use quarto, not Pluto to render pages

quarto/_freeze/integral_vector_calculus/review/execute-results/html.json

@@ -0,0 +1,11 @@
+{
+  "hash": "39bdf839ddda2ab0c04a02238aa01e6a",
+  "result": {
+    "markdown": "# Quick Review of Vector Calculus\n\n\n\n\n\nThis section considers functions from $R^n$ into $R^m$ where one or both of $n$ or $m$ is greater than $1$:\n\n\n  * functions $f:R \\rightarrow R^m$ are called univariate functions.\n  * functions $f:R^n \\rightarrow R$ are called scalar-valued functions.\n  * function $f:R \\rightarrow R$ are univariate, scalar-valued functions.\n  * functions $\\vec{r}:R\\rightarrow R^m$ are parameterized curves. The trace of a parameterized curve is a path.\n  * functions $F:R^n \\rightarrow R^m$, may be called vector fields in applications. They are also used to describe transformations.\n\n\nWhen $m>1$ a function is called *vector valued*.\n\n\nWhen $n>1$ the argument may be given in terms of components, e.g. $f(x,y,z)$; with a point as an argument, $F(p)$; or with a vector as an argument, $F(\\vec{a})$. The identification of a point with a vector is done frequently.\n\n\n## Limits\n\n\nLimits when $m > 1$ depend on the limits of each component existing.\n\n\nLimits when $n > 1$ are more complicated. One characterization is a limit at a point $c$ exists if and only if for *every* continuous path going to $c$ the limit along the path for every component exists in the univariate sense.\n\n\n## Derivatives\n\n\nThe derivative of a univariate function, $f$, at a point $c$ is defined by a limit:\n\n\n\n$$\nf'(c) = \\lim_{h\\rightarrow 0} \\frac{f(c+h)-f(c)}{h},\n$$\n\n\nand as a function by considering the mapping $c$ into $f'(c)$. A characterization is it is the value for which\n\n\n\n$$\n|f(c+h) - f(h) - f'(c)h| = \\mathcal{o}(|h|),\n$$\n\n\nThat is,  after dividing the left-hand side by $|h|$ the expression goes to $0$ as $|h|\\rightarrow 0$. This characterization will generalize with the norm replacing the absolute value, as needed.\n\n\n### Parameterized curves\n\n\nThe derivative of a function $\\vec{r}: R \\rightarrow R^m$, $\\vec{r}'(t)$, is found by taking the derivative of each component. (The function consisting of just one component is univariate.)\n\n\nThe derivative satisfies\n\n\n\n$$\n\\| \\vec{r}(t+h) - \\vec{r}(t) - \\vec{r}'(t) h \\| = \\mathcal{o}(|h|).\n$$\n\n\nThe derivative is *tangent* to the curve and indicates the direction of travel.\n\n\nThe **tangent** vector is the unit vector in the direction of $\\vec{r}'(t)$:\n\n\n\n$$\n\\hat{T} = \\frac{\\vec{r}'(t)}{\\|\\vec{r}(t)\\|}.\n$$\n\n\nThe path is parameterized by *arc* length if $\\|\\vec{r}'(t)\\| = 1$ for all $t$. In this case an \"$s$\" is used for the parameter, as a notational hint: $\\hat{T} = d\\vec{r}/ds$.\n\n\nThe **normal** vector is the unit vector in the direction of the derivative of the tangent vector:\n\n\n\n$$\n\\hat{N} = \\frac{\\hat{T}'(t)}{\\|\\hat{T}'(t)\\|}.\n$$\n\n\nIn dimension $m=2$, if $\\hat{T} = \\langle a, b\\rangle$ then $\\hat{N} = \\langle -b, a\\rangle$ or $\\langle b, -a\\rangle$ and $\\hat{N}'(t)$ is parallel to $\\hat{T}$.\n\n\nIn dimension $m=3$, the **binormal** vector, $\\hat{B}$, is the unit vector $\\hat{T}\\times\\hat{N}$.\n\n\nThe [Frenet-Serret]() formulas define the **curvature**, $\\kappa$,  and the **torsion**, $\\tau$, by\n\n\n\n$$\n\\begin{align}\n\\frac{d\\hat{T}}{ds} &=  & \\kappa \\hat{N} &\\\\\n\\frac{d\\hat{N}}{ds} &= -\\kappa\\hat{T} & & + \\tau\\hat{B}\\\\\n\\frac{d\\hat{B}}{ds} &= & -\\tau\\hat{N}&\n\\end{align}\n$$\n\n\nThese formulas apply in dimension $m=2$ with $\\hat{B}=\\vec{0}$.\n\n\nThe curvature, $\\kappa$, can be visualized by imagining a circle of radius $r=1/\\kappa$ best approximating the path at a point. (A straight line would have a circle of infinite radius and curvature $0$.)\n\n\nThe chain rule says $(\\vec{r}(g(t))' = \\vec{r}'(g(t)) g'(t)$.\n\n\n### Scalar functions\n\n\nA scalar function, $f:R^n\\rightarrow R$, $n > 1$ has a **partial derivative** defined. For $n=2$, these are:\n\n\n\n$$\n\\begin{align}\n\\frac{\\partial{f}}{\\partial{x}}(x,y) &=\n\\lim_{h\\rightarrow 0} \\frac{f(x+h,y)-f(x,y)}{h}\\\\\n\\frac{\\partial{f}}{\\partial{y}}(x,y) &=\n\\lim_{h\\rightarrow 0} \\frac{f(x,y+h)-f(x,y)}{h}.\n\\end{align}\n$$\n\n\nThe generalization to $n>2$ is clear - the partial derivative in $x_i$ is the derivative of $f$ when the *other* $x_j$ are held constant.\n\n\nThis may be viewed as the derivative of the univariate function $(f\\circ\\vec{r})(t)$ where $\\vec{r}(t) = p + t \\hat{e}_i$, $\\hat{e}_i$ being the unit vector of all $0$s except a $1$ in the $i$th component.\n\n\nThe **gradient** of $f$, when the limits exist, is the vector-valued function for $R^n$ to $R^n$:\n\n\n\n$$\n\\nabla{f} = \\langle\n\\frac{\\partial{f}}{\\partial{x_1}},\n\\frac{\\partial{f}}{\\partial{x_2}},\n\\dots\n\\frac{\\partial{f}}{\\partial{x_n}}\n\\rangle.\n$$\n\n\nThe gradient satisfies:\n\n\n\n$$\n\\|f(\\vec{x}+\\Delta{\\vec{x}}) - f(\\vec{x}) - \\nabla{f}\\cdot\\Delta{\\vec{x}}\\| = \\mathcal{o}(\\|\\Delta{\\vec{x}\\|}).\n$$\n\n\nThe gradient is viewed as a column vector. If the dot product above is viewed as matrix multiplicatioLine truncated
+    "supporting": [
+      "review_files/figure-html"
+    ],
+    "filters": [],
+    "includes": {}
+  }
+}