By Terence Tao

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**Additional resources for Analysis I (v. 1)**

**Example text**

T-2,-i]) where we have zero for the last term if m = 0. 2) is exact. -2]. Thts sheaf is a generalization of the sheaf of holomorphic left-handed massless free fields of helicity n/2 from [5]. 2. -n-2)) ^ ( l i , ! ^ ) . In the case that n = 0, m ^ 0, there is a canonical isomorphism 26 ROBERT POOL A: H 1 (U , , ,0(m J -2)) + r ( U , 0 ( B ' ' " E , ) [ - 1 ] . As discussed i n [5] the Penrose transform produces solutions to Maxwell's equations both directly—by giving the electromagnetic f i e l d - - a n d i n d i r e c t l y — via potentials.

10) The left-hand group acts on the set of second-order extensions of a E. We are now in a position to identify the curvature of the Yang-Mills 2 field F € r(U, ftw 8 End E ) . Given a point x e U we have H ^ L . 11) s Sl^M Checking that H 1 (L Y , End E|, 8 End E| x . 8 N*) = H 2 ( L . End E|. X 8 N*) = 0 we find x E|,L + L has a unique first-order extension into U n . Restricting x * E + U" and the trivial bundle over U" to the second-order neighborhood of L gives two second order extensions of E|, .

M ( x > r 0 s 0 ' 6 then d e l n m= ro = 0 gives conditions on {E°J m } . a ap * n»m We now form the coboundary of { £ - } : F L *= a3Y ' P ct£ + P e $y - P ay" I f we expand E°L = I I ^ ( x ^ V0n0V a3Y *=_™ m £_™ n»m we can calculate that n=-°° ro--00 E 01 m +. t F^ F 12 M n,m nm9 m E X + E 2 nm i m -02 n,m n m s>n 0 E^ nm 9 m — mo ) n < 0, m > 0 r012 _ J n,m n,m — n,m " ] £ 12 . E 02 n > 0, m < 0 n,m n,m - ' 12 E n,m „m n,m < 0 rOl , r 13 03 „ , 0 m . ft c E + E ^ ™ E~ « ~ ~ ™ n < -2, m > 0 n,m n,m n,m — — n n ; EJ 1 ™.