The compact classical Lie groups can be regarded as groups of $n \times n$ matrices over the real, complex, and quaternion fields $\mathbb{R,~ C},$ and $\mathbb{Q}$ that satisfy metric- and volume-conserving conditions. These groups, ${\rm SO}(n,\mathbb{R}), {\rm SU}(n,\mathbb{C})$, and ${\rm Sp}(n,\mathbb{Q})$, are not all independent. Homomorphisms exist among some of these groups for small dimension. We review these relations by describing the Lie algebras of the compact forms and their complex extensions. Other noncompact real forms of these Lie algebras are constructed by systematic methods. The relations among all distinct real forms is presented.

A concise discussion of the three-dimensional irreducible (1,0) and (0,1) representations of the restricted Lorentz group and their application to the description of the electromagnetic field is given. It is shown that a mass term is in conflict with relativistic invariance of a formalism using electric and magnetic fields only, contrasting the case of the two-component Majorana field equations. An important difference between the Dirac equation and the Dirac form of Maxwell's equations is highlighted by considering the coupling of the electromagnetic field to the electric current.

We pose a variant of the classical minimum surface problem inspired by a simple experiment with soap films: to find the surface of least area containing a cavity of given perimeter. We show that the equilibrium surface is governed by a system of two equations one of which is the zero mean curvature condition. The other equation states that the curvature of the cavity's contour is constant and that its principal normal lies in the plane tangential to the surface. A gradient descent simulation confirms the analytical equilibrium conditions and yields configurations qualitatively consistent with experiment.

Here we derive analytic expressions for the scalar parameters which appear in the generalized Euler decomposition of the rotational matrices in $\mathbb{R}^3$. The axes of rotations in the decomposition are almost arbitrary and they need only to obey a simple condition to guarantee that the problem is well posed. A special attention is given to the case when the rotation is decomposable using only two rotations and for this case quite elegant expressions for the parameters were derived. In certain cases one encounters infinite parameters due to the rotations by an angle $\pi$ (the so called half turns). We utilize both geometric and algebraic methods to obtain those conditions that can be used to predict and deal with various configurations of that kind and then, applying l'Hôpital's rule, we easily obtain the solutions in terms of linear fractional functions. The results are summarized in two Tables and a flowchart presenting in full details the procedure.

We consider some aspects of the geometry of surfaces of revolution in three-dimensional Minkowski space. First, we show that Clairaut's theorem, which gives a well-known characterization of geodesics on a surface of revolution in Euclidean space, has an analogous result in three-dimensional Minkowski space. We then illustrate the significant differences between the two cases which arise in spite of their formal similarity.