Rigorous mathematical treatments of engineering topics

Question

I started out as an engineering student and got interested in mathematics. So after some point (Rigorous analysis and linear algebra, some real analysis, basic measure theory and topology etc.) I thought it would be nice if I could turn all of my engineering knowledge into mathematical knowledge. So I'm looking for books with rigorous mathematical treatments of topics such as Mechanics, Thermodynamics, Control Systems, Dynamical Systems....

Ideally I'm looking for books that are not just a sequence of theorems and proofs in computer-testable form, but rather ones where the theorems and definitions are motivated. Also since each of the topics I mentioned is a mathematical field in itself I would say "nothing too advanced". For example when by "dynamical systems" I don't mean a book for graduate students in ergodic theory.

Thanks

Answer 1

I'm not very knowledgeable in some of the fields that you mention, but here's a quick recommendation.

Mechanics: You may benefit from reading Apostol's calculus books since they provide the rigorous background for things like kinematics and ODEs.

Thermodynamics: Here, I would recommend Stein and Shakarchi's Fourier Analysis book since it is motivated by the initial examples of the wave and heat equations. The methods they introduce for the solution of PDEs have applications in statistical mechanics. If you're interested in a more abstract, less applied treatment, try Folland or Papa Rudin.

Control theory:....I'm not sure about this one. On a related note, if you're interested in signal processing or information theory, I recommend reading Shannon's original paper on the subject. Also, I can't recommend highly enough Hernandez's and Weiss's excellent book on wavelets to anyone interested in signal processing.

Dynamical systems: More applied: Strogatz. More abstract: Glendinning

Answer 2

Control theory is quite rigorous, one of the more mathematical areas of engineering. Eduardo Sontag's "Mathematical Control Theory: Deterministic Finite Dimensional Systems" is a textbook written by a mathematician that covers a lot of the field. It is not too advanced or too specialized. I can't think of a better place to start. http://www.math.rutgers.edu/~sontag/mct.html

Answer 3

For (deterministic) control theory of linear systems, Brockett's Finite Dimensional Linear Systems is good.

For (nonlinear) mechanical systems, including an introduction to the required differential geometry, there is Bullo and Lewis's Geometric Control of Mechanical Systems.