WILSON AND LEWIS. — RELATIVITY. 455 



41. In the ordinary integral calculus of vectors the theorems clue 

 to Gauss and Stokes play an important role. In our notation we may 

 express these laws with great simplicity and generalize them to a 

 space of any dimensions. Let us consider first the form of these 

 theorems in the case of two dimensions, beginning with the more 

 familiar Euclidean case. 



Stokes's theorem states that the line integral of a vector function f 

 around a closed path is equal to the integral of the curl of f over the 

 area bounded by the curve. The analytic statement is 



f(1S'f= f fdS curl {, 



where ds is the vector element of arc, and dS the scalar element of 

 area. In our notation ^^ this becomes 



fdS'i= f fdS'Vxt, 



where dS is now the 2-vector element of area (a pseudo-scalar) and 

 V^f is a pseudo-scaldr (the complement of curl f , which itself is a 

 scalar in the two dimensional case). Transforming by (35), we may 

 also write 



A/s-f = - j j (dS'V)'f. • (59) 



Gauss's theorem states that the integral of the flux of a vector 

 through a closed curve is ecjual to the integral of the divergence of the 

 vector f over the area bounded by the curve. The analytic statement 

 is 



jfnds = j jdS div f , 



where /„ is the component of f normal to the curve. In our notation 

 this becomes 



. - / (dsxi)* = ffdS \/.t= j j dS*V-f, 

 or, by taking the complement of both sides, 



— fd&<t = f jdSV'i; 



36 One of the advantages of our system of notation is that if one term in an 

 equation is a vector of p dimensions, every other term is a vector of p dimen- 

 sions. This furnishes at once a check on the correctness of any equation. 



