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<h1 class="quarto-secondary-nav-title"><span class="chapter-number">63</span> <span class="chapter-title">Quick Review of Vector Calculus</span></h1>
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<a href="../precalc/calculator.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">1</span> <span class="chapter-title">From calculator to computer</span></a>
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<a href="../precalc/plotting.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">8</span> <span class="chapter-title">The Graph of a Function</span></a>
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<a href="../precalc/transformations.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">9</span> <span class="chapter-title">Function manipulations</span></a>
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<a href="../precalc/polynomial.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">11</span> <span class="chapter-title">Polynomials</span></a>
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<a href="../derivatives/linearization.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">29</span> <span class="chapter-title">Linearization</span></a>
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|
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||
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<a href="../integrals/center_of_mass.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">44</span> <span class="chapter-title">Center of Mass</span></a>
|
||
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|
||
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|
||
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||
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<a href="../integrals/volumes_slice.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">45</span> <span class="chapter-title">Volumes by slicing</span></a>
|
||
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||
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<a href="../integrals/arc_length.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">46</span> <span class="chapter-title">Arc length</span></a>
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<a href="../integrals/surface_area.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">47</span> <span class="chapter-title">Surface Area</span></a>
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<a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-5" aria-expanded="false">ODEs</a>
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<a href="../ODEs/euler.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">49</span> <span class="chapter-title">Euler’s method</span></a>
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<a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-6" aria-expanded="false">Differential vector calculus</a>
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<a href="../differentiable_vector_calculus/vector_valued_functions.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">54</span> <span class="chapter-title">Vector-valued functions, <span class="math inline">\(f:R \rightarrow R^n\)</span></span></a>
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|
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<a href="../differentiable_vector_calculus/scalar_functions_applications.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">56</span> <span class="chapter-title">Applications with scalar functions</span></a>
|
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||
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<a href="../differentiable_vector_calculus/vector_fields.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">57</span> <span class="chapter-title">Functions <span class="math inline">\(R^n \rightarrow R^m\)</span></span></a>
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||
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|
||
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<a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-7" aria-expanded="true">Integral vector calculus</a>
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<h2 id="toc-title">Table of contents</h2>
|
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<ul>
|
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<li><a href="#limits" id="toc-limits" class="nav-link active" data-scroll-target="#limits"> <span class="header-section-number">63.1</span> Limits</a></li>
|
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<li><a href="#derivatives" id="toc-derivatives" class="nav-link" data-scroll-target="#derivatives"> <span class="header-section-number">63.2</span> Derivatives</a>
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<ul class="collapse">
|
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<li><a href="#parameterized-curves" id="toc-parameterized-curves" class="nav-link" data-scroll-target="#parameterized-curves"> <span class="header-section-number">63.2.1</span> Parameterized curves</a></li>
|
||
<li><a href="#scalar-functions" id="toc-scalar-functions" class="nav-link" data-scroll-target="#scalar-functions"> <span class="header-section-number">63.2.2</span> Scalar functions</a></li>
|
||
<li><a href="#vector-valued-functions" id="toc-vector-valued-functions" class="nav-link" data-scroll-target="#vector-valued-functions"> <span class="header-section-number">63.2.3</span> Vector-valued functions</a></li>
|
||
<li><a href="#the-divergence-curl-and-their-vanishing-properties" id="toc-the-divergence-curl-and-their-vanishing-properties" class="nav-link" data-scroll-target="#the-divergence-curl-and-their-vanishing-properties"> <span class="header-section-number">63.2.4</span> The divergence, curl, and their vanishing properties</a></li>
|
||
</ul></li>
|
||
<li><a href="#integrals" id="toc-integrals" class="nav-link" data-scroll-target="#integrals"> <span class="header-section-number">63.3</span> Integrals</a>
