begin{itemize} + \item A curve is closed/simple/(piecewise) regular if it has a + (closed/simple)/(piecewise) regular parametrisation. + \item The implicit/explicit Cartesian representation of a curve is + a curve, as it describes a set of points. + \item The parametric representation of a curve is a path, as it + includes the timing of the points. + \item Converting from a curve to a path introduces a parameter. + \item Converting from a path to a curve eliminates a parameter. + \item \(\symbf{r}'\left( t \right)\) is the velocity vector of a path. + \item \(\norm*{\symbf{r}'\left( t \right)}\) is the speed of a path. + \item \(\symbf{r}''\left( t \right)\) is the acceleration vector of a path. + \item Parametrisations are not unique. +\end{itemize} +\subsubsection{Reparametrisation} +Let \(\mathscr{C}\) be a curve parametrised by \(\symbf{r}\left( t +\right)\) with \(t \in \interval{a}{b}\). If there exists a bijective +map \(t = \theta\left( u \right)\), defined by \(\theta : +\interval{c}{d} \to \interval{a}{b}\) such that \begin{equation*} - B_r\left( \symbf{p} \right) = \left\{ \symbf{x} \in \R^n : \norm{\symbf{x} - \symbf{p}} < r \right\}. + \symbf{r}\left( \theta\left( u \right) \right) = \tilde{\symbf{r}}\left( u \right), \end{equation*} -This region includes all points less than a distance \(r\) from the vector \(\symbf{p}\), -where the distance is typically defined by the \(L_2\)-norm: +where +\begin{itemize} + \item \(\theta\left( u \right)\) is continuously differentiable, and + \item \(\theta'\left( u \right) \neq 0\) for all \(u \in \interval{c}{d}\), +\end{itemize} +then \(\tilde{\symbf{r}}\left( u \right)\) is a reparametrisation of +\(\symbf{r}\left( t \right)\) for \(u \in \interval{c}{d}\), and a +parametrisation of \(\mathscr{C}\). The map \(\theta\left( u \right)\) +guarantees that the simple/regular properties of \(\symbf{r}\left( t \right)\) +are preserved in \(\tilde{\symbf{r}}\left( u \right)\). +\begin{itemize} + \item If \(\theta'\left( u \right) > 0\), then \(\symbf{r}\left( t + \right)\) and \(\tilde{\symbf{r}}\left( u \right)\) are + \textbf{equivalent} parametrisations. + \item If \(\theta'\left( u \right) < 0\), then \(\symbf{r}\left( t + \right)\) and \(\tilde{\symbf{r}}\left( u \right)\) are + \textbf{opposite} parametrisations. +\end{itemize} +The unit tangent vectors of \(\symbf{r}\left( t \right)\) and +\(\tilde{\symbf{r}}\left( u \right)\) are related by the chain rule: \begin{equation*} - \norm{\symbf{x} - \symbf{p}}_2 = \left( \sum_{i=1}^n \left( x_i - p_i \right)^2 \right)^{1/2}. + \tilde{\symbf{r}}'\left( u \right) = \symbf{r}'\left( \theta\left( u \right) \right) \theta'\left( u \right) \implies \norm*{\tilde{\symbf{r}}'\left( u \right)} = \norm*{\symbf{r}'\left( \theta\left( u \right) \right)} \abs*{\theta'\left( u \right)} \end{equation*} +so that by dividing the first result by the second, we obtain +\begin{align*} + \frac{\symbf{r}'\left( t \right)}{\norm*{\symbf{r}'\left( t \right)}} & = \frac{\tilde{\symbf{r}}'\left( u \right)}{\norm*{\tilde{\symbf{r}}'\left( u \right)}} \frac{\theta'\left( u \right)}{\abs*{\theta'\left( u \right)}} \\ + & = \sign{\left( \theta'\left( u \right) \right)} \frac{\tilde{\symbf{r}}'\left( u \right)}{\norm*{\tilde{\symbf{r}}'\left( u \right)}} \\ + & = + \begin{cases} + \displaystyle\frac{\tilde{\symbf{r}}'\left( u \right)}{\norm*{\tilde{\symbf{r}}'\left( u \right)}} & \text{if } \theta'\left( u \right) > 0 \\[2.5ex] + -\displaystyle\frac{\tilde{\symbf{r}}'\left( u \right)}{\norm*{\tilde{\symbf{r}}'\left( u \right)}} & \text{if } \theta'\left( u \right) < 0 + \end{cases} +\end{align*} \subsection{Mathematical Representation of Curves} -\subsubsection{Explicit Form} -A curve in \(\R^2\) can be represented in explicit form as +\begin{definition}[Degree of Freedom] + The degree of freedom of a curve is the difference between the + number of variables and the number of equations. +\end{definition} +\subsubsection{Explicit Form (2D)} +A curve \(\mathscr{C} \in \R^2\) can be represented explicitly as \begin{equation*} y = f\left( x \right) \end{equation*} -but this is not possible in \(\R^3\) as a 3D curve requires two equations. -For a 2D explicit curve: -\begin{itemize} - \item \(x\) is an independent variable such that we have 1 degree of freedom. -\end{itemize} +When \(n > 2\), it is not possible to describe a curve explicitly. +The degree of freedom for a 2D explicit curve is 1. \subsubsection{Implicit Form} -A curve in \(\R^2\) can be represented in implicit form as -\begin{equation*} - F\left( x,\: y \right) = 0. -\end{equation*} -In 3D, we must impose an additional equation that intersects a surface. +A curve \(\mathscr{C} \in \R^n\) can be represented implicitly as the +intersection of \(n - 1\) surfaces. \begin{equation*} \left\{ \begin{aligned} - F\left( x,\: y,\: z \right) & = 0 \\ - G\left( x,\: y,\: z \right) & = 0 + F_1\left( \symbf{x},\: z \right) & = 0 \\ + F_2\left( \symbf{x},\: z \right) & = 0 \\ + \vdots & \\ + F_{n - 1}\left( \symbf{x},\: z \right) & = 0 \end{aligned} \right. \end{equation*} -In both cases, we have 1 degree of freedom as the degrees of freedom is the -difference between the number of variables and the number of equations. +The degree of freedom is 1. \subsubsection{Parametric Form} -In parametric form, curves are parametrised in terms of a parameter -\(t\). In 