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Extra info for Encyclopedia Dictionary of Mathematics (Section C)
N - r. But these general solutions are not essentially different. For the partial equation of the second order F(x, y, z, p, 4, r, s, t) = 0, this deli- We can regard equation (8) as a relation between the point (x, y, z) on the integral surface S and the direction cosines of a tangent plane at that point. Therefore the tangent planes at all points of the surface form a one-parameter family. They envelop a cone (T) whose vertex is (x, y, z) on S. The tangent plane at a point M on the integral surface S is tangent to this cone (T) along one generating line G of (T).
Example 2. If PI, P2, . . , P,,, R are functions of independent variables x1, . . > =o G’ax,ax,‘“’ > that involves a function z of independent variables x1, x2, . . x,, its tpartial derivatives, and the independent variables x1, . ,x,. The definition of a system of partial differential equations is similar to that of a system of tordinary differential equations. ) The torder of the highest derivative appearing in a partial differential equation is called the order of the partial differential equation.
T. Schwartz, Linear operators II, Interscience, 1963.  B. M. Levitan and I. S. Sargsyan (Sargsjan), Introduction to spectral theory: Self-adjoint ordinary differential operators, Aml:r. Math. Sot. Transl. of Math. Monographs, 1975. 3) Ordinary Differential Equations (Initial Value Problems) A. General Consider equations Remarks a system of ordinary dy,ldx=fi(x,~,, . ,n. (1) A. L. , n, and their derivatives aJ/ay, are continuous in a neighborhood Iof a point (a, b,, . . , n. These conditions are called initial conditions, and the values a, b,, .
Encyclopedia Dictionary of Mathematics (Section C)