diff --git a/doc/source/tutorial/index.rst b/doc/source/tutorial/index.rst
index 243ef15fa37a8a1ef24aa5029822c7c1886d7096..aa78d2fcde96bdc8a0068f1445ffb43f09c80868 100644
--- a/doc/source/tutorial/index.rst
+++ b/doc/source/tutorial/index.rst
@@ -1,4 +1,4 @@
-Tutorial: Learning Kwant through examples
+Tutorial: learning Kwant through examples
=========================================
.. toctree::
diff --git a/doc/source/tutorial/spin_orbit.py b/doc/source/tutorial/spin_orbit.py
index 8f3a35a681557af5b7d89a09a5bf0f8a91160258..5b1ddbff6ccf5c7801154772e092c31259ee3047 100644
--- a/doc/source/tutorial/spin_orbit.py
+++ b/doc/source/tutorial/spin_orbit.py
@@ -38,14 +38,14 @@ def make_system(a=1, t=1.0, alpha=0.5, e_z=0.08, W=10, L=30):
#### Define the scattering region. ####
#HIDDEN_BEGIN_uxrm
- sys[(lat(x, y) for x in range(L) for y in range(W))] = 4 * t * sigma_0 + \
- e_z * sigma_z
+ sys[(lat(x, y) for x in range(L) for y in range(W))] = \
+ 4 * t * sigma_0 + e_z * sigma_z
# hoppings in x-direction
- sys[kwant.builder.HoppingKind((1, 0), lat, lat)] = -t * sigma_0 - \
- 1j * alpha * sigma_y
+ sys[kwant.builder.HoppingKind((1, 0), lat, lat)] = \
+ -t * sigma_0 - 1j * alpha * sigma_y
# hoppings in y-directions
- sys[kwant.builder.HoppingKind((0, 1), lat, lat)] = -t * sigma_0 + \
- 1j * alpha * sigma_x
+ sys[kwant.builder.HoppingKind((0, 1), lat, lat)] = \
+ -t * sigma_0 + 1j * alpha * sigma_x
#HIDDEN_END_uxrm
#### Define the left lead. ####
@@ -54,11 +54,11 @@ def make_system(a=1, t=1.0, alpha=0.5, e_z=0.08, W=10, L=30):
#HIDDEN_BEGIN_yliu
lead[(lat(0, j) for j in xrange(W))] = 4 * t * sigma_0 + e_z * sigma_z
# hoppings in x-direction
- lead[kwant.builder.HoppingKind((1, 0), lat, lat)] = -t * sigma_0 - \
- 1j * alpha * sigma_y
+ lead[kwant.builder.HoppingKind((1, 0), lat, lat)] = \
+ -t * sigma_0 - 1j * alpha * sigma_y
# hoppings in y-directions
- lead[kwant.builder.HoppingKind((0, 1), lat, lat)] = -t * sigma_0 + \
- 1j * alpha * sigma_x
+ lead[kwant.builder.HoppingKind((0, 1), lat, lat)] = \
+ -t * sigma_0 + 1j * alpha * sigma_x
#HIDDEN_END_yliu
#### Attach the leads and return the finalized system. ####
diff --git a/doc/source/tutorial/tutorial1.rst b/doc/source/tutorial/tutorial1.rst
index 570382fc0e044ac0d7b365de79be2681bb9cea87..7e0e7821bbc466efcb7ca4ce6ae215ec1d206d72 100644
--- a/doc/source/tutorial/tutorial1.rst
+++ b/doc/source/tutorial/tutorial1.rst
@@ -1,5 +1,5 @@
-First steps in Kwant: Setting up a simple system and computing transport
-------------------------------------------------------------------------
+First steps: setting up a simple system and computing conductance
+-----------------------------------------------------------------
Discretization of a Schrödinger Hamiltonian
...........................................
@@ -321,20 +321,19 @@ subbands that increases with energy.
.. [#] http://xkcd.com/353/
.. [#] Leads are numbered in the python convention, starting from 0.
-The same but different: Alternative system building
-...................................................
+Building the same system with less code
+.......................................
-Kwant is very flexible, and often allows you more than one way to
-build up your system. The reason is that `~kwant.builder.Builder`
-is essentially just a container, and allows for different
-ways to be filled. Here we present a more compact rewrite of
-the previous example (still with the same results).
+Kwant allows for more than one way to build a system. The reason is that
+`~kwant.builder.Builder` is essentially just a container that can be filled in
+different ways. Here we present a more compact rewrite of the previous example
+(still with the same results).
