Improve knowledge base

1 files changed, 197 insertions, 44 deletions

diff --git a/source/know/concept/orthogonal-curvilinear-coordinates/index.md b/source/know/concept/orthogonal-curvilinear-coordinates/index.md
index 4fb45b4..675b83a 100644
--- a/source/know/concept/orthogonal-curvilinear-coordinates/index.md
+++ b/source/know/concept/orthogonal-curvilinear-coordinates/index.md
@@ -43,12 +43,12 @@ $$\begin{aligned}
     &= z(c_1, c_2, c_3)
 \end{aligned}$$
 
-A useful attribute of a coordinate system is its **line element** $$\dd{\vu{\ell}}$$,
+A useful attribute of a coordinate system is its **line element** $$\dd{\vb{\ell}}$$,
 which represents the differential element of a line in any direction.
 Let $$\vu{e}_x$$, $$\vu{e}_y$$ and $$\vu{e}_z$$ be the Cartesian basis unit vectors:
 
 $$\begin{aligned}
-    \dd{\vu{\ell}}
+    \dd{\vb{\ell}}
     \equiv \vu{e}_x \dd{x} + \: \vu{e}_y \dd{y} + \: \vu{e}_z \dd{z}
 \end{aligned}$$
 
@@ -56,7 +56,7 @@ The Cartesian differential elements can be rewritten
 in $$(c_1, c_2, c_3)$$ with the chain rule:
 
 $$\begin{aligned}
-    \dd{\vu{\ell}}
+    \dd{\vb{\ell}}
     = \quad &\vu{e}_x \bigg( \pdv{x}{c_1} \dd{c_1} + \: \pdv{x}{c_2} \dd{c_2} + \: \pdv{x}{c_3} \dd{c_3} \!\bigg)
     \\
     + \: &\vu{e}_y \bigg( \pdv{y}{c_1} \dd{c_1} + \: \pdv{y}{c_2} \dd{c_2} + \: \pdv{y}{c_3} \dd{c_3} \!\bigg)
@@ -93,6 +93,7 @@ and orthogonal for any orthogonal curvilinear system.
 They are called *local* basis vectors
 because they generally depend on $$(c_1, c_2, c_3)$$,
 i.e. their directions vary from position to position.
+Their definitions can also be inverted:
 
 $$\begin{aligned}
     \boxed{
@@ -109,7 +110,6 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-
 In the following subsections, we use the scale factors $$h_1$$, $$h_2$$ and $$h_3$$
 to derive general formulae for converting vector calculus
 from Cartesian coordinates to $$(c_1, c_2, c_3)$$.
@@ -121,12 +121,12 @@ from Cartesian coordinates to $$(c_1, c_2, c_3)$$.
 The point of the scale factors $$h_1$$, $$h_2$$ and $$h_3$$,
 as can be seen from their derivation,
 is to correct for "distortions" of the coordinates compared to the Cartesian system,
-such that the line element $$\dd{\vu{\ell}}$$ retains its length.
+such that the line element $$\dd{\vb{\ell}}$$ retains its length.
 As was already established above:
 
 $$\begin{aligned}
     \boxed{
-        \dd{\vu{\ell}}
+        \dd{\vb{\ell}}
         = \vu{e}_1 h_1 \dd{c_1} + \: \vu{e}_2 h_2 \dd{c_2} + \: \vu{e}_3 h_3 \dd{c_3}
     }
 \end{aligned}$$
@@ -148,7 +148,7 @@ can be expressed in terms of $$\dd{}_1\!\vb{x}$$, $$\dd{}_2\!\vb{x}$$ and $$\dd{
 The differential normal vector element $$\dd{\vu{S}}$$ in a surface integral is hence given by:
 
 $$\begin{aligned}
-    \dd{\vu{S}}
+    \dd{\vb{S}}
     &= \dd{}_1\!\vb{x} \cross \dd{}_2\!\vb{x} + \dd{}_2\!\vb{x} \cross \dd{}_3\!\vb{x} + \dd{}_3\!\vb{x} \cross \dd{}_1\!\vb{x}
     \\
     &= (\vu{e}_1 \cross \vu{e}_2) \: h_1 h_2 \dd{c_1} \dd{c_2}
@@ -163,7 +163,7 @@ $$\vu{e}_3 \cross \vu{e}_1 = \vu{e}_2$$, so:
 
 $$\begin{aligned}
     \boxed{
-        \dd{\vu{S}}
+        \dd{\vb{S}}
         = \vu{e}_1 \: h_2 h_3 \dd{c_2} \dd{c_3} + \: \vu{e}_2 \: h_1 h_3 \dd{c_1} \dd{c_3} + \: \vu{e}_3 \: h_1 h_2 \dd{c_1} \dd{c_2}
     }
 \end{aligned}$$
@@ -357,8 +357,8 @@ because $$\ipdv{\vu{e}_1}{c_1}$$ must be orthogonal to $$\vu{e}_1$$.
 