|
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<ul class="collapse">
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<li><a href="#line-integrals" id="toc-line-integrals" class="nav-link" data-scroll-target="#line-integrals"> <span class="header-section-number">63.3.1</span> Line integrals</a></li>
|
||
<li><a href="#surface-integrals" id="toc-surface-integrals" class="nav-link" data-scroll-target="#surface-integrals"> <span class="header-section-number">63.3.2</span> Surface integrals</a></li>
|
||
<li><a href="#stokes-theorem-divergence-theorem" id="toc-stokes-theorem-divergence-theorem" class="nav-link" data-scroll-target="#stokes-theorem-divergence-theorem"> <span class="header-section-number">63.3.3</span> Stokes’ theorem, divergence theorem</a></li>
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<div class="quarto-title">
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<h1 class="title d-none d-lg-block"><span class="chapter-number">63</span> <span class="chapter-title">Quick Review of Vector Calculus</span></h1>
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<p>This section considers functions from <span class="math inline">\(R^n\)</span> into <span class="math inline">\(R^m\)</span> where one or both of <span class="math inline">\(n\)</span> or <span class="math inline">\(m\)</span> is greater than <span class="math inline">\(1\)</span>:</p>
|
||
<ul>
|
||
<li>functions <span class="math inline">\(f:R \rightarrow R^m\)</span> are called univariate functions.</li>
|
||
<li>functions <span class="math inline">\(f:R^n \rightarrow R\)</span> are called scalar-valued functions.</li>
|
||
<li>function <span class="math inline">\(f:R \rightarrow R\)</span> are univariate, scalar-valued functions.</li>
|
||
<li>functions <span class="math inline">\(\vec{r}:R\rightarrow R^m\)</span> are parameterized curves. The trace of a parameterized curve is a path.</li>
|
||
<li>functions <span class="math inline">\(F:R^n \rightarrow R^m\)</span>, may be called vector fields in applications. They are also used to describe transformations.</li>
|
||
</ul>
|
||
<p>When <span class="math inline">\(m>1\)</span> a function is called <em>vector valued</em>.</p>
|
||
<p>When <span class="math inline">\(n>1\)</span> the argument may be given in terms of components, e.g. <span class="math inline">\(f(x,y,z)\)</span>; with a point as an argument, <span class="math inline">\(F(p)\)</span>; or with a vector as an argument, <span class="math inline">\(F(\vec{a})\)</span>. The identification of a point with a vector is done frequently.</p>
|
||
<section id="limits" class="level2" data-number="63.1">
|
||
<h2 data-number="63.1" class="anchored" data-anchor-id="limits"><span class="header-section-number">63.1</span> Limits</h2>
|
||
<p>Limits when <span class="math inline">\(m > 1\)</span> depend on the limits of each component existing.</p>
|
||
<p>Limits when <span class="math inline">\(n > 1\)</span> are more complicated. One characterization is a limit at a point <span class="math inline">\(c\)</span> exists if and only if for <em>every</em> continuous path going to <span class="math inline">\(c\)</span> the limit along the path for every component exists in the univariate sense.</p>
|
||
</section>
|
||
<section id="derivatives" class="level2" data-number="63.2">
|
||
<h2 data-number="63.2" class="anchored" data-anchor-id="derivatives"><span class="header-section-number">63.2</span> Derivatives</h2>
|
||
<p>The derivative of a univariate function, <span class="math inline">\(f\)</span>, at a point <span class="math inline">\(c\)</span> is defined by a limit:</p>
|
||
<p><span class="math display">\[
|
||
f'(c) = \lim_{h\rightarrow 0} \frac{f(c+h)-f(c)}{h},
|
||
\]</span></p>
|
||
<p>and as a function by considering the mapping <span class="math inline">\(c\)</span> into <span class="math inline">\(f'(c)\)</span>. A characterization is it is the value for which</p>
|
||
<p><span class="math display">\[
|
||
|f(c+h) - f(h) - f'(c)h| = \mathcal{o}(|h|),
|
||
\]</span></p>
|
||
<p>That is, after dividing the left-hand side by <span class="math inline">\(|h|\)</span> the expression goes to <span class="math inline">\(0\)</span> as <span class="math inline">\(|h|\rightarrow 0\)</span>. This characterization will generalize with the norm replacing the absolute value, as needed.</p>
|
||
<section id="parameterized-curves" class="level3" data-number="63.2.1">
|
||
<h3 data-number="63.2.1" class="anchored" data-anchor-id="parameterized-curves"><span class="header-section-number">63.2.1</span> Parameterized curves</h3>
|
||
<p>The derivative of a function <span class="math inline">\(\vec{r}: R \rightarrow R^m\)</span>, <span class="math inline">\(\vec{r}'(t)\)</span>, is found by taking the derivative of each component. (The function consisting of just one component is univariate.)</p>
|
||
<p>The derivative satisfies</p>
|
||
<p><span class="math display">\[
|
||
\| \vec{r}(t+h) - \vec{r}(t) - \vec{r}'(t) h \| = \mathcal{o}(|h|).
|
||
\]</span></p>
|
||
<p>The derivative is <em>tangent</em> to the curve and indicates the direction of travel.</p>
|
||
<p>The <strong>tangent</strong> vector is the unit vector in the direction of <span class="math inline">\(\vec{r}'(t)\)</span>:</p>
|
||
<p><span class="math display">\[
|
||
\hat{T} = \frac{\vec{r}'(t)}{\|\vec{r}(t)\|}.
|
||
\]</span></p>
|
||
<p>The path is parameterized by <em>arc</em> length if <span class="math inline">\(\|\vec{r}'(t)\| = 1\)</span> for all <span class="math inline">\(t\)</span>. In this case an “<span class="math inline">\(s\)</span>” is used for the parameter, as a notational hint: <span class="math inline">\(\hat{T} = d\vec{r}/ds\)</span>.</p>
|
||
<p>The <strong>normal</strong> vector is the unit vector in the direction of the derivative of the tangent vector:</p>
|
||
<p><span class="math display">\[
|
||
\hat{N} = \frac{\hat{T}'(t)}{\|\hat{T}'(t)\|}.