2D, this is represented as -\begin{equation*} - \symbf{r}\left( t \right) = \abracket*{x\left( t \right),\: y\left( t \right)} -\end{equation*} -and similarly in 3D, +A curve \(\mathscr{C} \in \R^n\) can be represented parametrically as \begin{equation*} - \symbf{r}\left( t \right) = \abracket*{x\left( t \right),\: y\left( t \right),\: z\left( t \right)} + \symbf{r}\left( t \right) = + \begin{bmatrix} + x_1\left( t \right) \\ + x_2\left( t \right) \\ + \vdots \\ + x_n\left( t \right) + \end{bmatrix} \end{equation*} +where \(t\) is the parameter. The degree of freedom is \(n - 1\). \subsection{Converting Between Representations} \subsubsection{Explicit to Implicit} -The equation \(y = f\left( x \right)\) can always be converted to -implicit form by rewriting it as +The equation \(z = f\left( \symbf{x} \right)\) can always be converted +to implicit form by rewriting it as \begin{equation*} - F\left( x,\: y \right) = y - f\left( x \right) = 0. + F\left( \symbf{x},\: z \right) = z - f\left( \symbf{x} \right) = 0. \end{equation*} \subsubsection{Implicit to Explicit} -The equation \(F\left( x,\: y \right) = 0\) can be converted to -explicit form if we can solve for \(y\) (or \(x\)): +The equation \(F\left( \symbf{x},\: z \right) = 0\) can be converted to +explicit form if we can solve for \(z\) in terms of \(\symbf{x}\). \subsubsection{Parametric to Explicit/Implicit} -The equation \(\symbf{r}\left( t \right) = \abracket*{x\left( t -\right),\: y\left( t \right)}\) can be written in explicit or implicit -form, if the parameter \(t\) can be eliminated from the simultaneous -equations. +The equation \(\symbf{r}\left( t \right) = \abracket*{x_1\left( t +\right),\: \dots,\: x_n\left( t \right)}\) can be written in explicit +or implicit form, if the parameter \(t\) can be eliminated from the +simultaneous equations. \subsubsection{Explicit to Parametric} The equation \(y = f\left( x \right)\) can always be converted to parametric form by choosing the parameter \(t = x\), so that @@ -211,15 +334,15 @@ \subsubsection{Explicit to Parametric} \symbf{r}\left( t \right) = \abracket*{t,\: f\left( t \right)}. \end{equation*} \subsubsection{Implicit to Parametric} -The equation \(F\left( x,\: y \right) = 0\) can be converted to -parametric form if we can find \(x = p\left( t \right)\) and \(y = -q\left( t \right)\), such that \(F\left( p\left( t \right),\: q\left( t -\right) \right) = 0\), and +The equation \(F\left( \symbf{x} \right) = 0\) can be converted to +parametric form if we can find \(\symbf{x} = \symbf{r}\left( t +\right)\) such that \(F\left( \symbf{r}\left( t \right) \right) = 0\), +and \begin{equation*} - \symbf{r}\left( t \right) = \abracket*{p\left( t \right),\: q\left( t \right)} + \symbf{r}\left( t \right) = \abracket*{x_1\left( t \right),\: \dots,\: x_n\left( t \right)}. \end{equation*} for all \(t\). -\subsection{Paramaterisation} +\subsection{Common Parametrisations} To parametrise a curve, consider the following strategies: \begin{itemize} \item For a closed curve, consider the polar parametrisation in @@ -287,15 +410,19 @@ \subsubsection{Velocity Vectors} The velocity vector of a parametrised curve \(\symbf{r}\left( t \right) = \begin{bmatrix} - x\left( t \right) \\ - y\left( t \right) + x_1\left( t \right) \\ + x_2\left( t \right) \\ + \vdots \\ + x_n\left( t \right) \end{bmatrix} \) is defined as \begin{equation*} \symbf{v}\left( t \right) = \symbf{r}'\left( t \right) = \lim_{\Delta t \to 0} \frac{\symbf{r}\left( t + \Delta t \right) - \symbf{r}\left( t \right)}{\Delta t} = \begin{bmatrix} - x'\left( t \right) \\ - y'\left( t \right) + x_1'\left( t \right) \\ + x_2'\left( t \right) \\ + \vdots \\ + x_n'\left( t \right) \end{bmatrix} \end{equation*} where \(\symbf{v}\left( t \right)\) is a tangent vector to the curve at the point \(\symbf{r}\left( t \right)\), for all \(t\). @@ -304,7 +431,7 @@ \subsubsection{Tangent Vectors} vectors of a parametrised curve are unit vectors in the direction of the velocity vector. \begin{equation*} - \hat{\symbf{\tau}}\left( t \right) = \pm \frac{\symbf{v}\left( t \right)}{\norm{\symbf{v}\left( t \right)}} = \pm \hat{\symbf{v}\left( t \right)} + \hat{\symbf{\tau}}\left( t \right) = \pm \frac{\symbf{v}\left( t \right)}{\norm{\symbf{v}\left( t \right)}} = \pm \hat{\symbf{v}}\left( t \right) \end{equation*} For a curve given in explicit form \(y = f\left( x \right)\), the tangent vectors are given by \begin{equation*} @@ -341,7 +468,7 @@ \subsection{Multivariable Functions} \end{equation*} \end{definition} \begin{definition}[Graph] - The graph of \(f\) is the defined as the set + The graph of \(f\) is defined as the set \begin{equation*} G = \left\{ \abracket*{\symbf{x},\: f\left( \symbf{x} \right)} : \symbf{x} \in E \right\} \subset \R^{n + 1} \end{equation*} @@ -367,9 +494,13 @@ \subsection{Curves of Intersection} In 2D, level sets are called \textbf{level curves}. \end{definition} \begin{definition}[Contour Map] - The projection of all level curves onto the \(xy\)-plane is called the \textbf{contour map} of \(f\). T - he lines of a contour map are called \textbf{contours}. + The projection of all level curves onto the \(xy\)-plane is called the \textbf{contour map} of \(f\). + The lines of a contour map are called \textbf{contours}. \end{definition} +\begin{theorem}[Level Sets and Gradients] + The level set of a function \(f\) is perpendicular