400 Mr. H. T. Flint on Integration Theorems 



If F is of the special form given above since 

 dvV l = Y 5 dvB 1 + Y 1 dvD 1 



and Y (Y,dvJ) l ).F = 0, 



we have : YoiY^vB, . F} = Y (dvT) 1 F)=Y (dvY 2 T) 1 F), 

 .-. Y $dpF(p)=$$Y {(Y 2 V 1 F).dv}. 

 The curl of a vector is defined by the equation 



Y ^dpF(p)=^Y (cur\F.dv). 



Thus curlF^V^F. 



§ 10. The second theorem obtained by performing the 

 operation S^tF(p) through a small four-dimensional ele- 

 ment dv connects a three-dimensional to a four-dimensional 

 integral. Ity a similar treatment in this case as in the last : 



Xdr.F(p) = 



— Y z dp 2 dp z dp 4i . Y dp 1 D l . F + Ysdpzdp^dp! . Y dp 2 J) 1 . F 



— Y z dp 4: dp l dp 2 . Y dp z D 1 . F + Y- d dp^dp 2 dp 3 . Yodp^ . F. 



This is simplified by observing that : 



V 4 a 1 a 2 a 3 o 4 . a=r — V 3 a 2 «3 a 4^ r o a i a + V 3 a 3 a 4 c^Vo^a 



— V 3 a 4 u l c< 2 . Y ct 3 a + V 3 a x cl 2 a 3 . V « 4 ». 



ZdT .F(p) = Y i d Pl dp 2 dp 3 d Pi . DiF(p) 

 ^I^.DjF^). 

 .-. ^pT.F( /0 ) = JjjJlD 1 F(p).^ 5 . . . (10-1) 

 or fjjUr.F(p)=J , j , jfD 1 F(p)^. 



The same limitations as before are to be imposed on F 

 throughout the finite region. 



By taking the scalar product of each side and employing 

 again the special form for F(/>) we have : 



J f f (X dy dz du+Y dz du dx + Zdudxdy + TJ dx dy dz) 



^m 



+ ^- H -v — I- -^ — idxdyazdu. 



dy d- om/ 



Sommerf eld's extended Div F is equal to -VqDxF, or to: 

 Lim -V,,yi^T.F(p) 