 ## Gradient of a scalar
 
-In $$(c_1, c_2, c_3)$$, the gradient $$\nabla f$$ of a scalar field $$f$$
-has the following components:
+The gradient $$\nabla f$$ of a scalar field $$f$$
+has the following components in $$(c_1, c_2, c_3)$$:
 
 $$\begin{aligned}
     \boxed{
@@ -483,13 +483,13 @@ Now, to proceed, it is easiest to just write out the index notation:
 
 $$\begin{aligned}
     \nabla \cdot \vb{V}
-    &= \quad\: \sum_{j} \frac{1}{h_j} \pdv{V_j}{c_j} + \sum_{j} \sum_{k \neq j} \frac{V_k}{h_j h_k} \pdv{h_j}{c_k}
+    &= \sum_{j} \frac{1}{h_j} \pdv{V_j}{c_j} + \sum_{j} \sum_{k \neq j} \frac{V_k}{h_j h_k} \pdv{h_j}{c_k}
     \\
-    &= \quad\: \frac{1}{h_1} \pdv{V_1}{c_1} + \frac{V_1}{h_1 h_2} \pdv{h_2}{c_1} + \frac{V_1}{h_1 h_3} \pdv{h_3}{c_1}
+    &= \quad \frac{1}{h_1} \pdv{V_1}{c_1} + \frac{V_1}{h_1 h_2} \pdv{h_2}{c_1} + \frac{V_1}{h_1 h_3} \pdv{h_3}{c_1}
     \\
-    &\quad\:\: + \frac{V_2}{h_1 h_2} \pdv{h_1}{c_2} + \frac{1}{h_2} \pdv{V_2}{c_2} + \frac{V_2}{h_2 h_3} \pdv{h_3}{c_2}
+    &\quad\: + \frac{V_2}{h_1 h_2} \pdv{h_1}{c_2} + \frac{1}{h_2} \pdv{V_2}{c_2} + \frac{V_2}{h_2 h_3} \pdv{h_3}{c_2}
     \\
-    &\quad\:\: + \frac{V_3}{h_1 h_3} \pdv{h_1}{c_3} + \frac{V_3}{h_2 h_3} \pdv{h_2}{c_3} + \frac{1}{h_3} \pdv{V_3}{c_3}
+    &\quad\: + \frac{V_3}{h_1 h_3} \pdv{h_1}{c_3} + \frac{V_3}{h_2 h_3} \pdv{h_2}{c_3} + \frac{1}{h_3} \pdv{V_3}{c_3}
     \\
     &= \frac{1}{h_1 h_2 h_3} \bigg( h_2 h_3 \pdv{V_1}{c_1} + h_3 V_1 \pdv{h_2}{c_1} + h_2 V_1 \pdv{h_3}{c_1}
     \\
@@ -550,36 +550,6 @@ and simply add up the results to get the desired formula.
 
 
 
-## Laplacian of a scalar
-
-The Laplacian $$\nabla^2 f$$ of a scalar field $$f$$
-is calculated as follows in $$(c_1, c_2, c_3)$$:
-
-$$\begin{aligned}
-    \boxed{
-        \nabla^2 f
-        = \sum_{j} \frac{1}{H} \pdv{}{c_j} \bigg( \frac{H}{h_j^2} \pdv{f}{c_j} \bigg)
-    }
-\end{aligned}$$
-
-Where $$H \equiv h_1 h_2 h_3$$.
-When this index notation is written out in full, it becomes:
-
-$$\begin{aligned}
-    \nabla^2 f
-    = \frac{1}{h_1 h_2 h_3}
-    \bigg(
-        \pdv{}{c_1}\Big(\! \frac{h_2 h_3}{h_1} \pdv{f}{c_1} \!\Big)
-        + \pdv{}{c_2}\Big(\! \frac{h_1 h_3}{h_2} \pdv{f}{c_2} \!\Big)
-        + \pdv{}{c_3}\Big(\! \frac{h_1 h_2}{h_3} \pdv{f}{c_3} \!\Big)
-    \bigg)
-\end{aligned}$$
-
-This is trivial to prove: $$\nabla^2 f = \nabla \cdot (\nabla f)$$,
-so combining our previous formulas is enough.
-
-
-
 ## Curl of a vector
 