to the gradient + of \(f\). +\end{theorem} \subsection{Derivatives} \begin{definition}[Continuity] A function \(f\) is continuous at a point \(\symbf{x}_0\) if @@ -413,7 +544,7 @@ \subsection{Derivatives} For an arbitrary \(n\) and \(k \leqslant n\), if all partials of order \(\leqslant \! k\) are continuous in the neighbourhood of \(\symbf{x}_0\), then mixed partials of order \(k\) are equal for any permutation of indices: \begin{equation*} - \pdv[mixed-order=k]{f}{x_{i_1}...,x_{i_k}} = \pdv[mixed-order=k]{f}{x_{j_1}\ldots,x_{j_k}} + \pdv[mixed-order=k]{f}{x_{i_1}\ldots,x_{i_k}} = \pdv[mixed-order=k]{f}{x_{j_1}\ldots,x_{j_k}} \end{equation*} \end{remark} \subsection{Chain Rule} @@ -425,7 +556,7 @@ \subsection{Chain Rule} \odv{}{t} f\left( x\left( t \right),\: y\left( t \right) \right) = \pdv{f}{x} \odv{x}{t} + \pdv{f}{y} \odv{y}{t} \\ \pdv{}{u} f\left( x\left( u,\: v \right),\: y\left( u,\: v \right) \right) = \pdv{f}{x} \pdv{x}{u} + \pdv{f}{y} \pdv{y}{u} \end{gather*} -\begin{definition}[Total derivative] +\begin{definition}[Total Derivative] The total derivative of a function \(f : \R^n \to \R\) is defined as \begin{equation*} \odv{f}{t} = \sum_{i = 1}^n \pdv{f}{x_i} \odv{x_i}{t} = \pdv{f}{x_1} \odv{x_1}{t} + \pdv{f}{x_2} \odv{x_2}{t} + \cdots + \pdv{f}{x_n} \odv{x_n}{t} @@ -465,7 +596,7 @@ \subsection{Chain Rule} \begin{equation*} \odv{f}{t} = \symbf{\nabla} f \cdot \odv{\symbf{x}}{t} \end{equation*} -\subsection{Directional Derivative} +\subsection{Directional Derivatives} The directional derivative of a function \(f : E \subset \R^n \to \R\) at a point \(\symbf{x}_0 \in E\) in the direction of a unit vector \(\hat{\symbf{u}} \in \R^n\) is the slope of \(f\) in the direction of @@ -605,7 +736,7 @@ \subsubsection{Implicit Curves in 2D} \end{bmatrix} \end{equation*} \subsubsection{Implicit Curves in 3D} -Given the intersectin of two implicit curves \(F\left( x,\: y,\: z +Given the intersection of two implicit curves \(F\left( x,\: y,\: z \right) = 0\) and \(G\left( x,\: y,\: z \right) = 0\), the tangent vector along the intersection is given by \begin{equation*} @@ -647,7 +778,7 @@ \subsection{Multivariable Functions} continuous (i.e., \(\pdv{f}{x_i} : \Omega \subset \R^n \to R\) are continuous), then \(f\) is differentiable everywhere in \(\Omega\). \end{theorem} -The first result tells us that the derivative is unique, and that \(f\) is +The first result tells us that the derivative is unique and that \(f\) is differentiable at \(\symbf{a}\) if it has a tangent plane at \(\symbf{a}\). The second result gives us a sufficient condition for differentiability. \subsection{Taylor Series Expansion} @@ -655,7 +786,7 @@ \subsection{Taylor Series Expansion} \(\symbf{a}\) is given by \begin{multline*} f\left( \symbf{x} \right) = f\left( \symbf{a} \right) + \sum_{i = 0}^n \pdv{f\left( \symbf{a} \right)}{x_i} \left( x_i - a_i \right) + \frac{1}{2} \sum_{i = 1}^n \sum_{j = 1}^n \pdv{f\left( \symbf{a} \right)}{x_i,x_j} \left( x_i - a_i \right) \left( x_j - a_j \right) + \cdots \\ - + \frac{1}{k!} \sum_{i_1, \ldots, i_k}^{n} \pdv{f\left( \symbf{a} \right)}{x_{i_1}, \cdots, x_{i_k}} \left( x_{i_1} - a_{i_1} \right) \cdots \left( x_{i_k} - a_{i_k} \right) + \cdots + + \frac{1}{k!} \sum_{i_1, \ldots, i_k}^{n} \pdv{f\left( \symbf{a} \right)}{x_{i_1}, \ldots, x_{i_k}} \left( x_{i_1} - a_{i_1} \right) \cdots \left( x_{i_k} - a_{i_k} \right) + \cdots \end{multline*} This allows us to compute the tangent plane, paraboloid, and so on, of a function \(f\) at \(\symbf{a}\), by increasing the order \(k\) of the @@ -735,7 +866,7 @@ \subsubsection{Nonpositive Functions} \iint_{\R^2} f\left( x,\: y \right) \odif{A} = \int_{-\infty}^\infty \left[ \int_{-\infty}^\infty f\left( x,\: y \right) \odif{x} \right] \odif{y} = \int_{-\infty}^\infty \left[ \int_{-\infty}^\infty f\left( x,\: y \right) \odif{y} \right] \odif{x} \end{equation*} \end{theorem} -\subsection{Measure of a Region} +\subsection{Measures} \begin{definition}[Indicator Function] The indicator function of \(R\) is defined: \begin{equation*} @@ -921,7 +1052,7 @@ \section{Multiple Integrals} triple integrals. \begin{theorem}[Fubini's Theorem in \(n\)-dimensions] If \(f : \R^n \to \R^+\), or \(f : \R^n \to \R\) is integrable, then - all \(n!\) permutations of integrals are equal: + all \(n! \) permutations of integrals are equal: \begin{equation*} \int_{\R^n} f\left( x_1,\: \ldots,\: x_n \right) \odif{x_1} \cdots \odif{x_n} = \int_{-\infty}^\infty \left[ \cdots \left[ \int_{-\infty}^\infty f\left( x_1,\: \ldots,\: x_n \right) \odif{x_1} \right] \cdots \right] \odif{x_n} \end{equation*} @@ -1137,4 +1268,518 @@ \subsubsection*{Almost Equal Functions} \begin{equation*} \int_{\R^n} f\left( \symbf{x} \right) \odif{\symbf{x}} = \int_{\R^n} g\left( \symbf{x} \right) \odif{\symbf{x}} \end{equation*} +\section{Vector Calculus} +A vector-valued multivariable function is a function \(\symbfit{f} : E +\subset \R^n \to \R^m\) that maps a vector \(\symbf{x} \in \R^n\) to +the vector \(\symbfit{f}\left( \symbf{x} \right) \in \R^m\). \(f\) is +continuous if all components of \(\symbfit{f}\) are continuous, and +differentiable if all components of \(\symbfit{f}\) are differentiable. + +If \(\symbfit{f}\) is differentiable at a point \(\symbf{x}_0\), then +it's derivative \(\symbfit{f}'\left( \symbf{x}_0 \right)\) is uniquely +defined by the Jacobian