equation. The two-dimensional and three-dimensional wave equations also fall into the category of hyperbolic PDE. This type of second-order hyperbolic partial differential equation may be transformed to a hyperbolic system of first-order differential equations. Hyperbolic system of partial differential equations The following is a system of first order partial differential equations for unknown functions , , where : where are once continuously differentiable functions, nonlinear in general. Next, for each define the Jacobian matrix The system () is hyperbolic if for all the matrix has only real eigenvalues and is diagonalizable. If the matrix has s distinct real eigenvalues, it follows that it is diagonalizable. In this case the system () is called strictly hyperbolic. If the matrix is symmetric, it follows that it is diagonalizable and the eigenvalues are real. In this case the system () is called symmetric hyperbolic. Hyperbolic system and conservation laws There is a connection between a hyperbolic system and a conservation law. Consider a hyperbolic system of one partial differential equation for one unknown function . Then the system () has the form Here, can be interpreted as a quantity that moves around according to the flux given by . To see that the quantity is conserved, integrate () over a domain If and are sufficiently smooth functions, we can use the divergence theorem and change the order of the integration and to get a conservation law for the quantity in the general form which means that the time rate of change of in the domain is equal to the net flux of through its boundary . Since this is an equality, it can be concluded that is conserved within . See also Elliptic partial differential equation Hypoelliptic operator Parabolic partial differential equation References Further reading A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. External links Linear Hyperbolic Equations at EqWorld: The World of Mathematical Equations. Nonlinear Hyperbolic Equations at EqWorld: The World of Mathematical Equations.