 The curl of a vector field $$\vb{V}$$ has the following components in $$(c_1, c_2, c_3)$$,
@@ -703,6 +673,56 @@ and simply add up the results to get the desired formula.
 
 
 
+## Laplacian of a scalar
+
+The Laplacian $$\nabla^2 f$$ of a scalar field $$f$$
+is calculated as follows in $$(c_1, c_2, c_3)$$:
+
+$$\begin{aligned}
+    \boxed{
+        \nabla^2 f
+        = \sum_{j} \frac{1}{H} \pdv{}{c_j} \bigg( \frac{H}{h_j^2} \pdv{f}{c_j} \bigg)
+    }
+\end{aligned}$$
+
+Where $$H \equiv h_1 h_2 h_3$$.
+When this index notation is written out in full, it becomes:
+
+$$\begin{aligned}
+    \nabla^2 f
+    = \frac{1}{h_1 h_2 h_3}
+    \bigg(
+        \pdv{}{c_1}\Big(\! \frac{h_2 h_3}{h_1} \pdv{f}{c_1} \!\Big)
+        + \pdv{}{c_2}\Big(\! \frac{h_1 h_3}{h_2} \pdv{f}{c_2} \!\Big)
+        + \pdv{}{c_3}\Big(\! \frac{h_1 h_2}{h_3} \pdv{f}{c_3} \!\Big)
+    \bigg)
+\end{aligned}$$
+
+This is trivial to prove: $$\nabla^2 f = \nabla \cdot (\nabla f)$$,
+so combining our previous formulae is enough.
+
+
+
+## Gradient of a divergence
+
+The gradient of a divergence $$\nabla (\nabla \cdot \vb{V})$$
+has the following components in $$(c_1, c_2, c_3)$$:
+
+$$\begin{aligned}
+    \boxed{
+        \big( \nabla (\nabla \cdot \vb{V}) \big)_j
+        = \frac{1}{h_j} \pdv{}{c_j} \bigg( \sum_{k} \frac{1}{H} \pdv{}{c_k} \Big( \frac{H V_k}{h_k} \Big) \bigg)
+    }
+\end{aligned}$$
+
+Where $$H \equiv h_1 h_2 h_3$$.
+This is trivial to prove: $$\nabla \cdot \vb{V}$$ is a scalar,
+which we insert into our gradient formula.
+We no longer write out the index notation,
+as the formulae become quite long.
+
+
+
 ## Gradient of a vector
 
 It also possible to take the gradient of a vector
@@ -1015,6 +1035,139 @@ so we can sum over $$j \neq m$$ instead.
 
 
 