|
||
\]</span></p>
|
||
<p>In dimension <span class="math inline">\(m=2\)</span>, if <span class="math inline">\(\hat{T} = \langle a, b\rangle\)</span> then <span class="math inline">\(\hat{N} = \langle -b, a\rangle\)</span> or <span class="math inline">\(\langle b, -a\rangle\)</span> and <span class="math inline">\(\hat{N}'(t)\)</span> is parallel to <span class="math inline">\(\hat{T}\)</span>.</p>
|
||
<p>In dimension <span class="math inline">\(m=3\)</span>, the <strong>binormal</strong> vector, <span class="math inline">\(\hat{B}\)</span>, is the unit vector <span class="math inline">\(\hat{T}\times\hat{N}\)</span>.</p>
|
||
<p>The <a href="">Frenet-Serret</a> formulas define the <strong>curvature</strong>, <span class="math inline">\(\kappa\)</span>, and the <strong>torsion</strong>, <span class="math inline">\(\tau\)</span>, by</p>
|
||
<p><span class="math display">\[
|
||
\begin{align}
|
||
\frac{d\hat{T}}{ds} &= & \kappa \hat{N} &\\
|
||
\frac{d\hat{N}}{ds} &= -\kappa\hat{T} & & + \tau\hat{B}\\
|
||
\frac{d\hat{B}}{ds} &= & -\tau\hat{N}&
|
||
\end{align}
|
||
\]</span></p>
|
||
<p>These formulas apply in dimension <span class="math inline">\(m=2\)</span> with <span class="math inline">\(\hat{B}=\vec{0}\)</span>.</p>
|
||
<p>The curvature, <span class="math inline">\(\kappa\)</span>, can be visualized by imagining a circle of radius <span class="math inline">\(r=1/\kappa\)</span> best approximating the path at a point. (A straight line would have a circle of infinite radius and curvature <span class="math inline">\(0\)</span>.)</p>
|
||
<p>The chain rule says <span class="math inline">\((\vec{r}(g(t))' = \vec{r}'(g(t)) g'(t)\)</span>.</p>
|
||
</section>
|
||
<section id="scalar-functions" class="level3" data-number="63.2.2">
|
||
<h3 data-number="63.2.2" class="anchored" data-anchor-id="scalar-functions"><span class="header-section-number">63.2.2</span> Scalar functions</h3>
|
||
<p>A scalar function, <span class="math inline">\(f:R^n\rightarrow R\)</span>, <span class="math inline">\(n > 1\)</span> has a <strong>partial derivative</strong> defined. For <span class="math inline">\(n=2\)</span>, these are:</p>
|
||
<p><span class="math display">\[
|
||
\begin{align}
|
||
\frac{\partial{f}}{\partial{x}}(x,y) &=
|
||
\lim_{h\rightarrow 0} \frac{f(x+h,y)-f(x,y)}{h}\\
|
||
\frac{\partial{f}}{\partial{y}}(x,y) &=
|
||
\lim_{h\rightarrow 0} \frac{f(x,y+h)-f(x,y)}{h}.
|
||
\end{align}
|
||
\]</span></p>
|
||
<p>The generalization to <span class="math inline">\(n>2\)</span> is clear - the partial derivative in <span class="math inline">\(x_i\)</span> is the derivative of <span class="math inline">\(f\)</span> when the <em>other</em> <span class="math inline">\(x_j\)</span> are held constant.</p>
|
||
<p>This may be viewed as the derivative of the univariate function <span class="math inline">\((f\circ\vec{r})(t)\)</span> where <span class="math inline">\(\vec{r}(t) = p + t \hat{e}_i\)</span>, <span class="math inline">\(\hat{e}_i\)</span> being the unit vector of all <span class="math inline">\(0\)</span>s except a <span class="math inline">\(1\)</span> in the <span class="math inline">\(i\)</span>th component.</p>
|
||
<p>The <strong>gradient</strong> of <span class="math inline">\(f\)</span>, when the limits exist, is the vector-valued function for <span class="math inline">\(R^n\)</span> to <span class="math inline">\(R^n\)</span>:</p>
|
||
<p><span class="math display">\[
|
||
\nabla{f} = \langle
|
||
\frac{\partial{f}}{\partial{x_1}},
|
||
\frac{\partial{f}}{\partial{x_2}},
|
||
\dots
|
||
\frac{\partial{f}}{\partial{x_n}}
|
||
\rangle.
|
||
\]</span></p>
|
||
<p>The gradient satisfies:</p>
|
||
<p><span class="math display">\[
|
||
\|f(\vec{x}+\Delta{\vec{x}}) - f(\vec{x}) - \nabla{f}\cdot\Delta{\vec{x}}\| = \mathcal{o}(\|\Delta{\vec{x}\|}).
|
||
\]</span></p>
|
||
<p>The gradient is viewed as a column vector. If the dot product above is viewed as matrix multiplication, then it would be written <span class="math inline">\(\nabla{f}' \Delta{\vec{x}}\)</span>.</p>
|
||
<p><strong>Linearization</strong> is the <em>approximation</em></p>
|
||
<p><span class="math display">\[
|
||
f(\vec{x}+\Delta{\vec{x}}) \approx f(\vec{x}) + \nabla{f}\cdot\Delta{\vec{x}}.
