Why are certain PDE called elliptic, hyperbolic, or parabolic? $\begingroup$ @VivekanandMohapatra actually, the solutions to simple elliptical PDEs around a small pertubation tend to come out as “blobs”, ellipse-ish, to parabolic PDEs they disperse ever slower like the arms of a parabola, and for hyperbolic they wander off asymptotically straight towards infinity like a hyperbola
linear transformations - Is hyperbolic rotation really a rotation . . . A hyperbolic rotation is a rotation because of its effect on hyperbolic angles! Like the fact circular angles relate to the area of a (circular) wedge, hyperbolic angle is related to the area of a hyperbolic wedge: (source: WolframAlpha)
What are the interesting applications of hyperbolic geometry? the hyperbolic plane In particular, the hyperbolic plane is the universal cover of every Riemann surface of genus two or higher This fact is centrally important all over mathematics This is why you have to learn about hyperbolic geometry to study modular forms in number theory, for instance
partial differential equations - Mathematical precise definition of a . . . First there is the notion of strict hyperbolicity This is the direct analogue of the above characterisation using hyperbolic polynomials in the context of principal symbols In fact, we have that Theorem If the principal part of an operator is strictly hyperbolic, than regardless of what the lower order terms are, the operator is hyperbolic
trigonometry - Proof for hyperbolic trigonometric identities . . . And hence every trigonometric identity can be easily transformed into a hyperbolic identity and vice versa Once you prove that $\exp' = \exp$ , you can recover all the basic properties of $\exp$ and hence $\cosh,\sinh,\cos,\sin$ , including:
Connection between hyperbola and hyperbolic functions The hyperbolic angle is not included between any two lines $$ \theta_h= A a^2 = \frac12\; \log (\tan(π 4+θ_e)):$$ This area relation is found by direct integration It is difficult for me to justify full cross-validity without going into definitions of lengths and hence the dependent hyperbolic angles in any one model of the hyperbolic
metric spaces - Minkowski plane vs. hyperbolic plane - Mathematics . . . Hyperbolic geometry usually refers to negative curvature The Hyperbolic plane is a 2d surface with constant negative curvature and positive definite metric Whoever told you that the space with pseudo-metric with signature $(-,+)$ was "the hyperbolic plane" was very misleading