Merged everything back into develop (not best)

book/0_overview/schedule.md

@@ -1,3 +1,60 @@
 # Weekly schedule
 
-Click on the dropdown blocks below to find the schedule of each week's activities.
+Each unit runs over one week and consists of three lessons, numbered after the unit. Unit 1.4, for example, is made up of lessons 1.4.1, 1.4.2 and 1.4.3. The first two are lectures and the third is the practical, where the computer labs in this book are used.
+
+[My Timetable](https://mytimetable.tudelft.nl/) carries the authoritative schedule, with rooms and any changes.
+
+## Quarter 1
+
+| Unit | Topic | Lesson | Date | Time |
+| :--- | :--- | :--- | :--- | :--- |
+| 1.1 | Gradient and divergence | 1.1.1 | Tue 1 Sep 2026 | 08:45-10:30 |
+|  |  | 1.1.2 | Thu 3 Sep 2026 | 15:45-17:30 |
+|  |  | 1.1.3 | Fri 4 Sep 2026 | 13:45-15:30 |
+| 1.2 | Curl | 1.2.1 | Mon 7 Sep 2026 | 10:45-12:30 |
+|  |  | 1.2.2 | Tue 8 Sep 2026 | 13:45-15:30 |
+|  |  | 1.2.3 | Fri 11 Sep 2026 | 13:45-16:30 |
+| 1.3 | Potential fields: history and experiments | 1.3.1 | Mon 14 Sep 2026 | 10:45-12:30 |
+|  |  | 1.3.2 | Tue 15 Sep 2026 | 13:45-15:30 |
+|  |  | 1.3.3 | Fri 18 Sep 2026 | 13:45-15:30 |
+| 1.4 | Potential fields: gravity, magnetic field of the Earth | 1.4.1 | Mon 21 Sep 2026 | 10:45-12:30 |
+|  |  | 1.4.2 | Tue 22 Sep 2026 | 13:45-15:30 |
+|  |  | 1.4.3 | Fri 25 Sep 2026 | 13:45-16:30 |
+| 1.5 | Electric field. Diffusion fields: hot wire | 1.5.1 | Mon 28 Sep 2026 | 10:45-12:30 |
+|  |  | 1.5.2 | Tue 29 Sep 2026 | 13:45-15:30 |
+|  |  | 1.5.3 | Fri 2 Oct 2026 | 13:45-15:30 |
+| 1.6 | Diffusion fields: boundary conditions, heat in 2D and 3D | 1.6.1 | Mon 5 Oct 2026 | 10:45-12:30 |
+|  |  | 1.6.2 | Tue 6 Oct 2026 | 15:45-17:30 |
+|  |  | 1.6.3 | Fri 9 Oct 2026 | 13:45-16:30 |
+| 1.7 | Mechanical waves: strings, acoustic waves, 2D and 3D | 1.7.1 | Mon 12 Oct 2026 | 10:45-12:30 |
+|  |  | 1.7.2 | Tue 13 Oct 2026 | 13:45-15:30 |
+|  |  | 1.7.3 | Fri 16 Oct 2026 | 13:45-15:30 |
+| 1.8 | Mechanical waves: power flux | 1.8.1 | Mon 19 Oct 2026 | 10:45-12:30 |
+|  |  | 1.8.2 | Tue 20 Oct 2026 | 13:45-15:30 |
+|  |  | 1.8.3 | Fri 23 Oct 2026 | 13:45-16:30 |
+| 1.9 | Unsupervised study |  |  |  |
+| 1.10 | Midterm week: exam and discussion of solutions | 1.10.1 |  |  |
+|  |  | 1.10.2 |  |  |
+|  |  | 1.10.3 | Fri 6 Nov 2026 | 13:30-16:30 |
+
+## Quarter 2
+
+| Unit | Topic | Lesson | Date | Time |
+| :--- | :--- | :--- | :--- | :--- |
+| 2.1 | Electromagnetism: Maxwell's equations, plane waves, telegraph equation | 2.1.1 | Mon 9 Nov 2026 | 13:45-15:30 |
+|  |  | 2.1.2 | Tue 10 Nov 2026 | 13:45-15:30 |
+|  |  | 2.1.3 | Fri 13 Nov 2026 | 10:45-12:30 |
+| 2.2 | 3D waves, Poynting vector, polarisation, lossy and lossless media | 2.2.1 | Mon 16 Nov 2026 | 13:45-15:30 |
+|  |  | 2.2.2 | Tue 17 Nov 2026 | 13:45-15:30 |
+|  |  | 2.2.3 | Fri 20 Nov 2026 | 09:45-12:30 |
+| 2.3 | Reflection, transmission, refraction. Multi-layered media | 2.3.1 | Mon 23 Nov 2026 | 13:45-15:30 |
+|  |  | 2.3.2 | Tue 24 Nov 2026 | 13:45-15:30 |
+|  |  | 2.3.3 | Fri 27 Nov 2026 | 10:45-12:30 |
+| 2.4 | Trapped and surface waves. Phase and group velocity | 2.4.1 | Mon 30 Nov 2026 | 13:45-15:30 |
+|  |  | 2.4.2 | Tue 1 Dec 2026 | 08:45-10:30 |
+|  |  | 2.4.3 | Fri 4 Dec 2026 | 09:45-12:30 |
+| 2.8 | Unsupervised study |  |  |  |
+| 2.9 | Unsupervised study |  |  |  |
+| 2.10 | Resit exam |  | Wed 27 Jan 2027 | 13:30-16:30 |
+
+The exam is on Wed 16 Dec 2026, 13:30-16:30. The longer practicals, running to 16:30, end with a formative assessment.

