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[Analysis] Parametrizations
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RobinB27
@@ -105,6 +105,7 @@ $\forall x_1,x_2 \in X:$ Line seg. $x_1 \to x_2$ is in $X$\\
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$$
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\forall 1 \leq i \neq j \leq n: \frac{\partial f_i}{\partial x_j} = \frac{\partial f_j}{\partial x_i} \implies f \text{ conservative}
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$$
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\subtext{This is the most common way to prove/disprove $f$ being conservative.}
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\definition $\text{curl}(f) := \begin{bmatrix}
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\partial_y f_3 - \partial_z f_2 \\
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@@ -199,7 +200,9 @@ $$
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$$
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\underset{x \to \infty}{\lim}\int_{[a,x]\times I}f(x,y)\ dxdy = \underbrace{\int_a^\infty \Biggl( \int_I f(x,y)\ dy \Biggr) dx}_\text{Order of Integration may change}
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$$
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If this Limit is equal for both orders of Integration:
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\subtext{The improper integral on the right is simply a regular improper integral.}
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\smalltext{If this Limit is equal for both orders of Integration:}
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\definition \textbf{Improper Integral in $\R^2$}
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$$
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@@ -243,57 +246,7 @@ $$
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The change of variable is: $\displaystyle\int_{\bar{X}} f\Bigl( \varphi(x) \Bigr)\ dx = \frac{1}{\left\vert \det(\textbf{A}) \right\vert} \int_{\bar{Y}}f(y)\ dy$
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\end{footnotesize}
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\remark \textbf{Common Changes}
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\begin{enumerate}
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\item Polar Coordinates\\
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\smalltext{
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$\varphi(r, \theta) = \bigl(r\cos(\theta),\ r\sin(\theta)\bigr) \quad \color{gray} \theta \in [0, 2\pi) $\\
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$dxdy = r\ dr\ d\theta$
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}
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\item Cylindrical Coordinates\\
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\smalltext{
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$\varphi(r, \theta, z) = \bigl( r\cos(\theta),\ r\sin(\theta),\ z \bigr) \quad \color{gray} \theta \in [0, \pi), \phi \in [0, 2\pi)$\\
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$dxdydz = r\ dr\ d\theta\ dz$
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}
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\item Spherical Coordinates\\
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\smalltext{
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$\varphi(r, \theta, \phi) = (r\sin(\phi)\cos(\theta),\ r\sin(\phi)\sin(\theta),\ r\cos(\phi))$\\
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$dxdydz = r^2\ \sin(\phi)\ dr\ d\theta\ d\phi$
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}
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\end{enumerate}
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\begin{footnotesize}
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Corresponding Jacobians:
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$$
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\textbf{J}_1 = \begin{bmatrix}
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\cos(\theta) & -r\sin(\theta) \\
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\sin(\theta) & r\cos(\theta) \\
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\end{bmatrix}
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\qquad
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\textbf{J}_2 = \begin{bmatrix}
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\cos(\theta) & -r\sin(\theta) & 0 \\
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\sin(\theta) & r\cos(\theta) & 0 \\
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0 & 0 & z
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\end{bmatrix}
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$$
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$$
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\textbf{J}_3 = \begin{bmatrix}
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\cos(\phi)\sin(\theta) & -r\sin(\phi)\sin(\theta) & r\cos(\phi)\cos(\theta) \\
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\sin(\phi)\sin(\theta) & r\cos(\phi)\sin(\theta) & r\sin(\phi)\cos(\theta) \\
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\cos(\theta) & 0 & -r\sin(\theta)
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\end{bmatrix}
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$$
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\end{footnotesize}
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% https://en.wikipedia.org/wiki/Spherical_coordinate_system#/media/File:3D_Spherical.svg
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% \begin{center}
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% \includegraphics[width=0.3\linewidth]{res/spherical-coords.png}
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% \end{center}
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\newpage
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\subsection{Green's Theorem}
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\smalltext{An analogue of the Fundamental Theorem of Calculus in $\R^2$.}
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\definition \textbf{Simple Closed Parametrized Curve}\\
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$\gamma: [a,b] \to \R^2$ closed param. curve s.t.
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@@ -301,8 +254,7 @@ $\gamma: [a,b] \to \R^2$ closed param. curve s.t.
