diff --git a/src/1_complex_numbers.md b/src/1_complex_numbers.md
index bb163865b3e4241be15e04edbfe609c98f5991b4..ea01b4f70ae8fb969601c38f458ced421ba45ed7 100644
--- a/src/1_complex_numbers.md
+++ b/src/1_complex_numbers.md
@@ -182,7 +182,7 @@ Furthermore, we can define the sine and cosine in terms of complex exponentials:
$$\sin(x) = \frac{e^{{\rm i} x} - e^{-{\rm i} x}}{2i}$$
Most operations on complex numbers become easier when complex numbers are converted to their *polar form* using the complex exponential.
-Some functions and operations, which are common in real analysis, can be easily derived for their complex counterparts by sustituting the real variable $x$ with the complex variable $z$ in its polar form:
+Some functions and operations, which are common in real analysis, can be easily derived for their complex counterparts by substituting the real variable $x$ with the complex variable $z$ in its polar form:
!!! info "Examples of some complex functions stated using polar form"
$$z^{n} = \left(r e^{{\rm i} \varphi}\right)^{n} = r^{n} e^{{\rm i} n \varphi}$$
$$\sqrt[n]{z} = \sqrt[n]{r e^{{\rm i} \varphi} } = \sqrt[n]{r} e^{{\rm i}\varphi/n} $$
@@ -219,7 +219,7 @@ We can then regard the complex ${\rm i}$ as another constant, and use our usual
## 1.4. Bonus: the complex exponential function and trigonometry
-Let us show some tricks in the folloiwing examples where the simple properties of the exponential
+Let us show some tricks in the following examples where the simple properties of the exponential
function help in re-deriving trigonometric identities.
!!! example "Properties of the complex exponential function I"
@@ -300,7 +300,7 @@ function help in re-deriving trigonometric identities.
(b) Evaluate $$\left| \frac{a+b\rm i}{a-b\rm i} \right|$$
for real $a$ and $b$.
-5. [:sweat:] For any given complex number $z$, we can take the inverse $\frac{1}{z}$. Visualize taking the inverse in the complex plane. What geomtric operation does taking the inverse correspond to? (Hint: first consider what geometric operation $\frac{1}{z^*}$ corresponds to.)
+5. [:sweat:] For any given complex number $z$, we can take the inverse $\frac{1}{z}$. Visualize taking the inverse in the complex plane. What geometric operation does taking the inverse correspond to? (Hint: first consider what geometric operation $\frac{1}{z^*}$ corresponds to.)
6. [:grinning:] Compute (a)
$$\frac{d}{dt} e^{{\rm i} (kx-\omega t)},$$
diff --git a/src/2_coordinates.md b/src/2_coordinates.md
index 2abb3ae5c7607e0bd8ef3de0273eb43a0a73f0ad..1d7e964d703426f1beed86f499fe8ba85efe3e33 100644
--- a/src/2_coordinates.md
+++ b/src/2_coordinates.md
@@ -1,7 +1,7 @@
---
title: Coordinates
---
-# Coordinate systems
+# 2. Coordinate systems
The lecture on coordinate systems consists of 3 parts, each with their own video:
@@ -252,7 +252,7 @@ chosen in physical space, we have two coordinates which have the
dimension of a distance: $r$ and $z$. The other coordinate,
$\varphi$ is of course dimensionless.
-What is the distance travelled along a path when we express this in
+What is the distance traveled along a path when we express this in
cylindrical coordinates? Let’s consider an example shown in the figure below.
<figure markdown>
@@ -284,11 +284,11 @@ sphere which is centered at the origin:
<figure markdown>
<figcaption>The position of a point on the sphere is specified using the radius $r$ and two angles
-$\theta$ and $\phi</figcaption>
+$\theta$ and $\phi$</figcaption>
</figure>
!!! warning
- In mathematics, the angles are often labelled the other way
+ In mathematics, the angles are often labeled the other way
around: there, $\phi$ is used for the angle between a line running from
the origin to the point of interest and the $z$-axis, and $\theta$ for
the angle of the projection of that line with the $x$-axis. The
@@ -300,7 +300,7 @@ The relation between Cartesian and spherical coordinates is defined by:
$$y = r \sin\varphi \sin \vartheta$$ $$z = r \cos\vartheta$$
The inverse transformation is easy to find:
-!!! info "The inverse relatuion between Cartesian and spherical coordinates"
+!!! info "The inverse relation between Cartesian and spherical coordinates"
$$r = \sqrt{x^2+y^2+z^2}$$
$$\theta = \arccos(z/\sqrt{x^2+y^2+z^2})$$
$$\phi = \begin{cases} \arctan(y/x) &{\rm for ~} x>0; \\
@@ -341,7 +341,7 @@ From these arguments we can again also find the volume element, it is
here given as
!!! info "Infinitesimal volume element in spherical coordinates"
-$$dV = r^2 \sin\theta dr d\theta d\varphi.$$
+ $$dV = r^2 \sin\theta dr d\theta d\varphi.$$
## 2.4. Summary
@@ -370,7 +370,7 @@ We have discussed four different coordinate systems:
Infinitesimal volume: $$dV = r dr d\varphi dz.$$
4. !!! tip "Spherical coordinates"
- $${\bf r} = (r, \theta, \phi).$$ This sysytem can be
+ $${\bf r} = (r, \theta, \phi).$$ This system can be
used in three dimensions. It is particularly suitable for systems with spherical
symmetry or functions given in terms of these coordinates. <br/>
Infinitesimal distance:
diff --git a/src/3_vector_spaces.md b/src/3_vector_spaces.md
index ea74338166c0a4aeb7d98a9342dc64ec5a4a98e3..a53e6a6b6385d081e11aec95a350e248468c9cb5 100644
--- a/src/3_vector_spaces.md
+++ b/src/3_vector_spaces.md
@@ -74,8 +74,8 @@ We can consider one example in the two-dimensional real vector space $\mathbb{R}
In this figure, you can see how the same vector $\vec{v}$ can be expressed in two different bases. In the first one (left panel), the Cartesian basis is used and its components are $\vec{v}=(2,2)$. In the second basis (right panel), the components are different, namely $\vec{v}=(2.4 ,0.8)$, while the magnitude and direction of the vector remain unchanged.
-For many problems, both in mathematics and in physics, the appropiate choice of the vector space basis may significantly simplify the
-solution proces.
+For many problems, both in mathematics and in physics, the appropriate choice of the vector space basis may significantly simplify the
+solution process.
## 3.2. Properties of a vector space