+## Laplacian of a vector
+
+The Laplacian $$\nabla^2 \vb{V}$$ of a vector $$\vb{V}$$
+has the following components in $$(c_1, c_2, c_3)$$:
+
+$$\begin{aligned}
+    \boxed{
+        \begin{aligned}
+            (\nabla^2 \vb{V})_j
+            &= \sum_{k} \frac{1}{H} \pdv{}{c_k} \bigg( \frac{H}{h_k^2} \pdv{V_j}{c_k} \bigg)
+            \\
+            &\quad\: + \sum_{k \neq j} \frac{1}{H} \bigg( \pdv{}{c_j} \Big( \frac{H V_k}{h_j^2 h_k} \pdv{h_j}{c_k} \Big)
+            - \pdv{}{c_k} \Big( \frac{H V_k}{h_j h_k^2} \pdv{h_k}{c_j} \Big) \bigg)
+            \\
+            &\quad\: + \sum_{k \neq j} \frac{1}{h_j h_k} \bigg( \frac{1}{h_j} \pdv{V_k}{c_j} \pdv{h_j}{c_k}
+            - \frac{1}{h_k} \pdv{V_k}{c_k} \pdv{h_k}{c_j} \bigg)
+            \\
+            &\quad\:- \sum_{k \neq j} \frac{1}{h_j h_k} \bigg( \frac{V_j}{h_j h_k} \Big( \pdv{h_j}{c_k} \Big)^2
+            + \sum_{l \neq k} \frac{V_l}{h_k h_l} \pdv{h_k}{c_l} \pdv{h_k}{c_j} \bigg)
+        \end{aligned}
+    }
+\end{aligned}$$
+
+
+{% include proof/start.html id="proof-lap-vector" -%}
+We already know how to calculate the Laplacian $$\nabla^2 f$$ of a scalar.
+From that, we read out the $$\nabla^2$$-operator
+and apply it to a vector $$\vb{V}$$ instead:
+
+$$\begin{aligned}
+    \nabla^2 \vb{V}
+    &= \bigg( \sum_{j} \frac{1}{H} \pdv{}{c_j} \Big( \frac{H}{h_j^2} \pdv{}{c_j} \Big) \bigg)
+    \bigg( \sum_{k} V_k \vu{e}_k \bigg)
+    \\
+    &= \sum_{jk} \frac{1}{H} \pdv{}{c_j} \bigg( \frac{H}{h_j^2} \pdv{}{c_j} (V_k \vu{e}_k) \bigg)
+    \\
+    &= \sum_{jk} \frac{1}{H} \pdv{}{c_j} \bigg( \frac{H}{h_j^2} \pdv{V_k}{c_j} \vu{e}_k + \frac{H V_k}{h_j^2} \pdv{\vu{e}_k}{c_j} \bigg)
+    \\
+    &= \sum_{jk} \frac{1}{H} \pdv{}{c_j} \bigg( \frac{H}{h_j^2} \pdv{V_k}{c_j} \vu{e}_k
+    + \frac{H V_k}{h_j^2} \Big( \frac{1}{h_k} \pdv{h_j}{c_k} \vu{e}_j - \delta_{jk} \sum_{l} \frac{1}{h_l} \pdv{h_k}{c_l} \vu{e}_l \Big) \bigg)
+    \\
+    &= \sum_{j} \frac{1}{H} \pdv{}{c_j} \bigg( \sum_{k} \frac{H}{h_j^2} \pdv{V_k}{c_j} \vu{e}_k
+    + \sum_{k} \frac{H V_k}{h_j^2 h_k} \pdv{h_j}{c_k} \vu{e}_j
+    - \sum_{l} \frac{H V_j}{h_j^2 h_l} \pdv{h_j}{c_l} \vu{e}_l \bigg)
+    \\
+    &= \sum_{j} \frac{1}{H} \pdv{}{c_j} \bigg( \sum_{k} \frac{H}{h_j^2} \pdv{V_k}{c_j} \vu{e}_k
+    + \sum_{k \neq j} \frac{H V_k}{h_j^2 h_k} \pdv{h_j}{c_k} \vu{e}_j
+    - \sum_{k \neq j} \frac{H V_j}{h_j^2 h_k} \pdv{h_j}{c_k} \vu{e}_k \bigg)
+\end{aligned}$$
+
+Where we have noticed that the latter two terms cancel out if $$j = k$$.
+We expand according to the product rule of differentiation:
+
+$$\begin{aligned}
+    \nabla^2 \vb{V}
+    &= \sum_{jk} \frac{1}{H} \pdv{}{c_j} \bigg( \frac{H}{h_j^2} \pdv{V_k}{c_j} \bigg) \vu{e}_k
+    + \sum_{jk} \frac{1}{H} \frac{H}{h_j^2} \pdv{V_k}{c_j} \pdv{\vu{e}_k}{c_j}
+    \\
+    &\quad\: + \sum_{j} \sum_{k \neq j} \frac{1}{H} \pdv{}{c_j} \bigg( \frac{H V_k}{h_j^2 h_k} \pdv{h_j}{c_k} \bigg) \vu{e}_j
+    + \sum_{j} \sum_{k \neq j} \frac{1}{H} \frac{H V_k}{h_j^2 h_k} \pdv{h_j}{c_k} \pdv{\vu{e}_j}{c_j}
+    \\
+    &\quad\: - \sum_{j} \sum_{k \neq j} \frac{1}{H} \pdv{}{c_j} \bigg( \frac{H V_j}{h_j^2 h_k} \pdv{h_j}{c_k} \bigg) \vu{e}_k
+    - \sum_{j} \sum_{k \neq j} \frac{1}{H} \frac{H V_j}{h_j^2 h_k} \pdv{h_j}{c_k} \pdv{\vu{e}_k}{c_j}
+\end{aligned}$$
+
+Substituting our expression for the derivatives of the local basis vectors, we find:
+
+$$\begin{aligned}
+    \nabla^2 \vb{V}
+    &= \sum_{jk} \frac{1}{H} \pdv{}{c_j} \bigg( \frac{H}{h_j^2} \pdv{V_k}{c_j} \bigg) \vu{e}_k
+    + \sum_{jk} \frac{1}{h_j^2} \pdv{V_k}{c_j}
+    \bigg( \frac{1}{h_k} \pdv{h_j}{c_k} \vu{e}_j - \delta_{jk} \sum_{l} \frac{1}{h_l} \pdv{h_k}{c_l} \vu{e}_l \bigg)
+    \\
+    &\quad\: + \sum_{j} \sum_{k \neq j} \frac{1}{H} \pdv{}{c_j} \bigg( \frac{H V_k}{h_j^2 h_k} \pdv{h_j}{c_k} \bigg) \vu{e}_j
+    - \sum_{j} \sum_{k \neq j} \frac{V_k}{h_j^2 h_k} \pdv{h_j}{c_k} \bigg( \sum_{l \neq j} \frac{1}{h_l} \pdv{h_j}{c_l} \vu{e}_l \bigg)
+    \\
+    &\quad\: - \sum_{j} \sum_{k \neq j} \frac{1}{H} \pdv{}{c_j} \bigg( \frac{H V_j}{h_j^2 h_k} \pdv{h_j}{c_k} \bigg) \vu{e}_k
+    - \sum_{j} \sum_{k \neq j} \frac{V_j}{h_j^2 h_k} \pdv{h_j}{c_k} \bigg( \frac{1}{h_k} \pdv{h_j}{c_k} \vu{e}_j \bigg)
+    \\
+    &= \sum_{jk} \frac{1}{H} \pdv{}{c_j} \bigg( \frac{H}{h_j^2} \pdv{V_k}{c_j} \bigg) \vu{e}_k
+    + \sum_{jk} \frac{1}{h_j^2 h_k} \pdv{V_k}{c_j} \pdv{h_j}{c_k} \vu{e}_j
+    - \sum_{jl} \frac{1}{h_j^2 h_l} \pdv{V_j}{c_j} \pdv{h_j}{c_l} \vu{e}_l
+    \\
+    &\quad\: + \sum_{j} \sum_{k \neq j} \frac{1}{H} \pdv{}{c_j} \bigg( \frac{H V_k}{h_j^2 h_k} \pdv{h_j}{c_k} \bigg) \vu{e}_j
+    - \sum_{j} \sum_{k \neq j} \sum_{l \neq j} \frac{V_k}{h_j^2 h_k h_l} \pdv{h_j}{c_k} \pdv{h_j}{c_l} \vu{e}_l
+    \\
+    &\quad\: - \sum_{j} \sum_{k \neq j} \frac{1}{H} \pdv{}{c_j} \bigg( \frac{H V_j}{h_j^2 h_k} \pdv{h_j}{c_k} \bigg) \vu{e}_k
+    - \sum_{j} \sum_{k \neq j} \frac{V_j}{h_j^2 h_k^2} \bigg( \pdv{h_j}{c_k} \bigg)^2 \vu{e}_j
+    \\
+    &= \sum_{jk} \frac{1}{H} \pdv{}{c_j} \bigg( \frac{H}{h_j^2} \pdv{V_k}{c_j} \bigg) \vu{e}_k
+    + \sum_{j} \sum_{k \neq j} \frac{1}{h_j^2 h_k} \pdv{V_k}{c_j} \pdv{h_j}{c_k} \vu{e}_j