|
||
\]</span></p>
|
||
<p>The <strong>directional derivative</strong> of <span class="math inline">\(f\)</span> in the direction <span class="math inline">\(\vec{v}\)</span> is <span class="math inline">\(\vec{v}\cdot\nabla{f}\)</span>, which can be seen as the derivative of the univariate function <span class="math inline">\((f\circ\vec{r})(t)\)</span> where <span class="math inline">\(\vec{r}(t) = p + t \vec{v}\)</span>.</p>
|
||
<p>For the function <span class="math inline">\(z=f(x,y)\)</span> the gradient points in the direction of steepest ascent. Ascent is seen in the <span class="math inline">\(3\)</span>d surface, the gradient is <span class="math inline">\(2\)</span> dimensional.</p>
|
||
<p>For a function <span class="math inline">\(f(\vec{x})\)</span>, a <strong>level curve</strong> is the set of values for which <span class="math inline">\(f(\vec{x})=c\)</span>, <span class="math inline">\(c\)</span> being some constant. Plotted, this may give a curve or surface (in <span class="math inline">\(n=2\)</span> or <span class="math inline">\(n=3\)</span>). The gradient at a point <span class="math inline">\(\vec{x}\)</span> with <span class="math inline">\(f(\vec{x})=c\)</span> will be <em>orthogonal</em> to the level curve <span class="math inline">\(f=c\)</span>.</p>
|
||
<p>Partial derivatives are scalar functions, so will themselves have partial derivatives when the limits are defined. The notation <span class="math inline">\(f_{xy}\)</span> stands for the partial derivative in <span class="math inline">\(y\)</span> of the partial derivative of <span class="math inline">\(f\)</span> in <span class="math inline">\(x\)</span>. <a href="">Schwarz</a>’s theorem says the order of partial derivatives will not matter (e.g., <span class="math inline">\(f_{xy} = f_{yx}\)</span>) provided the higher-order derivatives are continuous.</p>
|
||
<p>The chain rule applied to <span class="math inline">\((f\circ\vec{r})(t)\)</span> says:</p>
|
||
<p><span class="math display">\[
|
||
\frac{d(f\circ\vec{r})}{dt} = \nabla{f}(\vec{r}) \cdot \vec{r}'.
|
||
\]</span></p>
|
||
</section>
|
||
<section id="vector-valued-functions" class="level3" data-number="63.2.3">
|
||
<h3 data-number="63.2.3" class="anchored" data-anchor-id="vector-valued-functions"><span class="header-section-number">63.2.3</span> Vector-valued functions</h3>
|
||
<p>For a function <span class="math inline">\(F:R^n \rightarrow R^m\)</span>, the <strong>total derivative</strong> of <span class="math inline">\(F\)</span> is the linear operator <span class="math inline">\(d_F\)</span> satisfying:</p>
|
||
<p><span class="math display">\[
|
||
\|F(\vec{x} + \vec{h})-F(\vec{x}) - d_F \vec{h}\| = \mathcal{o}(\|\vec{h}\|)
|
||
\]</span></p>
|
||
<p>For <span class="math inline">\(F=\langle f_1, f_2, \dots, f_m\rangle\)</span> the total derivative is the <strong>Jacobian</strong>, a <span class="math inline">\(m \times n\)</span> matrix of partial derivatives:</p>
|
||
<p><span class="math display">\[
|
||
J_f = \left[
|
||
\begin{align}{}
|
||
\frac{\partial f_1}{\partial x_1} &\quad \frac{\partial f_1}{\partial x_2} &\dots&\quad\frac{\partial f_1}{\partial x_n}\\
|
||
\frac{\partial f_2}{\partial x_1} &\quad \frac{\partial f_2}{\partial x_2} &\dots&\quad\frac{\partial f_2}{\partial x_n}\\
|
||
&&\vdots&\\
|
||
\frac{\partial f_m}{\partial x_1} &\quad \frac{\partial f_m}{\partial x_2} &\dots&\quad\frac{\partial f_m}{\partial x_n}
|
||
\end{align}
|
||
\right].
|
||
\]</span></p>
|
||
<p>This can be viewed as being comprised of row vectors, each being the individual gradients; or as column vectors each being the vector of partial derivatives for a given variable.</p>
|
||
<p>The <strong>chain rule</strong> for <span class="math inline">\(F:R^n \rightarrow R^m\)</span> composed with <span class="math inline">\(G:R^k \rightarrow R^n\)</span> is:</p>
|
||
<p><span class="math display">\[
|
||
d_{F\circ G}(a) = d_F(G(a)) d_G(a),
|
||
\]</span></p>
|
||
<p>That is the total derivative of <span class="math inline">\(F\)</span> at the point <span class="math inline">\(G(a)\)</span> times (matrix multiplication) the total derivative of <span class="math inline">\(G\)</span> at <span class="math inline">\(a\)</span>. The dimensions work out as <span class="math inline">\(d_F\)</span> is <span class="math inline">\(m\times n\)</span> and <span class="math inline">\(d_G\)</span> is <span class="math inline">\(n\times k\)</span>, so <span class="math inline">\(d_(F\circ G)\)</span> will be <span class="math inline">\(m\times k\)</span> and <span class="math inline">\(F\circ{G}: R^k\rightarrow R^m\)</span>.</p>
|
||
<p>A scalar function <span class="math inline">\(f:R^n \rightarrow R\)</span> and a parameterized curve <span class="math inline">\(\vec{r}:R\rightarrow R^n\)</span> composes to yield a univariate function. The total derivative of <span class="math inline">\(f\circ\vec{r}\)</span> satisfies:</p>
|
||
<p><span class="math display">\[
|
||
d_f(\vec{r}) d_\vec{r} = \nabla{f}(\vec{r}(t))' \vec{r}'(t) =
|
||
\nabla{f}(\vec{r}(t)) \cdot \vec{r}'(t),
|
||
\]</span></p>
|
||
<p>as above. (There is an identification of a <span class="math inline">\(1\times 1\)</span> matrix with a scalar in re-expressing as a dot product.)</p>
|
||
</section>
|
||
<section id="the-divergence-curl-and-their-vanishing-properties" class="level3" data-number="63.2.4">
|
||
<h3 data-number="63.2.4" class="anchored" data-anchor-id="the-divergence-curl-and-their-vanishing-properties"><span class="header-section-number">63.2.4</span> The divergence, curl, and their vanishing properties</h3>
|
||
<p>Define the <strong>divergence</strong> of a vector-valued function <span class="math inline">\(F:R^n \rightarrow R^n\)</span> by:</p>
|
||
<p><span class="math display">\[
|
||
\text{divergence}(F) =
|
||
\frac{\partial{F_{x_1}}}{\partial{x_1}} +
|
||
\frac{\partial{F_{x_2}}}{\partial{x_2}} + \cdots
|
||
\frac{\partial{F_{x_n}}}{\partial{x_n}}.