matrix of \(\symbfit{f}\) at \(\symbf{x}_0\): +\begin{equation*} + \symbfit{f}'\left( \symbf{x}_0 \right) = \symbf{\nabla} \symbfit{f}\left( \symbf{x}_0 \right) = \symbf{J}^\top \left( \symbf{x}_0 \right) +\end{equation*} +\begin{definition}[Scalar Field] + If \(m = 1\), then \(f : E \subset \R^n \to \R\) is a scalar field + that associates a scalar value to each point \(\symbf{x} \in E\). +\end{definition} +\begin{definition}[Vector Field] + If \(m = n\), then \(\symbf{F} : E \subset \R^n \to \R^n\) is a + vector field that associates a vector to each point \(\symbf{x} \in E\). + In this case, the Jacobian matrix is a square matrix and the + Jacobian may be defined. +\end{definition} +\begin{definition}[Field Lines] + The field lines of a vector field \(\symbf{F}\) are a family of + curves that are tangent to \(\symbf{F}\) for all \(\symbf{x}\). They + are defined as the solutions to the differential equation + \begin{equation*} + \symbf{r}'\left( t \right) = \symbf{F}\left( \symbf{r}\left( t \right) \right) + \end{equation*} + for all \(t\). When \(m = 2\), + \begin{align*} + \odv{x}{t} & = F_1\left( x,\: y \right) \\ + \odv{y}{t} & = F_2\left( x,\: y \right) + \end{align*} + and a field line may be defined in Cartesian coordinates as the + solution to + \begin{equation*} + \odv{y}{x} = \frac{F_2\left( x,\: y \right)}{F_1\left( x,\: y \right)}. + \end{equation*} +\end{definition} +\begin{lemma} + The field lines of a scalar multiple of a vector field are the same + as the field lines of that vector field. +\end{lemma} +\subsection{Differential Operators on Scalar Fields} +\begin{definition}[Gradient] + The gradient of a scalar field is defined as the derivative of the + scalar field in every direction: + \begin{equation*} + \grad{f} = \symbf{\nabla} f = \symbf{J}^\top = + \begin{bmatrix} + \pdv{f}{\symbf{x}_1} \\ + \vdots \\ + \pdv{f}{\symbf{x}_n} + \end{bmatrix} + \end{equation*} + The gradient measures the rate of change of a scalar field in all + directions at a given point. +\end{definition} +\begin{definition}[Laplacian] + The Laplacian of a scalar field is defined as the divergence of the + gradient: + \begin{equation*} + \symbf{\Delta} f = \divergence{\left( \grad{f} \right)} = \symbf{\nabla}^2 f = \symbf{\nabla} \cdot \symbf{\nabla} f = \sum_{i = 1}^n \pdv[order=2]{f}{\symbf{x}_i} + \end{equation*} + The Laplacian measures the curvature or convexity of the surface + \(z = f\left( \symbf{x} \right)\). +\end{definition} +\subsection{Differential Operators on Vector Fields} +\begin{definition}[Divergence] + The divergence of a vector field is defined as the dot product of + the gradient and the vector field: + \begin{equation*} + \divergence{\symbf{F}} = \symbf{\nabla} \cdot \symbf{F} = \pdv{\symbf{F}_1}{\symbf{x}_1} + \cdots + \pdv{\symbf{F}_n}{\symbf{x}_n} = \sum_{i = 1}^n \pdv{\symbf{F}_i}{\symbf{x}_i}. + \end{equation*} + Divergence measures the expansion of a vector field at a given point. +\end{definition} +\begin{itemize} + \item \(\divergence{\symbf{F}}\left( \symbf{x}_0 \right) > 0\) + indicates that \(\symbf{x}_0\) is a source, and the vector field is + diverging out from \(\symbf{x}_0\). + \item \(\divergence{\symbf{F}}\left( \symbf{x}_0 \right) < 0\) + indicates that \(\symbf{x}_0\) is a sink, and the vector field is + converging into \(\symbf{x}_0\). + \item \(\divergence{\symbf{F}}\left( \symbf{x}_0 \right) = 0\) + indicates that the net flow of the vector field at \(\symbf{x}_0\) is zero. +\end{itemize} +\begin{definition}[Curl] + The curl of a vector field is defined as the cross product of the + gradient and the vector field: + \begin{equation*} + \curl{\symbf{F}} = \symbf{\nabla} \times \symbf{F} = + \begin{vmatrix} + \symbf{i} & \symbf{j} & \symbf{k} \\ + \displaystyle\pdv{}{\symbf{x}_1} & \displaystyle\pdv{}{\symbf{x}_2} & \displaystyle\pdv{}{\symbf{x}_3} \\ + \symbf{F}_1 & \symbf{F}_2 & \symbf{F}_3 \\ + \end{vmatrix} + = + \begin{vmatrix} + \displaystyle\pdv{}{\symbf{x}_2} & \displaystyle\pdv{}{\symbf{x}_3} \\ + \symbf{F}_2 & \symbf{F}_3 \\ + \end{vmatrix} + \symbf{i} + - + \begin{vmatrix} + \displaystyle\pdv{}{\symbf{x}_1} & \displaystyle\pdv{}{\symbf{x}_3} \\ + \symbf{F}_1 & \symbf{F}_3 \\ + \end{vmatrix} + \symbf{j} + + + \begin{vmatrix} + \displaystyle\pdv{}{\symbf{x}_1} & \displaystyle\pdv{}{\symbf{x}_2} \\ + \symbf{F}_1 & \symbf{F}_2 \\ + \end{vmatrix} + \symbf{k} + \end{equation*} + Curl measures the rotation of a vector field at a given point. + \begin{itemize} + \item \(\curl{\symbf{F}}\left( \symbf{x}_0 \right) > 0\) + indicates that the vector field is rotating anticlockwise about + \(\symbf{x}_0\). + \item \(\curl{\symbf{F}}\left( \symbf{x}_0 \right) < 0\) + indicates that the vector field is rotating clockwise about + \(\symbf{x}_0\). + \item \(\curl{\symbf{F}}\left( \symbf{x}_0 \right) = 0\) + indicates that the net rotation about \(\symbf{x}_0\) is zero. + \end{itemize} +\end{definition} +\subsection{Conservative Fields} +A vector field \(\symbf{F}\) is conservative if it is the gradient of a +potential function \(\phi\): +\begin{equation*} + \symbf{F} = \symbf{\nabla} \phi. +\end{equation*} +Such a vector field represents a force field in which the total energy is +conserved. + +The contours of the scalar field \(\phi\) are called equipotential +lines (\(\phi\) is constant). These lines are perpendicular to the +field lines of \(\symbf{F}\). +\begin{equation*} + \left( \symbf{F} = \symbf{\nabla} \phi \right) \perp \left( \phi = \text{constant} \right) +\end{equation*} +This is because the contours of a scalar field are defined to be +perpendicular to the gradient of that scalar field. +\subsection{Line Integrals} +A line integral is an integral where the function to be integrated is +evaluated along a curve. This function may be a scalar field or a +vector field. Line integrals can be interpreted as a measure of the +total effect of a function along a curve. + +To evaluate line integrals, we must define a parametrisation of the arc +length of the curve. +\subsubsection{Arc Length} +Arc length is the distance travelled along a path or curve. When +\(\symbf{r}\left( t \right)\) is a path, the arc length between +\(\symbf{r}\left( a \right)\) to \(\symbf{r}\left( t \right)\) is given +by +\begin{equation*} + s\left( t \right) = \int_a^t \norm*{\symbf{r}'\left( \tau \right)} \odif{\tau} +\end{equation*} +where the integrand can be interpreted as the product of the speed of +along the path \(\symbf{r}\left( t \right)\) with a small time interval +\(\odif{\tau}\). +The length \(L\) of a path is therefore +\begin{equation*} + L = s\left( b \right) = \int_a^b \norm*{\symbf{r}'\left( \tau \right)} \odif{\tau}. +\end{equation*} +\begin{theorem}[Arc Length Reparametrisation] + A (piecewise) regular curve \(\mathscr{C}\) can always be + reparametrised by the arc length parametrisation \(s\left( t \right)\): + \begin{equation*} + \tilde{\symbf{r}}\left( s \right) = \symbf{r}\left( \theta\left( s \right) \right) + \end{equation*} + for \(s \in \interval{0}{L}\). +\end{theorem} +\begin{proof} + As \(r'\left( t \right) \neq 0\), \(\norm*{r'\left( t \right)} > 0\), + and therefore, \(s\left( t \right)\) is a strictly increasing function. + This means that \(s\left( t \right)\) is invertible, and therefore, + \(t = \theta\left( s \right)\) is a bijective map between \(\interval{0}{L}\) + and \(\interval{a}{b}\). +\end{proof} +\begin{remark} + The rate of change of \(s\left( t \right)\) is the speed of the + path \(\symbf{r}\left( t \right)\): + \begin{equation*} + \odv{s}{t} = \odv{}{t} \left[ \int_a^t \norm*{\symbf{r}'\left( \tau \right)} \odif{\tau} \right] = \norm*{\symbf{r}'\left( t \right)}. + \end{equation*} +\end{remark} +\begin{remark} + The speed of a path parametrised by arc length is always \(1\): + \begin{equation*} + \norm*{\tilde{\symbf{r}}'\left( s \right)} = \norm*{\symbf{r}'\left( \theta\left( s \right) \right)} \abs*{\odv{\theta\left( s \right)}{s}} = \norm*{\symbf{r}'\left( \theta\left( s \right) \right)} \frac{1}{\abs*{\odv{s}{\theta\left( s \right)}}} = \norm*{\symbf{r}'\left( \theta\left( s \right) \right)} \frac{1}{\norm*{\symbf{r}'\left( \theta\left( s \right) \right)}} = 1. + \end{equation*} +\end{remark} +\subsubsection{Line Integral of a Scalar Field} +Let \(\mathscr{C}\) be a simple and piecewise regular curve +parametrised by \(\symbf{r}\left( t \right)\) with \(t \in +\interval{a}{b}\). The line integral of a scalar field \(f\) along +\(\mathscr{C}\) is defined as +\begin{equation*} + \int_{\mathscr{C}} f \odif{s} = \int_a^b f\left( \symbf{r}\left( t \right) \right) \norm*{\symbf{r}'\left( t \right)} \odif{t} +\end{equation*} +where \(\odif{s} = \norm*{\symbf{r}'\left( t \right)} \odif{t}\) is the +differential arc length element along \(\mathscr{C}\). + +This integral represents the weighted sum of \(f\) along +\(\mathscr{C}\), where the weight is the speed of the path +\(\symbf{r}\left( t \right)\). This line integral can be interpreted as +the area under the surface defined by \(f\) along \(\mathscr{C}\). +\begin{lemma}[Equivalence of Parametrisations] + Let \(\tilde{\symbf{r}}\left( u \right)\) be a + reparametrisation of \(\symbf{r}\left( t \right)\) with \(u \in + \interval{c}{d}\). Then + \begin{equation*} + \int_{\mathscr{C}} f \odif{s} = \int_c^d f\left( \tilde{\symbf{r}}\left( u \right) \right) \norm*{\tilde{\symbf{r}}'\left( u \right)} \odif{u} + \end{equation*} + and the line integral is independent of the parametrisation of + \(\mathscr{C}\). +\end{lemma} +\begin{proof} + Consider the parametrisation \(t = \theta\left( u \right)\), + where the differential time element + \(\odif{t} = \abs*{\theta'\left( u \right)} \odif{u}\), by the + Change of Variables property. Then + \begin{align*} + \int_{\mathscr{C}} f \odif{s} & = \int_a^b f\left( \symbf{r}\left( t \right) \right) \norm*{\symbf{r}'\left( t \right)} \odif{t} \\ + & = \int_c^d f\left( \symbf{r}\left( \theta\left( u \right) \right) \right) \norm*{\symbf{r}'\left( \theta\left( u \right) \right)} \abs*{\theta'\left( u \right)} \odif{u} \\ + & = \int_c^d f\left( \tilde{\symbf{r}}\left( u \right) \right) \norm*{\tilde{\symbf{r}}'\left( u \right)} \odif{u}. + \end{align*} +\end{proof} +\subsubsection{Line Integral of a Vector Field} +Let \(\mathscr{C}\) be a simple and piecewise regular curve +parametrised by \(\symbf{r}\left( t \right)\) with \(t \in +\interval{a}{b}\). The line integral of a vector field \(\symbf{F}\) +along \(\mathscr{C}\) is defined as +\begin{equation*} + \int_{\mathscr{C}} \symbf{F} \cdot \odif{\symbf{r}} = \int_a^b \symbf{F}\left( \symbf{r}\left( t \right) \right) \cdot \symbf{r}'\left( t \right) \odif{t} +\end{equation*} +where \(\odif{\symbf{r}} = \symbf{r}'\left( t \right) \odif{t}\) is the +differential path element along \(\mathscr{C}\). + +This integral represents the weighted sum of \(\symbf{F}\) along +\(\mathscr{C}\), where the weight is the component of the velocity of +the path \(\symbf{r}\left( t \right)\) in the direction of +\(\symbf{F}\). This line integral can be interpreted as the work done +by the force field \(\symbf{F}\) along \(\mathscr{C}\). +\begin{lemma}[Equivalence of Parametrisations] + Let \(\tilde{\symbf{r}}\left( u \right)\) be a + reparametrisation of \(\symbf{r}\left( t \right)\) with \(u \in + \interval{c}{d}\). Then + \begin{equation*} + \int_{\mathscr{C}} \symbf{F} \cdot \odif{\symbf{r}} = + \begin{cases} + \displaystyle\int_c^d \symbf{F}\left( \tilde{\symbf{r}}\left( u \right) \right) \cdot \tilde{\symbf{r}}'\left( u \right) \odif{u} & \text{if } \theta'\left( u \right) > 0 \\[2.5ex] + -\displaystyle\int_c^d \symbf{F}\left( \tilde{\symbf{r}}\left( u \right) \right) \cdot \tilde{\symbf{r}}'\left( u \right) \odif{u} & \text{if } \theta'\left( u \right) < 0 + \end{cases} + \end{equation*} + and the line integral is independent of the parametrisation of + \(\mathscr{C}\). +\end{lemma} +\begin{proof} + Consider the parametrisation \(t = \theta\left( u \right)\), + where the differential time element + \(\odif{t} = \abs*{\theta'\left( u \right)} \odif{u}\), by the + Change of Variables property. Then + \begin{align*} + \int_{\mathscr{C}} \symbf{F} \cdot \odif{\symbf{r}} & = \int_a^b \symbf{F}\left( \symbf{r}\left( t \right) \right) \cdot \symbf{r}'\left( t \right) \odif{t} \\ + & = + \begin{cases} + \displaystyle\int_c^d \symbf{F}\left( \symbf{r}\left( \theta\left( u \right) \right) \right) \cdot \symbf{r}'\left( \theta\left( u \right) \right) \abs*{\theta'\left( u \right)} \odif{u} & \text{if } \theta'\left( u \right) > 0 \\[2.5ex] + -\displaystyle\int_c^d \symbf{F}\left( \symbf{r}\left( \theta\left( u \right) \right) \right) \cdot \symbf{r}'\left( \theta\left( u \right) \right) \abs*{\theta'\left( u \right)} \odif{u} & \text{if } \theta'\left( u \right) < 0 + \end{cases} + \\ + & = + \begin{cases} + \displaystyle\int_c^d \symbf{F}\left( \tilde{\symbf{r}}\left( u \right) \right) \cdot \tilde{\symbf{r}}'\left( u \right) \odif{u} & \text{if } \theta'\left( u \right) > 0 \\[2.5ex] + -\displaystyle\int_c^d \symbf{F}\left( \tilde{\symbf{r}}\left( u \right) \right) \cdot \tilde{\symbf{r}}'\left( u \right) \odif{u} & \text{if } \theta'\left( u \right) < 0 + \end{cases} + \end{align*} +\end{proof} +\begin{corollary}[Line Integrals in the Reverse Direction] + Taking the line integral of a path in the reverse direction is + equivalent to negating the line integral over the original path. + \begin{align*} + \int_{-\mathscr{C}} \symbf{F} \cdot \odif{\symbf{r}} & = -\int_{\mathscr{C}} \symbf{F} \cdot \odif{\symbf{r}} + \end{align*} + where \(-\mathscr{C}\) is the curve parametrised by + \(\symbf{r}\left( a + b - t \right)\). +\end{corollary} +\subsubsection{Relationship between Line Integrals over Scalar and Vector Fields} +Line integrals over scalar fields may be evaluated with respect to a +single variable \(x_i\): +\begin{equation*} + \int_{\mathscr{C}} f \odif{x_i} = \int_a^b f\left( \symbf{r}\left( t \right) \right) \symbf{r}'_i\left( t \right) \odif{t} +\end{equation*} +where \(\odif{x_i} = \symbf{r}'_i\left( t \right) \odif{t}\) is the +differential element in the direction of \(x_i\) along \(\mathscr{C}\). +By adding the line integrals over all \(x_i\), we obtain +\begin{equation*} + \int_{\mathscr{C}} \left( \symbf{F}_1 \odif{x_1} + \cdots + \symbf{F}_n \odif{x_n} \right) = \int_a^b \left( \symbf{F}_1 \symbf{r}'_1 \odif{t} + \cdots + \symbf{F}_n \symbf{r}'_n \odif{t} \right) = \int_a^b \symbf{F}\left( \symbf{r}\left( t \right) \right) \cdot \symbf{r}'\left( t \right) \odif{t} = \int_{\mathscr{C}} \symbf{F} \cdot \odif{\symbf{r}}. +\end{equation*} +Line integrals over vector fields may be evaluated using the unit +tangent vector of the path \(\symbf{r}\left( t \right)\), so that +\begin{equation*} + \int_{\mathscr{C}} \symbf{F} \cdot \odif{\symbf{r}} = \int_a^b \symbf{F}\left( \symbf{r}\left( t \right) \right) \cdot \frac{\symbf{r}'\left( t \right)}{\norm*{\symbf{r}'\left( t \right)}} \norm*{\symbf{r}'\left( t \right)} \odif{t} = \int_a^b \underbrace{\symbf{F}\left( \symbf{r}\left( t \right) \right) \cdot \hat{\symbf{\tau}}\left( t \right)}_f \norm*{\symbf{r}'\left( t \right)} \odif{t} = \int_{\mathscr{C}} f \odif{s}. +\end{equation*} +\subsubsection{Line Integrals over Multiple Paths} +Let \(\mathscr{C}_1\) and \(\mathscr{C}_2\) be two simple and piecewise +regular curves parametrised by \(\symbf{r}_1 : \interval{a}{b} \subset +\R \to \R^n\) and \(\symbf{r}_2 : \interval{b}{c} \subset \R \to \R^n\) +respectively, such that \(\symbf{r}_1\left( b \right) = +\symbf{r}_2\left( b \right)\). The line integrals over the +concatenation of \(\mathscr{C}_1\) and \(\mathscr{C}_2\) are defined as +\begin{align*} + \int_{\mathscr{C}_1 + \mathscr{C}_2} f \odif{s} & = \int_{\mathscr{C}_1} f \odif{s} + \int_{\mathscr{C}_2} f \odif{s} \\ + \int_{\mathscr{C}_1 + \mathscr{C}_2} \symbf{F} \cdot \odif{\symbf{r}} & = \int_{\mathscr{C}_1} \symbf{F} \cdot \odif{\symbf{r}} + \int_{\mathscr{C}_2} \symbf{F} \cdot \odif{\symbf{r}}. +\end{align*} +\subsubsection{Fundamental Theorem of Line Integrals} +Let \(\symbf{F} = \symbf{\nabla} \phi\) be a conservative vector