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_posts/2025-02-01-magneto.md
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_posts/2025-02-01-magneto.md
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---
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title: Magneto
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layout: post
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excerpt:
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draft: true
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# Just a helper for the path if the page has lots of images.
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images: /assets/images/2024
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assets: # I often use this for model paths
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# The thumbnail that goes on the blog or posts page. SVG if possible.
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thumbnail: /assets/blog/template/thumbnail.svg
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# The social media preview image, must be png or other raster.
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social_image: /assets/blog/template/thumbnail.png
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# The alt text for both images.
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alt: An image of the text "{...}" to suggest the idea of a template.
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image_class: invertable # For images that look good when inverted in dark modes by default they're dimmed
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mathjax: true
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model_viewer: true
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load_klipse: true
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head: |
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<meta name="robots" content="noindex">
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---
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| | Impedance | Diff. Eqn. |
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|-----------|-----------|------------|
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| Inductor | $i \omega L$ |$V = L\frac{dI}{dt}$ |
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| Resistor | R |$V = IR$|
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| Capacitor | $$\left( i\omega C \right)^{-1} $$|$V = \frac{1}{C}\int I dt$|
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## Math
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Stack overflow has a nice [mathjax summary](https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference)
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List of mathjax symbols [here](https://docs.mathjax.org/en/latest/input/tex/macros/index.html)
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So the setup is this: Imagine we draw a very short line vector $\vec{v}$ and let it flow along in a fluid with velocity field $u(\vec{x}, t)$.
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<figure style="max-width: 250px;">
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<img src="/assets/blog/astrophysical_fluids/line_element.svg" class="invertable">
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<figcaption>A line element $\delta \vec{v}$ being dragged aloung in a fluid with velocity field $u(\vec{x}, t)$</figcaption>
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</figure>
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Three things will happen, the vector will be translated along, it will change length and it will change direction. If we ignore the translation, we can ask what the equation would be for the change in length and direction of $\vec{v}$. I'll drop the vector symbols on $v$, $u$ and $x$ from now on.
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$$ D_t \; v = ? $$
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If we assume $v$ is very small we can think about expanding $u$ to first order along $v$
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$$ u(x + v, t) = u(x, t) + v \cdot \nabla u $$
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where $v \cdot \nabla$ is the directional derivative $v_x \partial_x + v_y \partial_y + v_y \partial_y$ and when $v$ is infinitesimal it just directly tells us how $u$ will change if we move from point $x$ to point $x + v$.
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So from this we can see that one end of our vector $v$ is moving along at $u(x, t)$ while the other end will move at $u(x, t) + v \cdot \nabla u$ hence:
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$$ D_t \; v = v \cdot \nabla u $$
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Below is a more “index‐by‐index” look at how one carries out **Step 3** in detail. We start from
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**math with color**:
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$${\color{red} x} + {\color{blue} y}$$
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$$
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\frac{D}{Dt}\,\delta S_i
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\;=\;
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\varepsilon_{i j k}\,\bigl(\tfrac{D}{Dt}\delta x_j^{(1)}\bigr)\,\delta x_k^{(2)}
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\;+\;
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\varepsilon_{i j k}\,\delta x_j^{(1)}\,\bigl(\tfrac{D}{Dt}\delta x_k^{(2)}\bigr),
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$$
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and then substitute
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$$
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\frac{D}{Dt}\,\delta x_j^{(1)}
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\;=\;
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\delta x_\ell^{(1)}\,\frac{\partial u_j}{\partial x_\ell},
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\quad
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\frac{D}{Dt}\,\delta x_k^{(2)}
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\;=\;
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\delta x_\ell^{(2)}\,\frac{\partial u_k}{\partial x_\ell}.
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$$
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I like these underbraces:
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$$
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\frac{D}{Dt}\,\delta S_i
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\;=\;
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\underbrace{\varepsilon_{i j k}\,\delta x_\ell^{(1)}\,\frac{\partial u_j}{\partial x_\ell}\,\delta x_k^{(2)}}
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_{T_{1}}
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\;+\;
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\underbrace{\varepsilon_{i j k}\,\delta x_j^{(1)}\,\delta x_\ell^{(2)}\,\frac{\partial u_k}{\partial x_\ell}}
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_{T_{2}}.
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$$
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Aligning equations:
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$$
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\begin{align}
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\sqrt{37} & = \sqrt{\frac{73^2-1}{12^2}} \\
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& = \sqrt{\frac{73^2}{12^2}\cdot\frac{73^2-1}{73^2}} \\
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& = \sqrt{\frac{73^2}{12^2}}\sqrt{\frac{73^2-1}{73^2}} \\
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& = \frac{73}{12}\sqrt{1 - \frac{1}{73^2}} \\
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& \approx \frac{73}{12}\left(1 - \frac{1}{2\cdot73^2}\right)
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\end{align}
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$$
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References:
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[This is a link to the subtitle heading at the top of the page](#subtitle)
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A [link][kramdown hp]
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to the homepage.
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[kramdown hp]: http://kramdown.gettalong.org "hp"
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This is a text with a
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footnote[^1].
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[^1]: And here is the definition.
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This is a text with a
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footnote[^2].
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[^2]:
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And here is the definition.
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> With a quote!
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and some math
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$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$
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* * *
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<figure class="multiple">
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<img src="/assets/blog/alpha_test/original.jpg" class = "no-dim">
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<img src="/assets/blog/alpha_test/white_subtracted.png" class = "no-dim">
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<img src="/assets/blog/alpha_test/white_subtracted.png" class = "no-dim" style="filter: brightness(2);">
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<img src="/assets/blog/alpha_test/ai_subtracted.png">
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<figcaption> Here are some images, (top left) original, (top right) white subtracted and replaced with alpha, (bottom left) same but brightened, (bottom right) ai background removal tool (loses shadow) </figcaption>
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</figure>
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## 3D models
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<figure>
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<img class="outline-model-poster no-wc" src = "/assets/projects/bike_lights/thumbnail.svg">
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<outline-model-viewer model = "/assets/blog/vector_magnet/vector_magnet.glb" zoom=500 camera='{"position":[3.118,3.203,10.1],"rotation":[-0.3104,0.2858,0.0902],"zoom":428.68750000000136,"target":[0,0,0]}'>
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</outline-model-viewer>
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<figcaption class="no-wc">If you have JS enabled this is interactive.</figcaption>
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<figcaption class="has-wc">An interactive point cloud view. Should show a billboard if JS is disabled.</figcaption>
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</figure>
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<figure>
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<img class="no-wc" src="{{page.assets}}/rear_stereo/point_cloud_preview.png">
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<point-cloud-viewer model="/assets/blog/heic_depth_map/rear_stereo/pointcloud.pcd" camera = '{"type":"perspective","position":[-3.598,-0.4154,1.971],"rotation":[0.2078,-1.06,0.1819],"zoom":1,"target":[0,0,0]}'>
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</point-cloud-viewer>
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<figcaption class="no-wc">If you have JS enabled this is interactive.</figcaption>
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<figcaption class="has-wc">An interactive point cloud view. Should show a billboard if JS is disabled.</figcaption>
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</figure>
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