What is Applied Mathematics?

Applied mathematics is the practical language for modeling reality. It bridges the gap between abstract mathematical concepts and real-world problems in engineering, physics, economics, biology, and beyond. Unlike pure mathematics, which focuses on theoretical structures and proofs, applied mathematics is fundamentally concerned with solving real problems using mathematical tools and computational methods.

In modern practice, applied mathematics is inseparable from computation. The problems we face in engineering and science (designing structures, optimizing processes, predicting system behavior, analyzing data) are typically too complex for hand calculation. This is not a limitation; it is an opportunity to focus our human effort where it matters most: modeling and interpretation.

The Computational Thinking Paradigm

The approach we take in this course is built on a fundamental principle: humans do what humans do best, and computers do what computers do best. You will model real-world problems, translating messy physical situations into precise mathematical form. The computer will calculate, solving equations, performing integrations, and running simulations. You will then interpret the results, analyzing their validity, exploring parameter sensitivity, and making decisions based on computational evidence.

This is not about learning to program computers to mimic hand calculations. It is about thinking computationally. This is a modern mindset where mathematical problem-solving is computer-augmented from the start. We delegate mechanical manipulation to machines, freeing our cognitive resources for creative modeling and critical interpretation. Critically, when we say mechanical manipulation, we mean both numerical computation AND symbolic algebraic manipulation. Computer Algebra Systems like SymPy will perform the integration, differentiation, equation solving, and algebraic simplification that traditional courses spend the majority of time teaching students to do by hand.

Cf. [3] and [4] for a detailed definition on computational thinking in a mathematical context in education.

A New Mindset

This approach may feel different from mathematics courses you have taken before. You might be accustomed to memorizing formulas and procedures, solving contrived textbook problems by hand, focusing on getting “the right answer,” and treating computer work as a separate “lab” activity. We ask you to adopt a different mindset.

We want you to understand principles, not formulas. When you understand the underlying physics and mathematics, you can derive what you need rather than searching through formula collections. We want you to model realistic problems from the start, not simplified toy problems designed for hand calculation. We want you to focus on the process (formulation, solution strategy, validation) not just the final numerical answer. We want you to use computers as your primary mathematical tool, not an afterthought.

This shift can be disorienting initially, but it is also liberating. You will solve problems that would be impossible by hand. You will discover patterns through exploration rather than memorization. You will build intuition through visualization and interaction. You will develop skills directly applicable to the challenges you will face as a professional engineer or scientist.

Most importantly, you will understand that applied mathematics is not a collection of formulas to memorize but a way of thinking about and analyzing physical systems. It is a language for translating between physical reality, mathematical models, and computational implementations.

Welcome to applied mathematics in the computational age.

Bridging Education and Practice

The shift to computational thinking is a direct response to industrial reality, particularly in fields like computational mechanics. Relying on simulation software without understanding the underlying applied mathematics reduces an engineer to a ‘black-box’ user: one who cannot validate results, debug models, or innovate beyond a tool’s presets. The methods in this course bridge that gap, providing the mathematical fluency required to build, understand, and critically evaluate the computational tools used in modern professional practice.

The methods you learn here are not academic exercises but the daily tools of modern engineering and science. We carefully chosen problems and examples for you to study based on the vast number of problems you will encounter in the following courses! But these skills go way beyond the academic setting and your years at school; Product development requires simulation and optimization that cannot be done by hand. Data analysis demands numerical methods for fitting, interpolation, and statistical inference. System design relies on solving differential equations that govern physical behavior. Decision-making depends on optimization under constraints. Validation requires comparing models with experimental data through numerical techniques.

By learning applied mathematics as a computational discipline from the start, you close the gap between education and professional practice. You develop computational thinking: the ability to formulate problems precisely, leverage computational tools effectively, and interpret results critically. This is not about becoming a programmer, though you will write code. It is about becoming a modern engineer or scientist who can think mathematically and act computationally to solve real problems.

The industrial reality is clear: development times are shorter, problems are more complex, and computational tools are ubiquitous. Traditional education that emphasizes hand calculation leaves graduates unprepared for professional practice. Computer-based mathematics education, by contrast, reflects how mathematics is actually applied in modern engineering and science.

The Modern Workflow

For every applied mathematics problem, we follow a consistent pattern

1. We formulate the problem by defining all relationships systematically. We write governing equations, specify constraints, and establish boundary conditions. This formulation phase requires careful thought about what equations apply, what assumptions we make, and what we seek to find.

