BOOK IV in print

We are thrilled to announce that the fourth and final analysis volume in our series Matematisk analys & linjär algebra (Mathematical Analysis & Linear Algebra) is now in print and shipping to bookstores! The book is co-authored with my colleagues Stig Larsson and Axel Målqvist. Its full Swedish title is Flervariabelanalys och partiella differentialekvationer.

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A new book series for engineering maths education

Today’s engineering problems are solved in a digital environment where programming, numerical methods, and simulation are as natural as pencil-and-paper calculations. Our aim with this series is to bridge the gap between traditional calculus courses and the realities students meet later in their studies and careers.

• Integrated analytical and numerical viewpoints from the very first page
• Consistent use of Python as a natural computational tool
• Examples inspired by real and relevant engineering applications
• Rigorous treatment of key concepts (limits, convergence, proofs) balanced with hands-on programming and visualisation
• An extensive digital companion at https://mathbook.se with video lectures, interactive quizzes, full solutions, teaching slides, and project material

After successful classroom trials at Chalmers University of Technology, the approach has proved to boost student activity, motivation, and results – without abandoning the elegance of classical mathematics.

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What’s inside Book IV?

Volume IV brings multivariable calculus and partial differential equations under one roof, tying together theory, computation, and modern applications.

1. Differential Calculus in Several Variables – limits, continuity, Jacobians, the chain rule, Taylor’s theorem, and numerical differentiation.
2. Equation Solving and Optimisation – Newton’s method in ℝⁿ, gradient methods, constrained optimisation, and the implicit/inverse function theorems.
3. Integral Calculus in Several Variables – double and triple integrals, change of variables, moments, and improper integrals.
4. Line & Surface Integrals – vector calculus, Green/Gauss/Stokes theorems, and the differential operators grad, div, and curl.
5. Discretisation & Approximation – meshing, basis construction, interpolation, projection, and numerical quadrature in higher dimensions.
6. Numerical Solution of PDEs – a self-contained introduction to the finite-element method (FEM) for stationary and time-dependent problems, with full Python implementations.
7. Applications – from advection-diffusion and Schrödinger’s equation to an introductory chapter on AI & machine learning viewed through the lens of PDE-based optimisation.

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Sample pages

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What’s next?

While Book IV completes the analysis track, we still have some work to do. We are already hard at work on Part III – Linear Algebra, where we will bring the same integration of proof, computation, and application to vector spaces, eigenproblems, and modern data-driven engineering. Watch this space!

As always, we are grateful for feedback from teachers, students, and practitioners. Happy reading – and happy coding!