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Additional info for A ''sup+ c inf'' inequality for Liouville-type equations with singular potentials

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2 F i n d also t h e v a l u e of t h e integral when C is t h e circle x 2 2 once in t h e positive sense. 6. If A = (2xy + 3 z ) i + (x 2 -f y = 1 described 2 + 4yz) j + (2y -f 6xz) k , e v a l u a t e / A - d r c 2 where C is (a) t h e curve w i t h p a r a m e t r i c e q u a t i o n s x ••— t, y -— £ , ζ = / Λ (δ) t h e straight line # = y — z; t h e curve in each case joining t h e points ( 0 , 0 , 0 ) a n d ( 1 , 1 , 1 ) . 7 . D e t e r m i n e w h e t h e r t h e f o l l o w i n g v e c t o r fields are expressible as g r a d i e n t s : (i) (iii) (a-r)r, (ii) λ2 τ~ [χ Λ - f (a · r) r ] , (iv) (a · r) a , 12 r~ \r a — (a · r) r ] , where a is a c o n s t a n t non-zero v e c t o r .

W e illustrate t h e proofs of these, which are a l l vector relations, b y p r o v i n g t h e second. S u p p o s e , instead of a in ( 1 . 4 0 ) , w e use e χ a where e is a constant, arbitrary vec- tor. T h e l . h . side is fye χ a · d S = e ·

Side of the last equation into an integral involving u, v, the parameters of the surface χ χ (Η , ν), y — y(u, v). where R{L2) denotes R evaluated at L2 in the upper element of area. Since the elementary column enters the surface at Lx, ^ , as shown in the figure, makes an obtuse angle with the ζ -axis, so that at Lx, and therefore Therefore # x being the lower portion and # 2 the upper portion of S. 38). If the surface S is n o t c o n v e x , we divide the v o l u m e into a finite number of sections each b o u n d e d b y a c o n v e x surface, and add the various contributions.