field +and \(\mathscr{C}\) be a simple and piecewise regular curve +parametrised by \(\symbf{r}\left( t \right)\) with \(t \in +\interval{a}{b}\). Then +\begin{equation*} + \int_{\mathscr{C}} \symbf{F} \cdot \odif{\symbf{r}} = \int_{\mathscr{C}} \symbf{\nabla} \phi \cdot \odif{\symbf{r}} = \phi\left( \symbf{b} \right) - \phi\left( \symbf{a} \right) +\end{equation*} +where \(\symbf{a} = \symbf{r}\left( a \right)\) and \(\symbf{b} = \symbf{r}\left( b \right)\). +This result demonstrates that a line integral in a conservative field +is path independent, and depends only on the endpoints of the path. +\subsubsection{Circulation} +Let \(\mathscr{C}\) be a simple closed curve parametrised by +\(\symbf{r}\left( t \right)\) with \(t \in \interval{a}{b}\), where +\(\symbf{r}\left( a \right) = \symbf{r}\left( b \right)\). The line +integral of a vector field \(\symbf{F}\) around \(\mathscr{C}\) is +called the circulation of \(\symbf{F}\) around \(\mathscr{C}\), and is +notated by a circle inside the integral sign: +\begin{equation*} + \Gamma = \oint_{\mathscr{C}} \symbf{F} \cdot \odif{\symbf{r}}. +\end{equation*} +\subsubsection{Theorems in Conservative Fields} +Let \(\symbf{F}\) be a continuous conservative vector field in an open +and connected region \(\Omega \subset \R^n\). Then the following +theorems hold: +\begin{theorem}[Circulation in a Conservative Field]\label{thm:circulation_in_conservative_field} + The circulation of \(\symbf{F}\) around any closed path + \(\mathscr{C} \in \Omega\) is zero: + \begin{equation*} + \oint_{\mathscr{C}} \symbf{F} \cdot \odif{\symbf{r}} = 0. + \end{equation*} +\end{theorem} +\begin{proof} + Let \(\mathscr{C}\) be a closed path parametrised by + \(\symbf{r}\left( t \right)\) with \(t \in \interval{a}{b}\), where + \(\symbf{r}\left( a \right) = \symbf{r}\left( b \right)\). Then + \begin{equation*} + \oint_{\mathscr{C}} \symbf{F} \cdot \odif{\symbf{r}} = \oint_a^b \symbf{F}\left( \symbf{r}\left( t \right) \right) \cdot \symbf{r}'\left( t \right) \odif{t} = \oint_a^b \symbf{\nabla} \phi\left( \symbf{r}\left( t \right) \right) \cdot \symbf{r}'\left( t \right) \odif{t} = \phi\left( \symbf{r}\left( b \right) \right) - \phi\left( \symbf{r}\left( a \right) \right) = 0. + \end{equation*} +\end{proof} +\begin{theorem}[Path Independence in a Conservative Field] + The line integral of \(\symbf{F}\) between two points \(\symbf{a}\) + and \(\symbf{b}\) in \(\Omega\) is independent of the path connecting + \(\symbf{a}\) and \(\symbf{b}\): + \begin{equation*} + \int_{\mathscr{C}_1} \symbf{F} \cdot \odif{\symbf{r}} = \int_{\mathscr{C}_2} \symbf{F} \cdot \odif{\symbf{r}}. + \end{equation*} +\end{theorem} +\begin{proof} + Let \(\mathscr{C}_1\) and \(\mathscr{C}_2\) be two different paths + parametrised by \(\symbf{r}_1\left( t \right)\) and \(\symbf{r}_2\left( t \right)\) + respectively, with \(t \in \interval{a}{b}\), where + \(\symbf{r}_1\left( a \right) = \symbf{r}_2\left( a \right)\) and + \(\symbf{r}_1\left( b \right) = \symbf{r}_2\left( b \right)\). + Consider the opposite reparametrisation of \(\symbf{r}_2\left( t \right)\) + such that the concatenation of \(\symbf{r}_1\) and \(\tilde{\symbf{r}}_2\) + is a closed path. Then, by Theorem~\ref{thm:circulation_in_conservative_field}, + \begin{equation*} + 0 = \oint_{\mathscr{C}_1 - \mathscr{C}_2} \symbf{F} \cdot \odif{\symbf{r}} = \int_{\mathscr{C}_1} \symbf{F} \cdot \odif{\symbf{r}} + \int_{-\mathscr{C}_2} \symbf{F} \cdot \odif{\symbf{r}} = \int_{\mathscr{C}_1} \symbf{F} \cdot \odif{\symbf{r}} - \int_{\mathscr{C}_2} \symbf{F} \cdot \odif{\symbf{r}} + \end{equation*} + which implies that + \begin{equation*} + \int_{\mathscr{C}_1} \symbf{F} \cdot \odif{\symbf{r}} = \int_{\mathscr{C}_2} \symbf{F} \cdot \odif{\symbf{r}}. + \end{equation*} +\end{proof} +Using this theorem, it is possible to show that a conservative field +is the gradient of a potential function. +\begin{proof}[Proof for the Definition of a Conservative Field] + Consider the line integral of \(\symbf{F}\) from a fixed point + \(\symbf{x}_0\) to a variable point \(\symbf{x}\) in \(\Omega\): + \begin{equation*} + \phi\left( \symbf{x} \right) = \int_{\symbf{x}_0}^{\symbf{x}} \symbf{F} \cdot \odif{\symbf{r}}. + \end{equation*} + The derivative of \(\phi\left( \symbf{x} \right)\) with + respect to \(\symbf{x}_i\) is given by + \begin{align*} + \pdv{\phi}{\symbf{x}_i} & = \lim_{h \to 0} \frac{1}{h} \left[ \phi\left( \symbf{x} + h \symbf{e}_i \right) - \phi\left( \symbf{x} \right) \right] \\ + & = \lim_{h \to 0} \frac{1}{h} \left[ \int_{\symbf{x}_0}^{\symbf{x} + h \symbf{e}_i} \symbf{F} \cdot \odif{\symbf{r}} - \int_{\symbf{x}_0}^{\symbf{x}} \symbf{F} \cdot \odif{\symbf{r}} \right] \\ + & = \lim_{h \to 0} \frac{1}{h} \int_{\symbf{x}}^{\symbf{x} + h \symbf{e}_i} \symbf{F} \cdot \odif{\symbf{r}} + \end{align*} + The path of integration is a straight line from \(\symbf{x}\) to + \(\symbf{x} + h \symbf{e}_i\), and can be parametrised by + \(\symbf{r}\left( t \right) = \symbf{x} + t h \symbf{e}_i\) for + \(t \in \interval{0}{1}\). Then + \begin{align*} + \pdv{\phi}{\symbf{x}_i} & = \lim_{h \to 0} \frac{1}{h} \int_0^1 \symbf{F}\left( \symbf{r}\left( t \right) \right) \cdot \symbf{r}'\left( t \right) \odif{t} \\ + & = \lim_{h \to 0} \frac{1}{h} \int_0^1 \symbf{F}\left( \symbf{x} + t h \symbf{e}_i \right) \cdot h \symbf{e}_i \odif{t} \\ + & = \lim_{h \to 