2. We solve using appropriate computational methods. For analytical solutions, we use SymPy’s symbolic capabilities. For numerical solutions, we employ SciPy’s solvers. For simulations, we implement time-stepping schemes. The specific solving method changes (differential equation solver, optimizer, integrator) but the principle remains: delegate calculation to the computer.

3. We analyze the results. We visualize solutions through plots and animations. We vary parameters to understand sensitivity. We check dimensions to verify consistency. We test extreme cases to validate physical behavior. This analysis phase builds intuition and reveals whether our model captures the essential physics or requires refinement.

This three-step pattern (formulate, solve, analyze) appears throughout applied mathematics. Master this workflow, and you can tackle diverse problems across engineering and science by adapting the same fundamental approach.

Guiding Principles

As you work through these chapters, keep several principles in mind. First, start simple.

Börja enkelt! Det kanske räcker! (Start simple! It might be enough!) - Bertil Nilsson

Begin with the simplest model that captures the essential physics. If it proves insufficient, refine it in the next iteration. Einstein’s principle applies: make it as simple as possible, but not simpler.

Second, avoid relying on ready-made formulas from textbooks or formula collections. Verifying whether a formula applies to your specific problem is often harder than solving the problem from first principles. Instead, formulate governing equations systematically and let the computer solve them. This approach builds understanding and prevents the “färdiga-formler-tänkandet” (ready-made formula thinking) that substitutes memorization for comprehension.

Third, trust the mathematics. Define quantities with consistent physical conventions, and the mathematics will automatically handle signs, units, and algebraic manipulations. Don’t manually track these details; trust the computational process.

“Matematiken gör jobbet!” (Mathematics does the work!). - Bertil Nilsson

Fourth, dimensional analysis always applies. Every equation must be dimensionally consistent. Every result must be dimensionally checked. This is not optional; it is fundamental to correctness and physical understanding. Dimensional analysis catches errors that purely mathematical checks might miss.

Finally, keep solutions in symbolic form when possible. A result like \(F_{Ax} = -\frac{10F_A}{\sqrt{d^2+100}}\) is more informative than just a number. Self-documenting work helps you understand relationships between quantities and debug when things go wrong. Python libraries like SymPy make this natural and straightforward.

What You Will Learn

This section of the course covers fundamental methods of applied mathematics. We begin with numeric differentiation, where you will approximate derivatives from discrete data or functions when analytical differentiation is impractical. Numeric integration follows, enabling you to compute integrals numerically when analytical solutions are unavailable or intractable. Time integration methods address the solution of ordinary differential equations that describe dynamic systems evolving in time. The chapter on differential equations examines the formulation and solution of ODEs and PDEs that govern physical phenomena across engineering and science. Root finding methods locate zeros of functions and solve nonlinear equations that arise throughout applied mathematics. Interpolation methods construct functions that pass through given data points, essential for data analysis and curve fitting. Finally, optimization methods find maxima and minima of objective functions subject to constraints, addressing design and decision problems fundamental to engineering.

Each topic emphasizes computational implementation and practical application. You will not memorize formulas or perform tedious hand calculations. Instead, you will understand the underlying mathematical principles, implement methods using Python and its scientific libraries, apply these methods to realistic engineering problems, interpret and validate computational results, and build intuition through interactive exploration. The goal is to develop computational thinking: the ability to formulate problems precisely, select appropriate methods, implement solutions efficiently, and interpret results critically.

Why This Approach Matters

Without the constraint of hand calculability, we can tackle realistic problems: higher-dimensional systems, nonlinear differential equations, large-scale optimization, data-driven models, and time-dependent simulations. These are the problems you will encounter in professional practice. Learning to solve toy problems by hand does not prepare you for this reality.

When computers handle calculations, you gain time for what truly builds understanding. Interactive visualization lets you see how solutions change as parameters vary. Parameter studies reveal sensitivity and expose patterns. Rapid iteration allows you to test ideas quickly, fail fast, and refine models efficiently. Multiple approaches (analytical, numerical, graphical) provide complementary insights into the same physical system. This exploratory process is where conceptual understanding emerges, not from memorizing formulas or performing tedious manipulations.

Counterintuitively, a computational approach creates the need for more mathematics, not less. You can use a much larger portion of what you’ve learned because computational tools make complex methods accessible. Mathematical concepts that seemed abstract become useful when you can apply them to realistic problems. Real problems demand deeper mathematical sophistication than the simplified examples solvable by hand. Students in computer-based mathematics courses consistently report needing and wanting more advanced mathematical education because they see its value in solving problems that matter.