|
||
\]</span></p>
|
||
<p>The divergence is a scalar function. For a vector field <span class="math inline">\(F\)</span>, it measures the microscopic flow out of a region.</p>
|
||
<p>A vector field whose divergence is identically <span class="math inline">\(0\)</span> is called <strong>incompressible</strong>.</p>
|
||
<p>Define the <strong>curl</strong> of a <em>two</em>-dimensional vector field, <span class="math inline">\(F:R^2 \rightarrow R^2\)</span>, by:</p>
|
||
<p><span class="math display">\[
|
||
\text{curl}(F) = \frac{\partial{F_y}}{\partial{x}} -
|
||
\frac{\partial{F_x}}{\partial{y}}.
|
||
\]</span></p>
|
||
<p>The curl for <span class="math inline">\(n=2\)</span> is a scalar function.</p>
|
||
<p>For <span class="math inline">\(n=3\)</span> define the <strong>curl</strong> of <span class="math inline">\(F:R^3 \rightarrow R^3\)</span> to be the <em>vector field</em>:</p>
|
||
<p><span class="math display">\[
|
||
\text{curl}(F) =
|
||
\langle \
|
||
\frac{\partial{F_z}}{\partial{y}} - \frac{\partial{F_y}}{\partial{z}},
|
||
\frac{\partial{F_x}}{\partial{z}} - \frac{\partial{F_z}}{\partial{x}},
|
||
\frac{\partial{F_y}}{\partial{x}} - \frac{\partial{F_x}}{\partial{y}}
|
||
\rangle.
|
||
\]</span></p>
|
||
<p>The curl measures the circulation in a vector field. In dimension <span class="math inline">\(n=3\)</span> it <em>points</em> in the direction of the normal of the plane of maximum circulation with direction given by the right-hand rule.</p>
|
||
<p>A vector field whose curl is identically of magnitude <span class="math inline">\(0\)</span> is called <strong>irrotational</strong>.</p>
|
||
<p>The <span class="math inline">\(\nabla\)</span> operator is the <em>formal</em> vector</p>
|
||
<p><span class="math display">\[
|
||
\nabla = \langle
|
||
\frac{\partial}{\partial{x}},
|
||
\frac{\partial}{\partial{y}},
|
||
\frac{\partial}{\partial{z}}
|
||
\rangle.
|
||
\]</span></p>
|
||
<p>The gradient is then scalar “multiplication” on the left: <span class="math inline">\(\nabla{f}\)</span>.</p>
|
||
<p>The divergence is the dot product on the left: <span class="math inline">\(\nabla\cdot{F}\)</span>.</p>
|
||
<p>The curl is the the cross product on the left: <span class="math inline">\(\nabla\times{F}\)</span>.</p>
|
||
<p>These operations satisfy two vanishing properties:</p>
|
||
<ul>
|
||
<li>The curl of a gradient is the zero vector: <span class="math inline">\(\nabla\times\nabla{f}=\vec{0}\)</span></li>
|
||
<li>The divergence of a curl is <span class="math inline">\(0\)</span>: <span class="math inline">\(\nabla\cdot(\nabla\times F)=0\)</span></li>
|
||
</ul>
|
||
<p><a href="">Helmholtz</a> decomposition theorem says a vector field (<span class="math inline">\(n=3\)</span>) which vanishes rapidly enough can be expressed in terms of <span class="math inline">\(F = -\nabla\phi + \nabla\times{A}\)</span>. The left term will be irrotational (no curl) and the right term will be incompressible (no divergence).</p>
|
||
</section>
|
||
</section>
|
||
<section id="integrals" class="level2" data-number="63.3">
|
||
<h2 data-number="63.3" class="anchored" data-anchor-id="integrals"><span class="header-section-number">63.3</span> Integrals</h2>
|
||
<p>The definite integral, <span class="math inline">\(\int_a^b f(x) dx\)</span>, for a bounded univariate function is defined in terms Riemann sums, <span class="math inline">\(\lim \sum f(c_i)\Delta{x_i}\)</span> as the maximum <em>partition</em> size goes to <span class="math inline">\(0\)</span>. Similarly the integral of a bounded scalar function <span class="math inline">\(f:R^n \rightarrow R\)</span> over a box-like region <span class="math inline">\([a_1,b_1]\times[a_2,b_2]\times\cdots\times[a_n,b_n]\)</span> can be defined in terms of a limit of Riemann sums. A Riemann integrable function is one for which the upper and lower Riemann sums agree in the limit. A characterization of a Riemann integrable function is that the set of discontinuities has measure <span class="math inline">\(0\)</span>.</p>
|
||
<p>If <span class="math inline">\(f\)</span> and the partial functions (<span class="math inline">\(x \rightarrow f(x,y)\)</span> and <span class="math inline">\(y \rightarrow f(x,y)\)</span>) are Riemann integrable, then Fubini’s theorem allows the definite integral to be performed iteratively:</p>
|
||
<p><span class="math display">\[
|
||
\iint_{R\times S}fdV = \int_R \left(\int_S f(x,y) dy\right) dx
|
||
= \int_S \left(\int_R f(x,y) dx\right) dy.
