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Diffstat (limited to 'source/know/concept/curvilinear-coordinates')
-rw-r--r-- | source/know/concept/curvilinear-coordinates/index.md | 35 |
1 files changed, 14 insertions, 21 deletions
diff --git a/source/know/concept/curvilinear-coordinates/index.md b/source/know/concept/curvilinear-coordinates/index.md index cb22e43..48a5a72 100644 --- a/source/know/concept/curvilinear-coordinates/index.md +++ b/source/know/concept/curvilinear-coordinates/index.md @@ -48,6 +48,7 @@ we derive general formulae to convert expressions from Cartesian coordinates to the new orthogonal system $$(x_1, x_2, x_3)$$. + ## Basis vectors Consider the the vector form of the line element $$\dd{\ell}$$, @@ -86,6 +87,7 @@ $$\begin{aligned} \end{aligned}$$ + ## Gradient In an orthogonal coordinate system, @@ -102,11 +104,8 @@ $$\begin{gathered} } \end{gathered}$$ -<div class="accordion"> -<input type="checkbox" id="proof-grad"/> -<label for="proof-grad">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-grad">Proof.</label> + +{% include proof/start.html id="proof-grad" -%} For a direction $$\dd{\ell}$$, we know that $$\idv{f}{\ell}$$ is the component of $$\nabla f$$ in that direction: @@ -127,9 +126,8 @@ $$\begin{gathered} + \vu{e}_2 \dv{x_2}{\ell} \pdv{f}{x_2} + \vu{e}_3 \dv{x_3}{\ell} \pdv{f}{x_3} \end{gathered}$$ +{% include proof/end.html id="proof-grad" %} -</div> -</div> ## Divergence @@ -145,11 +143,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-div"/> -<label for="proof-div">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-div">Proof.</label> + +{% include proof/start.html id="proof-div" -%} As preparation, we rewrite $$\vb{V}$$ as follows to introduce the scale factors: @@ -222,8 +217,8 @@ $$\begin{aligned} After repeating this procedure for the other components of $$\vb{V}$$, we get the desired general expression for the divergence. -</div> -</div> +{% include proof/end.html id="proof-div" %} + ## Laplacian @@ -246,6 +241,7 @@ $$\begin{aligned} \end{aligned}$$ + ## Curl The curl of a vector $$\vb{V}$$ is as follows @@ -264,11 +260,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-curl"/> -<label for="proof-curl">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-curl">Proof.</label> + +{% include proof/start.html id="proof-curl" -%} The curl is found in a similar way as the divergence. We rewrite $$\vb{V}$$ like so: @@ -317,8 +310,8 @@ $$\begin{aligned} If we go through the same process for the other components of $$\vb{V}$$ and add up the results, we get the desired expression for the curl. -</div> -</div> +{% include proof/end.html id="proof-curl" %} + ## Differential elements |