0} \int_0^1 \symbf{F}\left( \symbf{x} + t h \symbf{e}_i \right) \odif{t} + \end{align*} + As the integrand is a continuous function of \(t\) and \(\symbf{x}\), + the mean value theorem for integrals implies that there exists a + time \(t_0 \in \interval{0}{1}\) where the integral is equal to the + mean value of \(\symbf{F}\) on \(\interval{0}{1}\): + \begin{equation*} + \pdv{\phi}{\symbf{x}_i} = \lim_{h \to 0} \symbf{F}\left( \symbf{x} + t_0 h \symbf{e}_i \right) \cdot \symbf{e}_i = \symbf{F}\left( \symbf{x} \right) \cdot \symbf{e}_i. + \end{equation*} + Without loss of generality, this can be extended to all \(i \in \interval{1}{n}\), + so that: + \begin{equation*} + \symbf{F}\left( \symbf{x} \right) = \symbf{\nabla} \phi\left( \symbf{x} \right). + \end{equation*} +\end{proof} +\begin{theorem}[Antiderivative of a Conservative Field] + The line integral of \(\symbf{F}\) from a fixed point \(\symbf{x}_0\) + to a variable point \(\symbf{x}\) in \(\Omega\) is precisely the + potential function \(\phi\left( \symbf{x} \right)\) evaluated at + \(\symbf{x}\): + \begin{equation*} + \phi\left( \symbf{x} \right) = \int_{\symbf{x}_0}^{\symbf{x}} \symbf{F} \cdot \odif{\symbf{r}}. + \end{equation*} +\end{theorem} +\begin{remark} + When \(\symbf{F}\) is not a conservative field, the line integral + may still be evaluated, but the result will not be path independent. +\end{remark} +\begin{theorem}[Curl Criterion for Conservative Fields] + For \(n = 2\) and \(n = 3\), if \(\symbf{F} \subset C^1\) is a + continuous vector field in an open and simply-connected region + \(\Omega \subset \R^n\), then + \begin{equation*} + \left( \exists \phi \subset C^2 : \symbf{F} = \symbf{\nabla}\phi \right) \iff \symbf{\nabla} \times \symbf{F} = \symbf{0}. + \end{equation*} +\end{theorem} +\begin{proof} + The forward direction uses Schwartz theorem: + \begin{equation*} + \left( \exists \phi \subset C^2 : \symbf{F} = \symbf{\nabla}\phi \right) \implies \symbf{\nabla} \times \symbf{F} = \symbf{\nabla} \times \symbf{\nabla} \phi = + \begin{bmatrix} + \partial_x \\ + \partial_y \\ + \partial_z + \end{bmatrix} + \times + \begin{bmatrix} + \phi_x \\ + \phi_y \\ + \phi_z + \end{bmatrix} + = + \begin{bmatrix} + \phi_{zy} - \phi_{yz} \\ + -\left( \phi_{zx} - \phi_{xz} \right) \\ + \phi_{yx} - \phi_{xy} + \end{bmatrix} + = \symbf{0}. + \end{equation*} +\end{proof} +\subsubsection{Energy Theorems} +\begin{theorem}[Work Energy Theorem] + The total work done by a field \(\symbf{F}\) on a particle moving + along a curve \(\mathscr{C}\) from point \(\symbf{a}\) to \(\symbf{b}\) + is equal to the change in kinetic energy \(T\) of the particle along + \(\mathscr{C}\): + \begin{equation*} + W = T_{\symbf{b}} - T_{\symbf{a}}. + \end{equation*} +\end{theorem} +\begin{proof} + From Newton's Second Law + \begin{align*} + m \symbf{r}''\left( t \right) & = \symbf{F}\left( \symbf{r}\left( t \right) \right) \\ + m \symbf{r}'\left( t \right) \cdot \symbf{r}''\left( t \right) & = \symbf{F}\left( \symbf{r}\left( t \right) \right) \cdot \symbf{r}'\left( t \right) \\ + \int_a^b m \symbf{r}'\left( t \right) \cdot \symbf{r}''\left( t \right) \odif{t} & = \int_a^b \symbf{F}\left( \symbf{r}\left( t \right) \right) \cdot \symbf{r}'\left( t \right) \odif{t} \\ + \int_a^b \odv*{\left[ \frac{1}{2} m \symbf{r}'\left( t \right) \cdot \symbf{r}'\left( t \right) \right]}{t} \odif{t} & = \int_{\mathscr{C}} \symbf{F} \cdot \odif{\symbf{r}} \\ + \int_a^b \odv*{\left[ \frac{1}{2} m \norm*{\symbf{v}\left( t \right)}^2 \right]}{t} \odif{t} & = \int_{\mathscr{C}} \symbf{F} \cdot \odif{\symbf{r}} \\ + \frac{1}{2} m \norm*{\symbf{v}\left( b \right)}^2 - \frac{1}{2} m \norm*{\symbf{v}\left( a \right)}^2 & = \int_{\mathscr{C}} \symbf{F} \cdot \odif{\symbf{r}} \\ + T_{\symbf{b}} - T_{\symbf{a}} & = W + \end{align*} +\end{proof} +\begin{definition}[Total Energy] + The total energy of a particle is given by \(E = T + V\), where \(V\) + is the potential energy of conservative forces acting on the particle: + \(\symbf{F}_{\text{c}} = -\symbf{\nabla} V\). +\end{definition} +\begin{theorem}[Conservation of Energy] + The work done by non-conservative forces on a particle moving along + a curve \(\mathscr{C}\) from point \(\symbf{a}\) to \(\symbf{b}\) is + equal to the change in total energy of the particle along \(\mathscr{C}\): + \begin{equation*} + W_{\mathrm{nc}} = E_{\symbf{b}} - E_{\symbf{a}}. + \end{equation*} +\end{theorem} +\begin{proof} + From the Work Energy Theorem, + \begin{align*} + T_{\symbf{b}} - T_{\symbf{a}} & = W \\ + T_{\symbf{b}} - T_{\symbf{a}} & = \int_{\mathscr{C}} \left( \symbf{F}_{\text{c}} + \symbf{F}_{\text{nc}} \right) \cdot \odif{\symbf{r}} \\ + T_{\symbf{b}} - T_{\symbf{a}} & = \int_{\mathscr{C}} \left( -\symbf{\nabla} V + \symbf{F}_{\text{nc}} \right) \cdot \odif{\symbf{r}} \\ + T_{\symbf{b}} - T_{\symbf{a}} & = -\int_{\mathscr{C}} \symbf{\nabla} V \cdot \odif{\symbf{r}} + \int_{\mathscr{C}} \symbf{F}_{\text{nc}} \cdot \odif{\symbf{r}} \\ + T_{\symbf{b}} - T_{\symbf{a}} & = -\left( V_{\symbf{b}} - V_{\symbf{a}} \right) + \int_{\mathscr{C}} \symbf{F}_{\text{nc}} \cdot \odif{\symbf{r}} \\ + \left( T_{\symbf{b}} + V_{\symbf{b}} \right) - \left( T_{\symbf{a}} + V_{\symbf{a}} \right) & = \int_{\mathscr{C}} \symbf{F}_{\text{nc}} \cdot \odif{\symbf{r}} \\ + E_{\symbf{b}} - E_{\symbf{a}} & = W_{\text{nc}} + \end{align*} +\end{proof} \end{document}