From cc391ce3b9867d88d124e147931d33be34e756fc Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Mon, 14 Sep 2026 18:11:38 +0200
Subject: Improve knowledge base

---
 source/know/concept/bernoullis-theorem/index.md | 29 ++++++-------------------
 1 file changed, 7 insertions(+), 22 deletions(-)

(limited to 'source/know/concept/bernoullis-theorem/index.md')

diff --git a/source/know/concept/bernoullis-theorem/index.md b/source/know/concept/bernoullis-theorem/index.md
index 867c443..2795f22 100644
--- a/source/know/concept/bernoullis-theorem/index.md
+++ b/source/know/concept/bernoullis-theorem/index.md
@@ -12,36 +12,21 @@ layout: "concept"
 For inviscid fluids, **Bernouilli's theorem** states
 that an increase in flow velocity $$\va{v}$$ is paired
 with a decrease in pressure $$p$$ and/or potential energy.
-For a qualitative argument, look no further than
-one of the [Euler equations](/know/concept/euler-equations/),
-with a [material derivative](/know/concept/material-derivative/):
-
-$$\begin{aligned}
-    \frac{\mathrm{D} \va{v}}{\mathrm{D} t}
-    = \pdv{\va{v}}{t} + (\va{v} \cdot \nabla) \va{v}
-    = \va{g} - \frac{\nabla p}{\rho}
-\end{aligned}$$
-
-Assuming that $$\va{v}$$ is constant in $$t$$,
-it becomes clear that a higher $$\va{v}$$ requires a lower $$p$$.
-
-
-## Simple form
-
-For an incompressible fluid
+Quantitatively, for an incompressible fluid
 with a time-independent velocity field $$\va{v}$$ (i.e. **steady flow**),
-Bernoulli's theorem formally states that the
-**Bernoulli head** $$H$$ is constant along a streamline:
+Bernoulli's theorem states that
+the **Bernoulli head** $$H$$ is constant along every streamline:
 
 $$\begin{aligned}
     \boxed{
         H
-        = \frac{1}{2} \va{v}^2 + \Phi + \frac{p}{\rho}
+        \equiv \frac{1}{2} |\va{v}|^2 + \Phi + \frac{p}{\rho}
     }
 \end{aligned}$$
 
 Where $$\Phi$$ is the gravitational potential, such that $$\va{g} = - \nabla \Phi$$.
-To prove this theorem, we take the material derivative of $$H$$:
+To prove this theorem, we take the 
+[material derivative](/know/concept/material-derivative/) of $$H$$:
 
 $$\begin{aligned}
     \frac{\mathrm{D} H}{\mathrm{D} t}
@@ -50,7 +35,7 @@ $$\begin{aligned}
     + \frac{1}{\rho} \frac{\mathrm{D} p}{\mathrm{D} t}
 \end{aligned}$$
 
-In the first term we insert the Euler equation,
+In the first term we insert the [Euler equation](/know/concept/euler-equations/),
 and in the other two we expand the derivatives:
 
 $$\begin{aligned}
-- 
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