-Also, the previous example was written in the form of a pythons script
-with little structure, and everything governed by global variables.
-This is OK for such a simple example, but for larger projects it makes
-sense to structure different functionality into different functional
-entities. In this example we therefore also aim at more structure.
+Also, the previous example was written in the form of a Python script with
+little structure, and with everything governed by global variables. This is OK
+for such a simple example, but for larger projects it makes sense to partition
+the code into separate entities. In this example we therefore also aim at more
+structure.
We begin the program collecting all imports in the beginning of the
file and put the build-up of the system into a separate function
diff --git a/doc/source/tutorial/tutorial2.rst b/doc/source/tutorial/tutorial2.rst
index 3fb0eeba879665333868584df7cec4ec2c2f5f10..8cdf7c1a4d53c11768c8b96918e791b8217b3d6b 100644
--- a/doc/source/tutorial/tutorial2.rst
+++ b/doc/source/tutorial/tutorial2.rst
@@ -1,15 +1,16 @@
-Adding more structure to the problem
-------------------------------------
+More interesting systems: spin, potential, shape
+------------------------------------------------
+
+Each of the following three examples highlights different ways to go beyond the
+very simple examples of the previous section.
.. _tutorial_spinorbit:
Matrix structure of on-site and hopping elements
................................................
-In the next examples, we will extend the previous examples and add more
-structure. We begin by extending the simple 2DEG-Hamiltonian by
-a Rashba spin-orbit coupling and a Zeeman splitting due to
-an external magnetic field:
+We begin by extending the simple 2DEG-Hamiltonian by a Rashba spin-orbit
+coupling and a Zeeman splitting due to an external magnetic field:
.. math::
diff --git a/doc/source/tutorial/tutorial3.rst b/doc/source/tutorial/tutorial3.rst
index 67e3007b5e7f864e5ab9de0aa354f5e65a22b001..be8f23b7e6385126ccfa791367cc59032fac8c01 100644
--- a/doc/source/tutorial/tutorial3.rst
+++ b/doc/source/tutorial/tutorial3.rst
@@ -1,5 +1,5 @@
-Beyond transport: Band structures and closed systems
-----------------------------------------------------
+Beyond transport: Band structure and closed systems
+---------------------------------------------------
Band structure calculations
...........................
diff --git a/doc/source/tutorial/tutorial4.rst b/doc/source/tutorial/tutorial4.rst
index 35cbfaf14aebda8fd0eb4753f3f2ff9ca7571695..111cf76099d3d632420a6e68d10ea1306c6c730a 100644
--- a/doc/source/tutorial/tutorial4.rst
+++ b/doc/source/tutorial/tutorial4.rst
@@ -1,7 +1,7 @@
.. _tutorial-graphene:
-Using a more complicated lattice (graphene)
--------------------------------------------
+Beyond square lattices: graphene
+--------------------------------
In the following example, we are going to calculate the
conductance through a graphene quantum dot with a p-n junction
diff --git a/doc/source/tutorial/tutorial5.rst b/doc/source/tutorial/tutorial5.rst
index 63e891bce6c0f35247dd9c07fda273e71f0e2fb1..f723f8d2f1fd5572aee2731513eabed473954578 100644
--- a/doc/source/tutorial/tutorial5.rst
+++ b/doc/source/tutorial/tutorial5.rst
@@ -1,5 +1,5 @@
-Superconductors: orbital vs lattice degrees of freedom
-------------------------------------------------------
+Superconductors: orbital vs. lattice degrees of freedom
+-------------------------------------------------------
This example deals with superconductivity on the level of the
Bogoliubov-de Gennes (BdG) equation. In this framework, the Hamiltonian
@@ -21,7 +21,7 @@ For this we restrict ourselves to a simple spin-less system without
magnetic field, so that :math:`\Delta` is just a number (which we
choose real), and :math:`\mathcal{T}H\mathcal{T}^{-1}=H_0^*=H_0`.
-"Orbital description": Using matrices
+"Orbital description": using matrices
.....................................
We begin by computing the band structure of a superconducting wire.
@@ -48,7 +48,7 @@ he?
:download:`tutorial/superconductor_band_structure.py <../../../tutorial/superconductor_band_structure.py>`
-"Lattice description": Using different lattices
+"Lattice description": using different lattices
...............................................
While it seems most natural to implement the BdG Hamiltonian