book/1_gradient_divergence_curl/curl.md

@@ -0,0 +1,106 @@
+# Curl
+
+The curl is a peculiar operator because it turns a vector field into another vector field. Often a determinant expression is used, but we can give it in terms of even and odd permutations of the triplet $(x,y,z)$, which is easy to remember. Let $\boldsymbol v(\boldsymbol r,t)$ be a continuously differentiable space-time function. Then
+
+$$
+\nabla\times\boldsymbol v = \hat{\boldsymbol x}(\partial_y v_z - \partial_z v_y) + \hat{\boldsymbol y}(\partial_z v_x - \partial_x v_z) + \hat{\boldsymbol z}(\partial_x v_y - \partial_y v_x).
+$$ (eq:curl)
+
+Notice that the three terms have the triplet pairs $(xyz)-(xzy)$, $(yzx)-(yxz)$, and $(zxy)-(zyx)$ in (1) direction, (2) differentiation, and (3) vector component. The first in each pair is an even permutation of $(xyz)$ and the second is an odd permutation. It can be seen that in each term the direction of the resulting vector $\boldsymbol w = \nabla\times\boldsymbol v$ is perpendicular both to the direction of differentiation and to the component of $\boldsymbol v$ that is being differentiated. The curl is interpreted as the rate of circulation of the vector field $\boldsymbol v$.
+
+## The circulation integral
+
+Let us first look at the situation of an elementary area with centre point $\boldsymbol r_p = \left(\tfrac{1}{2}\mathrm{d}x,\ \tfrac{1}{2}\mathrm{d}y,\ \tfrac{1}{2}\mathrm{d}z\right)$. The circulation integral for the path from $y=0$ to $y=\mathrm{d}y$ at $x=\tfrac{1}{2}\mathrm{d}x$ and $z=0$ is given by
+
+$$
+\int_{y=0}^{\mathrm{d}y} v_y\!\left(\tfrac{1}{2}\mathrm{d}x,\,y,\,0\right)\mathrm{d}l ,
+$$
+
+and completing the circulation for all four sides of the elementary area as shown in {numref}`fig-circulation`, we find
+
+```{figure} figures/circulation.png
+:name: fig-circulation
+:width: 55%
+
+Net circulation integral on a rectangle around the point $\boldsymbol r_p$.
+```
+
+$$
+\begin{aligned}
+\oint_{\boldsymbol r}\boldsymbol\tau\cdot\boldsymbol v\,\mathrm{d}l
+&= \int_{y=0}^{\mathrm{d}y} v_y\!\left(\tfrac{1}{2}\mathrm{d}x,\,y,\,0\right)\mathrm{d}l
+ + \int_{z=0}^{\mathrm{d}z} v_z\!\left(\tfrac{1}{2}\mathrm{d}x,\,\mathrm{d}y,\,z\right)\mathrm{d}l \\
+&\quad - \int_{y=0}^{\mathrm{d}y} v_y\!\left(\tfrac{1}{2}\mathrm{d}x,\,y,\,\mathrm{d}z\right)\mathrm{d}l
+ - \int_{z=0}^{\mathrm{d}z} v_z\!\left(\tfrac{1}{2}\mathrm{d}x,\,0,\,z\right)\mathrm{d}l ,
+\end{aligned}
+$$ (eq:circint)
+
+where the symbol $\oint$ stands for a closed loop integral, $\boldsymbol\tau$ is the local unit tangent vector along the edges of $\mathbb{S}$, and $l$ is the arclength along the path.
+
+Earlier we introduced the Taylor series expansion for a function in one variable, and we give it here for each vector component of $\boldsymbol v$ around the point $\boldsymbol r_p$,
+
+$$
+\begin{aligned}
+v_x(\boldsymbol r) &= v_x(\boldsymbol r_p) + \partial_x v_x\left(x-\tfrac{1}{2}\mathrm{d}x\right) + \partial_y v_x\left(y-\tfrac{1}{2}\mathrm{d}y\right) + \partial_z v_x\left(z-\tfrac{1}{2}\mathrm{d}z\right) + \text{higher order terms}, \\
+v_y(\boldsymbol r) &= v_y(\boldsymbol r_p) + \partial_x v_y\left(x-\tfrac{1}{2}\mathrm{d}x\right) + \partial_y v_y\left(y-\tfrac{1}{2}\mathrm{d}y\right) + \partial_z v_y\left(z-\tfrac{1}{2}\mathrm{d}z\right) + \text{higher order terms}, \\
+v_z(\boldsymbol r) &= v_z(\boldsymbol r_p) + \partial_x v_z\left(x-\tfrac{1}{2}\mathrm{d}x\right) + \partial_y v_z\left(y-\tfrac{1}{2}\mathrm{d}y\right) + \partial_z v_z\left(z-\tfrac{1}{2}\mathrm{d}z\right) + \text{higher order terms},
+\end{aligned}
+$$
+
+which is the Taylor series expansion to first order. Substituting these results in {eq}`eq:circint` leads to
+
+$$