|
||
\]</span></p>
|
||
<p>The integral satisfies linearity and monotonicity properties that follow from the definitions:</p>
|
||
<ul>
|
||
<li>For integrable <span class="math inline">\(f\)</span> and <span class="math inline">\(g\)</span> and constants <span class="math inline">\(a\)</span> and <span class="math inline">\(b\)</span>:</li>
|
||
</ul>
|
||
<p><span class="math display">\[
|
||
\iint_R (af(x) + bg(x))dV = a\iint_R f(x)dV + b\iint_R g(x) dV.
|
||
\]</span></p>
|
||
<ul>
|
||
<li>If <span class="math inline">\(R\)</span> and <span class="math inline">\(R'\)</span> are <em>disjoint</em> rectangular regions (possibly sharing a boundary), then the integral over the union is defined by linearity:</li>
|
||
</ul>
|
||
<p><span class="math display">\[
|
||
\iint_{R \cup R'} f(x) dV = \iint_R f(x)dV + \iint_{R'} f(x) dV.
|
||
\]</span></p>
|
||
<ul>
|
||
<li>As <span class="math inline">\(f\)</span> is bounded, let <span class="math inline">\(m \leq f(x) \leq M\)</span> for all <span class="math inline">\(x\)</span> in <span class="math inline">\(R\)</span>. Then</li>
|
||
</ul>
|
||
<p><span class="math display">\[
|
||
m V(R) \leq \iint_R f(x) dV \leq MV(R).
|
||
\]</span></p>
|
||
<ul>
|
||
<li>If <span class="math inline">\(f\)</span> and <span class="math inline">\(g\)</span> are integrable <em>and</em> <span class="math inline">\(f(x) \leq g(x)\)</span>, then the integrals have the same property, namely <span class="math inline">\(\iint_R f dV \leq \iint_R gdV\)</span>.</li>
|
||
<li>If <span class="math inline">\(S \subset R\)</span>, both closed rectangles, then if <span class="math inline">\(f\)</span> is integrable over <span class="math inline">\(R\)</span> it will be also over <span class="math inline">\(S\)</span> and, when <span class="math inline">\(f\geq 0\)</span>, <span class="math inline">\(\iint_S f dV \leq \iint_R fdV\)</span>.</li>
|
||
<li>If <span class="math inline">\(f\)</span> is bounded and integrable, then <span class="math inline">\(|\iint_R fdV| \leq \iint_R |f| dV\)</span>.</li>
|
||
</ul>
|
||
<p>In two dimensions, we have the following interpretations:</p>
|
||
<p><span class="math display">\[
|
||
\begin{align}
|
||
\iint_R dA &= \text{area of } R\\
|
||
\iint_R \rho dA &= \text{mass with constant density }\rho\\
|
||
\iint_R \rho(x,y) dA &= \text{mass of region with density }\rho\\
|
||
\frac{1}{\text{area}}\iint_R x \rho(x,y)dA &= \text{centroid of region in } x \text{ direction}\\
|
||
\frac{1}{\text{area}}\iint_R y \rho(x,y)dA &= \text{centroid of region in } y \text{ direction}
|
||
\end{align}
|
||
\]</span></p>
|
||
<p>In three dimensions, we have the following interpretations:</p>
|
||
<p><span class="math display">\[
|
||
\begin{align}
|
||
\iint_VdV &= \text{volume of } V\\
|
||
\iint_V \rho dV &= \text{mass with constant density }\rho\\
|
||
\iint_V \rho(x,y) dV &= \text{mass of volume with density }\rho\\
|
||
\frac{1}{\text{volume}}\iint_V x \rho(x,y)dV &= \text{centroid of volume in } x \text{ direction}\\
|
||
\frac{1}{\text{volume}}\iint_V y \rho(x,y)dV &= \text{centroid of volume in } y \text{ direction}\\
|
||
\frac{1}{\text{volume}}\iint_V z \rho(x,y)dV &= \text{centroid of volume in } z \text{ direction}
|
||
\end{align}
|
||
\]</span></p>
|
||
<p>To compute integrals over non-box-like regions, Fubini’s theorem may be utilized. Alternatively, a <strong>transformation</strong> of variables</p>
|
||
<section id="line-integrals" class="level3" data-number="63.3.1">
|
||
<h3 data-number="63.3.1" class="anchored" data-anchor-id="line-integrals"><span class="header-section-number">63.3.1</span> Line integrals</h3>
|
||
<p>For a parameterized curve, <span class="math inline">\(\vec{r}(t)\)</span>, the <strong>line integral</strong> of a scalar function between <span class="math inline">\(a \leq t \leq b\)</span> is defined by: <span class="math inline">\(\int_a^b f(\vec{r}(t)) \| \vec{r}'(t)\| dt\)</span>. For a path parameterized by arc-length, the integral is expressed by <span class="math inline">\(\int_C f(\vec{r}(s)) ds\)</span> or simply <span class="math inline">\(\int_C f ds\)</span>, as the norm is <span class="math inline">\(1\)</span> and <span class="math inline">\(C\)</span> expresses the path.</p>
|
||
<p>A Jordan curve in two dimensions is a non-intersecting continuous loop in the plane. The Jordan curve theorem states that such a curve divides the plane into a bounded and unbounded region. The curve is <em>positively</em> parameterized if the the bounded region is kept on the left. A line integral over a Jordan curve is denoted <span class="math inline">\(\oint_C f ds\)</span>.</p>
|
||
<p>Some interpretations: <span class="math inline">\(\int_a^b \| \vec{r}'(t)\| dt\)</span> computes the <em>arc-length</em>. If the path represents a wire with density <span class="math inline">\(\rho(\vec{x})\)</span> then <span class="math inline">\(\int_a^b \rho(\vec{r}(t)) \|\vec{r}'(t)\| dt\)</span> computes the mass of the wire.</p>