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\item $\gamma(t) \neq \gamma(s)\quad$ unless $s = t$, or $\{s,t\} = \{a,b\}$
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\item $\gamma'(t) \neq 0\quad \forall a < t < b$
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\end{enumerate}
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\subtext{Example: $\varphi(t) = \Bigl( x_0 + r\cos(t),\ y_0 + r\sin(t) \Bigr)$\\
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(A circle, traversed \textit{once}, i.e. for $0 \leq t \leq 2\pi$)}
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\subtext{Example: $\varphi(t) = \Bigl( x_0 + r\cos(t),\ y_0 + r\sin(t) \Bigr)$}
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\begin{subbox}{Green's Theorem}
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\smalltext{$X \subset \R^2 \text{ compact with Boundary } \partial X = \displaystyle\underset{1 \leq i \leq n}{\bigcup} \gamma_i$ as above}\\
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@@ -338,4 +290,127 @@ $$
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$$
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\text{Area}(X) = \int_X 1\ dxdy = \int_X \bigg(\partial_x f_y - \partial_y f_x \bigg) \ dxdy = \int_\gamma f(s)\ ds
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$$
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\end{footnotesize}
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\end{footnotesize}
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\subsection{Useful Parametrizations}
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\begin{footnotesize}
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\textbf{Circle:} $x^2 + y^2 = r^2$
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$$
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\gamma: [0, 2\pi] \mapsto \begin{pmatrix}
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r\cos(t) \\
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r\sin(t)
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\end{pmatrix}
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\qquad
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\gamma: [0, 2\pi] \mapsto \begin{pmatrix}
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r\cos(t) \\
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-r\sin(t)
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\end{pmatrix}
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$$
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\color{gray} Clockwise \& Counterclockwise \color{black}
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\textbf{Ellipsoid:} $\displaystyle\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
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$$
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\gamma: [0, 2\pi] \mapsto \begin{pmatrix}
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a\cos(t) \\
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b\sin(t)
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\end{pmatrix}
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\qquad
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\gamma: [0, 2\pi] \mapsto \begin{pmatrix}
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a\cos(t) \\
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-b\sin(t)
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\end{pmatrix}
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$$
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\color{gray} Clockwise \& Counterclockwise \color{black}
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\textbf{Piecewise Continuous function:} $f(x)$
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$$
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\gamma: [a, b] \mapsto \bigl(t, f(t) \bigr)
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$$
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\textbf{Line Segment:} $(x_0,y_0,z_0) \to (x_1,y_1,z_1)$
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$$
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\gamma: [0, 1] \mapsto (1-t)
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\begin{pmatrix}
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x_0 \\
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y_0 \\
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z_0
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\end{pmatrix}
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+ t
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\begin{pmatrix}
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x_1 \\
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y_1 \\
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z_1
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\end{pmatrix}
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$$
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\color{gray} This is simply linear interpolation in $\R^3$ \color{black}
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\end{footnotesize}
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\subsection{Common Changes}
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\begin{footnotesize}
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\textbf{Polar Coordinates} ($\R^2$)\\
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$$
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\varphi(r, \theta) = \begin{pmatrix}
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r\cos(\theta)\\
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r\sin(\theta)
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\end{pmatrix} \qquad \color{gray} \theta \in [0, 2\pi)
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$$
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$dxdy = r\ dr\ d\theta$
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\textbf{Elliptic Coordinates} ($\R^2$)\\
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$$
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\varphi(r, \theta) = \begin{pmatrix}
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ra\cos(\theta) \\
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rb\sin(\theta)
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\end{pmatrix} \qquad \color{gray} \theta \in [0, 2\pi)
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$$
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$dxdy = a\cdot b\cdot r\ dr\ d\varphi$
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\textbf{Cylindrical Coordinates} ($\R^3$)\\
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$$
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\varphi(r, \theta, z) = \begin{pmatrix}
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r\cos(\theta) \\
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r\sin(\theta) \\
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z
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\end{pmatrix}
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\qquad \color{gray} \theta \in [0, \pi), \phi \in [0, 2\pi)
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$$
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$dxdydz = r\ dr\ d\theta\ dz$
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\textbf{Spherical Coordinates} ($\R^3$)\\
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$$
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\varphi(r, \theta, \phi) = \begin{pmatrix}
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r\sin(\phi)\cos(\theta) \\
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r\sin(\phi)\sin(\theta) \\
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r\cos(\phi)
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\end{pmatrix}
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\qquad \color{gray} \theta \in [0, \pi), \phi \in [0, 2\pi)
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$$
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$dxdydz = r^2\cdot\sin(\phi)\ dr\ d\theta\ d\phi$
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\textbf{Corresponding Jacobians}:
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$$
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\textbf{J}_1 = \begin{bmatrix}
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\cos(\theta) & -r\sin(\theta) \\
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\sin(\theta) & r\cos(\theta) \\
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\end{bmatrix}
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\qquad
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\textbf{J}_2 = \begin{bmatrix}
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\cos(\theta) & -r\sin(\theta) & 0 \\
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\sin(\theta) & r\cos(\theta) & 0 \\
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0 & 0 & z
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\end{bmatrix}
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$$
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$$
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\textbf{J}_3 = \begin{bmatrix}
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\cos(\phi)\sin(\theta) & -r\sin(\phi)\sin(\theta) & r\cos(\phi)\cos(\theta) \\
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\sin(\phi)\sin(\theta) & r\cos(\phi)\sin(\theta) & r\sin(\phi)\cos(\theta) \\
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\cos(\theta) & 0 & -r\sin(\theta)
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\end{bmatrix}
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$$
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% https://en.wikipedia.org/wiki/Spherical_coordinate_system#/media/File:3D_Spherical.svg
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% \begin{center}
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% \includegraphics[width=0.3\linewidth]{res/spherical-coords.png}
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% \end{center}
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\end{footnotesize}
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