From 96eb823c9fb82d95bd7c5df167b9f36e7e0b633e Mon Sep 17 00:00:00 2001
From: Maciej Topyla <m.m.topyla@student.tudelft.nl>
Date: Sat, 3 Sep 2022 14:30:20 +0000
Subject: [PATCH] Update src/2_coordinates.md, src/1_complex_numbers.md
---
src/1_complex_numbers.md | 6 +++---
src/2_coordinates.md | 24 +++++++++++-------------
2 files changed, 14 insertions(+), 16 deletions(-)
diff --git a/src/1_complex_numbers.md b/src/1_complex_numbers.md
index 5edd355..6af7cfe 100644
--- a/src/1_complex_numbers.md
+++ b/src/1_complex_numbers.md
@@ -6,9 +6,9 @@ title: Complex Numbers
The lecture on complex numbers consists of three parts, each with their own video:
-- [1.1. Definition and basic operations](#definition-and-basic-operations)
-- [1.2. Complex functions](#complex-functions)
-- [1.3. Differentiation and integration](#differentiation-and-integration)
+- [1.1. Definition and basic operations](#1.1.-definition-and-basic-operations)
+- [1.2. Complex functions](#1.2.-complex-functions)
+- [1.3. Differentiation and integration](#1.3.-differentiation-and-integration)
**Total video length: 38 minutes and 53 seconds**
diff --git a/src/2_coordinates.md b/src/2_coordinates.md
index 83249d4..d955feb 100644
--- a/src/2_coordinates.md
+++ b/src/2_coordinates.md
@@ -5,13 +5,13 @@ title: Coordinates
The lecture on coordinate systems consists of 3 parts, each with their own video:
-- [Introduction to coordinate systems: Cartesian and polar](#introduction-to-coordinate-systems-cartesian-and-polar)
-- [Converting derivatives between coordinate systems](#converting-derivatives-between-coordinate-systems)
-- [Coordinate systems in 3D](#coordinate-systems-in-3d)
+- [2.1. Introduction to coordinate systems: Cartesian and polar](#2.1.-introduction-to-coordinate-systems-cartesian-and-polar)
+- [2.2. Converting derivatives between coordinate systems](#2.2.-converting-derivatives-between-coordinate-systems)
+- [2.3. Coordinate systems in 3D](#2.3.-coordinate-systems-in-3d)
**Total video length: 35 minutes and 13 seconds**
-## Introduction to coordinate systems: Cartesian and polar
+## 2.1. Introduction to coordinate systems: Cartesian and polar
<iframe width="100%" height=315 src="https://www.youtube-nocookie.com/embed/CPMrsQlNxS8?rel=0" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
@@ -21,11 +21,11 @@ The most common coordinates are *Cartesian coordinates*, where we use a
number $n$ of perpendicular axes. The coordinates corresponding to these
axes are $x_j$ where $j=1, \ldots, n$.
-Cartesian coordinates are simple, as the coordinate axis are simply
-straight lines and perpendicular to each other. Due to this, it is
+Cartesian coordinates are simple to describe and operate in. The coordinate axes are
+straight lines perpendicular to each other. It is therefore
very easy to do calculations in Cartesian coordinates. For example,
the distance $\Delta s$ between two points $(x_1, x_2, \ldots, x_n)$
-and $(x'_1, x'_2, \ldots, x'_n)$ is easily computed as
+and $(x'_1, x'_2, \ldots, x'_n)$ can be quickly computed using a general formula for n-dimensions:
$$\Delta s^2 = (x'_1 - x_1)^2 + (x'_2 - x_2)^2 + \ldots + (x'_n - x_n)^2.$$
@@ -34,14 +34,12 @@ space*.)
In mathematics, we are often dealing with so-called *infinitesimally* small
distances, for example in the definition of derivatives and integrals.
-In Cartesion coordinates the expressions for infinitesimal distances $ds$ and
+In Cartesian coordinates, the expressions for infinitesimal distances $ds$ and
infinitesimal volumes $dV$ are given as
-$$ds = \sqrt{dx_1^2 + dx_2^2 + \ldots + dx_n^2}$$
-
-and
-
-$$dV = dx_1 dx_2 \ldots dx_N.$$
+!!! info "Segment and volume element in n-dimensional Cartesian coordinates"
+ $$ds = \sqrt{dx_1^2 + dx_2^2 + \ldots + dx_n^2}$$
+ $$dV = dx_1 dx_2 \ldots dx_N.$$
The formula for $dV$ also indicates that in Cartesian coordinates, the integral
over a volume can be expressed as individual integrals over all coordinate directions:
--
GitLab