|
||
<p>The line integral is also defined for a vector field <span class="math inline">\(F:R^n \rightarrow R^n\)</span> through <span class="math inline">\(\int_a^b F(\vec{r}(t)) \cdot \vec{r}'(t) dt\)</span>. When parameterized by arc length, this becomes <span class="math inline">\(\int_C F(\vec{r}(s)) \cdot \hat{T} ds\)</span> or more simply <span class="math inline">\(\int_C F\cdot\hat{T}ds\)</span>. In dimension <span class="math inline">\(n=2\)</span> if <span class="math inline">\(\hat{N}\)</span> is the normal, then this line integral (the flow) is also of interest <span class="math inline">\(\int_a^b F(\vec{r}(t)) \cdot \hat{N} dt\)</span> (this is also expressed by <span class="math inline">\(\int_C F\cdot\hat{N} ds\)</span>).</p>
|
||
<p>When <span class="math inline">\(F\)</span> is a <em>force field</em>, then the interpretation of <span class="math inline">\(\int_a^b F(\vec{r}(t)) \cdot \vec{r}'(t) dt\)</span> is the amount of <em>work</em> to move an object from <span class="math inline">\(\vec{r}(a)\)</span> to <span class="math inline">\(\vec{r}(b)\)</span>. (Work measures force applied times distance moved.)</p>
|
||
<p>A <strong>conservative force</strong> is a force field within an open region <span class="math inline">\(R\)</span> with the property that the total work done in moving a particle between two points is independent of the path taken. (Similarly, integrals over Jordan curves are zero.)</p>
|
||
<p>The gradient theorem or <strong>fundamental theorem of line integrals</strong> states if <span class="math inline">\(\phi\)</span> is a scalar function then the vector field <span class="math inline">\(\nabla{\phi}\)</span> (if continuous in <span class="math inline">\(R\)</span>) is a conservative field. That is if <span class="math inline">\(q\)</span> and <span class="math inline">\(p\)</span> are points, <span class="math inline">\(C\)</span> any curve in <span class="math inline">\(R\)</span>, and <span class="math inline">\(\vec{r}\)</span> a parameterization of <span class="math inline">\(C\)</span> over <span class="math inline">\([a,b]\)</span> that <span class="math inline">\(\phi(p) - \phi(q) = \int_a^b \nabla{f}(\vec{r}(t)) \cdot \vec{r}'(t) dt\)</span>.</p>
|
||
<p>If <span class="math inline">\(\phi\)</span> is a scalar function producing a field <span class="math inline">\(\nabla{\phi}\)</span> then in dimensions <span class="math inline">\(2\)</span> and <span class="math inline">\(3\)</span> the curl of <span class="math inline">\(\nabla{\phi}\)</span> is zero when the functions involved are continuous. Conversely, if the curl of a force field, <span class="math inline">\(F\)</span>, is zero <em>and</em> the derivatives are continuous in a <em>simply connected</em> domain, then there exists a scalar potential function, <span class="math inline">\(\phi,\)</span> with <span class="math inline">\(F = -\nabla{\phi}\)</span>.</p>
|
||
<p>In dimension <span class="math inline">\(2\)</span>, if <span class="math inline">\(F\)</span> describes a flow field, the integral <span class="math inline">\(\int_C F \cdot\hat{N}ds\)</span> is interpreted as the flow across the curve <span class="math inline">\(C\)</span>; when <span class="math inline">\(C\)</span> is a closed curve <span class="math inline">\(\oint_C F\cdot\hat{N}ds\)</span> is interpreted as the flow out of the region, when <span class="math inline">\(C\)</span> is positively parameterized.</p>
|
||
<p><strong>Green’s theorem</strong> states if <span class="math inline">\(C\)</span> is a positively oriented Jordan curve in the plane bounding a region <span class="math inline">\(D\)</span> and <span class="math inline">\(F\)</span> is a vector field <span class="math inline">\(F:R^2 \rightarrow R^2\)</span> then <span class="math inline">\(\oint_C F\cdot\hat{T}ds = \iint_D \text{curl}(F) dA\)</span>.</p>
|
||
<p>Green’s theorem can be re-expressed in flow form: <span class="math inline">\(\oint_C F\cdot\hat{N}ds=\iint_D\text{divergence}(F)dA\)</span>.</p>
|
||
<p>For <span class="math inline">\(F=\langle -y,x\rangle\)</span>, Green’s theorem says the area of <span class="math inline">\(D\)</span> is given by <span class="math inline">\((1/2)\oint_C F\cdot\vec{r}' dt\)</span>. Similarly, if <span class="math inline">\(F=\langle 0,x\rangle\)</span> or <span class="math inline">\(F=\langle -y,0\rangle\)</span> then the area is given by <span class="math inline">\(\oint_C F\cdot\vec{r}'dt\)</span>. The above follows as <span class="math inline">\(\text{curl}(F)\)</span> is <span class="math inline">\(2\)</span> or <span class="math inline">\(1\)</span>. Similar formulas can be given to compute the centroids, by identifying a vector field with <span class="math inline">\(\text{curl}(F) = x\)</span> or <span class="math inline">\(y\)</span>.</p>
