By Freeman, Stone, Martin
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Extra info for Borderline Personality Disorder: A Practitioner's Guide to Comparative Treatments (Springer Series on Comparative Treatments for Psychological Disorders)
23 GEOMETRIC INTERPRETATION + example, for p 2 q2 = 1 the family is p = cos t , q = sin t with the parameter t ) . If F is linear in p and q then this family of possible tangent planes forms an axial pencil of planes through a straight line called the “Monge axis”. ’ Thus, in a domain of the x, y, u-space the differential equation is represented geometrically by a “cone field” just as an ordinary differential equation of the first order is represented by a direction field. T o find a solution means to find a surface which a t each of its points touches the corresponding Monge cone (or “fits” into the cone field).
We obtain n third exurtmple of an overdetermined system if we int,ro- 17 SYSTEMS OF DIFFERENTIAL EQUATIONS + duce n 1 “homogeneous variables” z1, z 2 , - .. ), which is homogeneous of degree zero in the new variables and which therefore satisfies the Euler homogeneity relation zmz, + x w z * + * . = 0. ‘l’he first partial derivatives of the function u(x,y, - - with respect to x, y, . - can be expressed in terms of the derivatives of the function w(z1 , zr , - -): a) , ‘11, = x1wz3 , .. .. . .
Then, we have found infinitely many solutions of the potential equation but, in contrast to the previous examples, so far only denumerably many solutions. In polar coordinates T , 8 defined by 2 = r cos 8, JI = r sill 8, we have (6) P,,(x, 3 ) = r n cos 7 4 Qn(x, y) = r” sin ne. For any arbitrary real a, the functions Pa($, y) = ra cos a8, Qa(z,y) = ra sin a8 also satisfy the potential equation in any region of the x, y-plane excluding the origin z = y = 0. This is immediately verified after transforming Au into polar coordinates (cf.
Borderline Personality Disorder: A Practitioner's Guide to Comparative Treatments (Springer Series on Comparative Treatments for Psychological Disorders) by Freeman, Stone, Martin