+\oint_{\boldsymbol r}\boldsymbol\tau\cdot\boldsymbol v\,\mathrm{d}l = (\partial_y v_z - \partial_z v_y)\,\mathrm{d}y\,\mathrm{d}z + \text{higher order terms}.
+$$
+
+Similarly, the net circulation integral for the elementary surface area perpendicular to the $\hat{\boldsymbol y}$-axis is $(\partial_z v_x - \partial_x v_z)\,\mathrm{d}x\,\mathrm{d}z$, and the net circulation integral for the elementary surface area perpendicular to the $\hat{\boldsymbol z}$-axis is $(\partial_x v_y - \partial_y v_x)\,\mathrm{d}x\,\mathrm{d}y$.
+
+For this reason, the physical interpretation of the curl is the net circulation per unit of surface area at the point $\boldsymbol r_p$, and it can be expressed as
+
+$$
+\hat{\boldsymbol n}\cdot(\nabla\times\boldsymbol v) = \lim_{S\rightarrow 0}\frac{\oint_{\boldsymbol r}\boldsymbol\tau\cdot\boldsymbol v\,\mathrm{d}l}{A},
+$$ (eq:circ1)
+
+in which
+
+$$
+A = \int_{\boldsymbol r\in\mathbb{S}}\mathrm{d}S
+$$
+
+is the surface area of the surface $\mathbb{S}$.
+
+:::{admonition} The curl in words
+:class: tip
+The curl of a vector field is the net circulation of that field per unit of surface area, and it points along the normal of the surface for which that circulation is largest.
+:::
+
+## Stokes' integral theorem
+
+The unit normal $\hat{\boldsymbol n}$ in {eq}`eq:circ1` is the normal vector to the surface area $\mathbb{S}$ and is oriented to the side of advance of a right-hand screw as it is turned in the direction of $\boldsymbol\tau$ around the boundary of $\mathbb{S}$. This leads to Stokes' integral theorem, given by
+
+$$
+\oint_{\boldsymbol r}\boldsymbol\tau\cdot\boldsymbol v\,\mathrm{d}l = \int_{\boldsymbol r\in\mathbb{S}}\hat{\boldsymbol n}\cdot(\nabla\times\boldsymbol v)\,\mathrm{d}S .
+$$ (eq:stokes)
+
+## Exercises
+
+1. Use Stokes' theorem to evaluate $\oint_{\boldsymbol r}\boldsymbol\tau\,\mathrm{d}l$, where $\boldsymbol\tau$ is the unit tangent along the closed boundary of the area $\mathbb{S}$. The integration runs in the direction of circulation that forms a right-handed system with the unit normal vector on $\mathbb{S}$.
+2. Show that when $\boldsymbol v(\boldsymbol r) = \boldsymbol a\,p(\boldsymbol r)$, where $\boldsymbol a$ is an arbitrary constant vector and $p(\boldsymbol r)$ is a continuously differentiable scalar function, Stokes' integral theorem gives
+
+    $$
+    \oint_{\boldsymbol r}\boldsymbol\tau\,p\,\mathrm{d}l = \int_{\boldsymbol r\in\mathbb{S}}(\hat{\boldsymbol n}\times\nabla)p\,\mathrm{d}S,
+    $$
+
+    which is Stokes' theorem for the gradient.
+3. Show that when $\boldsymbol v(\boldsymbol r) = \boldsymbol a\times\boldsymbol w(\boldsymbol r)$, where $\boldsymbol a$ is an arbitrary constant vector and $\boldsymbol w(\boldsymbol r)$ is a continuously differentiable vector function, Stokes' integral theorem gives
+
+    $$
+    \oint_{\boldsymbol r}\boldsymbol\tau\times\boldsymbol w\,\mathrm{d}l = \int_{\boldsymbol r\in\mathbb{S}}(\hat{\boldsymbol n}\times\nabla)\times\boldsymbol w\,\mathrm{d}S .
+    $$
+4. Ampère's law is given by $\nabla\times\boldsymbol H = \boldsymbol J$, which states that the electric current is equal to the curl of the magnetic field. Convert this to integral form using Stokes' theorem, by using a flat surface $\mathbb{S}$ and choosing a unit normal vector $\hat{\boldsymbol n}$ on $\mathbb{S}$. You should find
+
+    $$
+    \oint_{\boldsymbol r}\boldsymbol H\cdot\boldsymbol\tau\,\mathrm{d}l = \int_{\boldsymbol r\in\mathbb{S}}\hat{\boldsymbol n}\cdot\boldsymbol J\,\mathrm{d}S .
+    $$
+
+    Show in which direction the unit tangent vector $\boldsymbol\tau$ is circulating at the boundary of $\mathbb{S}$, and in which direction through the surface a positive current should run.