From 966048bd3594eac4d3398992c8ad3143e290303b Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 8 Apr 2021 16:49:46 +0200
Subject: Expand knowledge base, add /sources/

---
 content/know/concept/material-derivative/index.pdc | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

(limited to 'content/know/concept/material-derivative')

diff --git a/content/know/concept/material-derivative/index.pdc b/content/know/concept/material-derivative/index.pdc
index af65ca0..1c6bfdc 100644
--- a/content/know/concept/material-derivative/index.pdc
+++ b/content/know/concept/material-derivative/index.pdc
@@ -57,7 +57,7 @@ then we can rewrite this expression like so:
 
 $$\begin{aligned}
     \dv{t} f\big(x(t), y(t), z(t), t\big)
-    &= \pdv{f}{t} + \va{v} \cdot \nabla f
+    &= \pdv{f}{t} + (\va{v} \cdot \nabla) f
 \end{aligned}$$
 
 Note that $\va{v} = \va{v}(\va{r}, t)$,
@@ -73,7 +73,7 @@ and is known as the **material derivative** or **comoving derivative**:
 $$\begin{aligned}
     \boxed{
         \frac{\mathrm{D}f}{\mathrm{D}t}
-        \equiv \pdv{f}{t} + \va{v} \cdot \nabla f
+        \equiv \pdv{f}{t} + (\va{v} \cdot \nabla) f
     }
 \end{aligned}$$
 
@@ -89,14 +89,14 @@ but in fact the definition also works for vector fields $\va{U}(\va{r}, t)$:
 $$\begin{aligned}
     \boxed{
         \frac{\mathrm{D} \va{U}}{\mathrm{D}t}
-        \equiv \pdv{f}{t} + \va{v} \cdot \nabla \va{U}
+        \equiv \pdv{\va{U}}{t} + (\va{v} \cdot \nabla) \va{U}
     }
 \end{aligned}$$
 
 Where the advective term is to be evaluated in the following way in Cartesian coordinates:
 
 $$\begin{aligned}
-    \va{v} \cdot \nabla \va{U}
+    (\va{v} \cdot \nabla) \va{U}
     =
     \begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix}
     \cdot
-- 
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