|
||
</section>
|
||
<section id="surface-integrals" class="level3" data-number="63.3.2">
|
||
<h3 data-number="63.3.2" class="anchored" data-anchor-id="surface-integrals"><span class="header-section-number">63.3.2</span> Surface integrals</h3>
|
||
<p>A surface in <span class="math inline">\(3\)</span> dimensions can be described by a scalar function <span class="math inline">\(z=f(x,y)\)</span>, a parameterization <span class="math inline">\(F:R^2 \rightarrow R^3\)</span> or as a level curve of a scalar function <span class="math inline">\(f(x,y,z)\)</span>. The second case, covers the first through the parameterization <span class="math inline">\((x,y) \rightarrow (x,y,f(x,y)\)</span>. For a parameterization of a surface, <span class="math inline">\(\Phi(u,v) = \langle \Phi_x, \Phi_y, \Phi_z\rangle\)</span>, let <span class="math inline">\(\partial{\Phi}/\partial{u}\)</span> be the <span class="math inline">\(3\)</span>-d vector <span class="math inline">\(\langle \partial{\Phi_x}/\partial{u}, \partial{\Phi_y}/\partial{u}, \partial{\Phi_z}/\partial{u}\rangle\)</span>, similarly define <span class="math inline">\(\partial{\Phi}/\partial{v}\)</span>. As vectors, these lie in the tangent plane to the surface and this plane has normal vector <span class="math inline">\(\vec{N}=\partial{\Phi}/\partial{u}\times\partial{\Phi}/\partial{v}\)</span>. For a closed surface, the parametrization is positive if <span class="math inline">\(\vec{N}\)</span> is an outward pointing normal. Let the <em>surface element</em> be defined by <span class="math inline">\(\|\vec{N}\|\)</span>.</p>
|
||
<p>The surface integral of a scalar function <span class="math inline">\(f:R^3 \rightarrow R\)</span> for a parameterization <span class="math inline">\(\Phi:R \rightarrow S\)</span> is defined by</p>
|
||
<p><span class="math display">\[
|
||
\iint_R f(\Phi(u,v))
|
||
\|\frac{\partial{\Phi}}{\partial{u}} \times \frac{\partial{\Phi}}{\partial{v}}\|
|
||
du dv
|
||
\]</span></p>
|
||
<p>If <span class="math inline">\(F\)</span> is a vector field, the surface integral may be defined as a flow across the boundary through</p>
|
||
<p><span class="math display">\[
|
||
\iint_R F(\Phi(u,v)) \cdot \vec{N} du dv =
|
||
\iint_R (F \cdot \hat{N}) \|\frac{\partial{\Phi}}{\partial{u}} \times \frac{\partial{\Phi}}{\partial{v}}\| du dv = \iint_S (F\cdot\hat{N})dS
|
||
\]</span></p>
|
||
</section>
|
||
<section id="stokes-theorem-divergence-theorem" class="level3" data-number="63.3.3">
|
||
<h3 data-number="63.3.3" class="anchored" data-anchor-id="stokes-theorem-divergence-theorem"><span class="header-section-number">63.3.3</span> Stokes’ theorem, divergence theorem</h3>
|
||
<p><strong>Stokes’ theorem</strong> states that in dimension <span class="math inline">\(3\)</span> if <span class="math inline">\(S\)</span> is a smooth surface with boundary <span class="math inline">\(C\)</span> – <em>oriented</em> so the right-hand rule gives the choice of normal for <span class="math inline">\(S\)</span> – and <span class="math inline">\(F\)</span> is a vector field with continuous partial derivatives then:</p>
|
||
<p><span class="math display">\[
|
||
\iint_S (\nabla\times{F}) \cdot \hat{N} dS = \oint_C F ds.
|
||
\]</span></p>
|
||
<p>Stokes’ theorem has the same formulation as Green’s theorem in dimension <span class="math inline">\(2\)</span>, where the surface integral is just the <span class="math inline">\(2\)</span>-dimensional integral.</p>
|
||
<p>Stokes’ theorem is used to show a vector field <span class="math inline">\(F\)</span> with zero curl is conservative if <span class="math inline">\(F\)</span> is continuous in a simply connected region.</p>
|
||
<p>Stokes’ theorem is used in Physics, for example, to relate the differential and integral forms of <span class="math inline">\(2\)</span> of Maxwell’s equations.</p>
|
||
<hr>
|
||
<p>The <strong>divergence theorem</strong> states if <span class="math inline">\(V\)</span> is a compact volume in <span class="math inline">\(R^3\)</span> with piecewise smooth boundary <span class="math inline">\(S=\partial{V}\)</span> and <span class="math inline">\(F\)</span> is a vector field with continuous partial derivatives then:</p>
|
||
<p><span class="math display">\[
|
||
\iint_V (\nabla\cdot{F})dV = \oint_S (F\cdot\hat{N})dS.
|
||
\]</span></p>
|
||
<p>The divergence theorem is available for other dimensions. In the <span class="math inline">\(n=2\)</span> case, it is the alternate (flow) form of Green’s theorem.</p>
|
||
<p>The divergence theorem is used in Physics to express physical laws in either integral or differential form.</p>
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