Add updated maths and fitting tutorials to the documentation

docs/docs/index.md

@@ -11,11 +11,11 @@ Here is a brief overview of the main documentation sections:
   – Guides users through system requirements, environment configuration,
   and the installation process.
 - [:material-book-open-variant: User Guide](user-guide/index.md) –
-  Covers core concepts, key terminology, workflow steps, and essential
-  parameters for effective use of EasyScience.
+  Explains the core concepts of EasyScience and how to control its log
+  output.
 - [:material-school: Tutorials](tutorials/index.md) – Offers practical,
-  step-by-step examples demonstrating common workflows and data analysis
-  tasks.
+  step-by-step examples, from the basics of working with parameters and
+  maths functions to common workflows and data analysis tasks.
 - [:material-code-braces-box: API Reference](api-reference/index.md) –
   An auto-generated reference detailing the available functions and
   modules in EasyScience.

docs/docs/introduction/index.md

@@ -17,6 +17,53 @@ making data analysis easier.
 [EasyScience framework](https://easyscience.org).
 -->
 
+## Who is it for
+
+**EasyScience** is the foundation for scientific libraries and
+applications. It is for developers who build on its numbers, which carry
+a **unit**, an **uncertainty** and **bounds**, and for scientists who
+use them directly to analyse data. It provides:
+
+- numbers (`Parameter`, `DescriptorNumber`) that know their unit,
+  uncertainty and bounds, with arithmetic and mathematical functions
+  that carry all three along,
+- fitting of a model to data with several minimizers, and Bayesian
+  sampling of the posterior.
+
+## A first example
+
+Bragg's law gives the lattice spacing from a wavelength and the Bragg
+angle. Each number has an uncertainty, and the result has one too:
+
+```python
+import easyscience as es
+from easyscience import DescriptorNumber
+
+wavelength = DescriptorNumber(name='wavelength', value=1.54, unit='angstrom', variance=0.01**2)
+angle = DescriptorNumber(name='angle', value=22.5, unit='deg', variance=0.1**2)
+
+d = wavelength / (2 * es.math.sin(angle))
+d.convert_unit('nm')
+print(f'd = {d.value:.4f} ± {d.error:.4f} {d.unit}')
+```
+
+```text
+d = 0.2012 ± 0.0016 nm
+```
+
+## Where to start
+
+- **New to EasyScience?** Follow the
+  [Installation & Setup](../installation-and-setup/index.md), then the
+  [Basics](../tutorials/basics/index.md) tutorials.
+- **Want to fit data?** Continue with the
+  [Fitting](../tutorials/fitting/index.md) tutorials.
+- **Building on EasyScience?** Read the
+  [Concepts](../user-guide/concepts.md) and the
+  [API Reference](../api-reference/index.md).
+- **Following a workshop?** See
+  [Workshops & Schools](../tutorials/workshops/index.md).
+
 ## License
 
 **EasyScience** library is released under the

docs/docs/tutorials/advanced/index.md

@@ -0,0 +1,16 @@
+---
+icon: material/tune-vertical
+---
+
+# :material-tune-vertical: Advanced Topics
+
+These notebooks go beyond a single least-squares fit. They assume that
+you are familiar with the [Fitting](../fitting/index.md) notebooks.
+
+- [Bayesian analysis](../fitting-bayesian.ipynb) – Run Bayesian MCMC
+  sampling on a model with EasyScience to obtain full posterior
+  distributions for the parameters. For derived quantities from the
+  posterior, see [Bayesian QENS](../fitting/bayesian-qens.ipynb).
+- [Progress Callback](../progress-callback.ipynb) – Monitor fitting
+  progress across minimizer backends and update a notebook interface
+  while a fit is running.

docs/docs/tutorials/basics/index.md

@@ -0,0 +1,59 @@
+---
+icon: material/calculator-variant
+---
+
+# :material-calculator-variant: Basics
+
+These short notebooks show how to work with EasyScience numbers: values
+that carry a **unit**, an **uncertainty** and, for fit parameters,
+**bounds**. You need Python and a little NumPy; no earlier knowledge of
+EasyScience is required.
+
+Every notebook stands alone, so you can read them in any order. If you
+are new, follow the path below.
+
+## A suggested path
+
+**1. Numbers with units and uncertainties**
+
+- [Arithmetic](maths-arithmetic.ipynb) – Add, subtract, multiply and
+  divide numbers with units, uncertainties and bounds, and see how each
+  of them carries over to the result. _Start here._
+
+**2. Functions**
+
+Read the ones you need; each stands alone.
+
+- [Exponential and logarithm](maths-exp-log.ipynb) – `exp`, `log`,
+  `log10`, `sqrt` and the error functions `erf` and `erfc`.
+- [Trigonometric functions](maths-trigonometric.ipynb) – `sin`, `cos`,
+  `tan` and `sinc`, with Bragg's law as the running example.
+- [Inverse trigonometric functions](maths-inverse-trigonometric.ipynb) –
+  `arcsin`, `arccos`, `arctan` and `arctan2`, and the values they
+  refuse.
+- [Hyperbolic functions](maths-hyperbolic.ipynb) – `sinh`, `cosh`,
+  `tanh` and their inverses.
+
+**3. Trusting a result**
+
+- [Error propagation](maths-error-propagation.ipynb) – How the choice of
+  formula changes the uncertainty of a result, with the area of a table
+  as the running example. _A step further._
+- [Correlation](maths-correlation.ipynb) – Correlated measurements and
+  fit parameters, and what ignoring the correlation costs. _A step
+  further._
+
+When you are comfortable with these, continue with the
+[Fitting introduction](../fitting/fitting-intro.ipynb), which uses the
+same numbers on the result of a fit.
+
+## How the notebooks work
+
+- Each part ends with a few **exercises**. The hints and the solution
+  are folded away and open on a click, so you can read straight through
+  or try each exercise first.
+- A solution is a worked answer: its code runs and the result is shown.
+- The charts are interactive, and they need an internet connection to
+  load.
+- Each notebook can be downloaded and run on your computer; see
+  [how to run the tutorials](../../installation-and-setup/index.md#how-to-run-tutorials).