+    - \sum_{j} \sum_{k \neq j} \frac{1}{h_j^2 h_k} \pdv{V_j}{c_j} \pdv{h_j}{c_k} \vu{e}_k
+    \\
+    &\quad\: + \sum_{j} \sum_{k \neq j} \frac{1}{H} \pdv{}{c_j} \bigg( \frac{H V_k}{h_j^2 h_k} \pdv{h_j}{c_k} \bigg) \vu{e}_j
+    - \sum_{j} \sum_{k \neq j} \sum_{l \neq j} \frac{V_k}{h_j^2 h_k h_l} \pdv{h_j}{c_k} \pdv{h_j}{c_l} \vu{e}_l
+    \\
+    &\quad\: - \sum_{j} \sum_{k \neq j} \frac{1}{H} \pdv{}{c_j} \bigg( \frac{H V_j}{h_j^2 h_k} \pdv{h_j}{c_k} \bigg) \vu{e}_k
+    - \sum_{j} \sum_{k \neq j} \frac{V_j}{h_j^2 h_k^2} \bigg( \pdv{h_j}{c_k} \bigg)^2 \vu{e}_j
+\end{aligned}$$
+
+Where we have once again noticed that terms #2 and #3 cancel out if $$j = k$$.
+Next, we isolate the $$c_m$$-component by dot-multiplying with $$\vu{e}_m$$:
+
+$$\begin{aligned}
+    (\nabla^2 \vb{V})_m
+    &= (\nabla^2 \vb{V}) \cdot \vu{e}_m
+    \\
+    &= \sum_{jk} \frac{\delta_{km}}{H} \pdv{}{c_j} \bigg( \frac{H}{h_j^2} \pdv{V_k}{c_j} \bigg)
+    + \sum_{j} \sum_{k \neq j} \frac{\delta_{jm}}{h_j^2 h_k} \pdv{V_k}{c_j} \pdv{h_j}{c_k}
+    - \sum_{j} \sum_{k \neq j} \frac{\delta_{km}}{h_j^2 h_k} \pdv{V_j}{c_j} \pdv{h_j}{c_k}
+    \\
+    &\quad\: + \sum_{j} \sum_{k \neq j} \delta_{jm} \frac{1}{H} \pdv{}{c_j} \bigg( \frac{H V_k}{h_j^2 h_k} \pdv{h_j}{c_k} \bigg)
+    - \sum_{j} \sum_{k \neq j} \sum_{l \neq j} \delta_{lm} \frac{V_k}{h_j^2 h_k h_l} \pdv{h_j}{c_k} \pdv{h_j}{c_l}
+    \\
+    &\quad\: - \sum_{j} \sum_{k \neq j} \delta_{km} \frac{1}{H} \pdv{}{c_j} \bigg( \frac{H V_j}{h_j^2 h_k} \pdv{h_j}{c_k} \bigg)
+    - \sum_{j} \sum_{k \neq j} \delta_{jm} \frac{V_j}{h_j^2 h_k^2} \bigg( \pdv{h_j}{c_k} \bigg)^2
+    \\
+    &= \sum_{j} \frac{1}{H} \pdv{}{c_j} \bigg( \frac{H}{h_j^2} \pdv{V_m}{c_j} \bigg)
+    + \sum_{k \neq m} \frac{1}{h_k h_m^2} \pdv{V_k}{c_m} \pdv{h_m}{c_k}
+    - \sum_{j \neq m} \frac{1}{h_j^2 h_m} \pdv{V_j}{c_j} \pdv{h_j}{c_m}
+    \\
+    &\quad\: + \sum_{k \neq m} \frac{1}{H} \pdv{}{c_m} \bigg( \frac{H V_k}{h_m^2 h_k} \pdv{h_m}{c_k} \bigg)
+    - \sum_{j \neq m} \sum_{k \neq j} \frac{V_k}{h_j^2 h_k h_m} \pdv{h_j}{c_k} \pdv{h_j}{c_m}
+    \\
+    &\quad\: - \sum_{j \neq m} \frac{1}{H} \pdv{}{c_j} \bigg( \frac{H V_j}{h_j^2 h_m} \pdv{h_j}{c_m} \bigg)
+    - \sum_{k \neq m} \frac{V_m}{h_m^2 h_k^2} \bigg( \pdv{h_m}{c_k} \bigg)^2
+\end{aligned}$$
+
+Which gives the desired formula after some simple index renaming and rearranging.
+{% include proof/end.html id="proof-lap-vector" %}
+
+
+
 ## References
 1.  M.L. Boas,
     *Mathematical methods in the physical sciences*, 2nd edition,