VECTOK ANALYSIS. 31 



Or, if r and o> are two single-valued continuous vector functions 

 of any number of scalar or vector variables, and 



then T 



where a is a vector constant. 



When the above hypotheses are not satisfied in general, but will be 

 satisfied if the variations of the independent variables are confined 

 within certain limits, then the conclusions will hold within those limits, 

 provided that we can pass by continuous variation of the independent 

 variables from any values within the limits to any other values 

 within them, without transgressing the limits. 



49. So far, it will be observed, all operations have been entirely 

 analogous to those of the ordinary calculus. 



Functions of Position in Space. 



50. Def. If u is any scalar function of position in space (i.e., any 

 scalar quantity having continuously varying values in space), Vu is 

 the vector function of position in space which has everywhere the 

 direction of the most rapid increase of u, and a magnitude equal to 

 the rate of that increase per unit of length. Vu may be called the 

 derivative of u, and u, the primitive of Vu. 



We may also take any one of the Nos. 51, 52, 53 for the definition 

 of Vu. 



51. If p is the vector defining the position of a point in space, 



__ .du.du tl du 

 o2. vu = ^J h?-j \-K-J-. 



dx ' dy dz 



du . _ du . du 

 53. -T-=I.VU, -f-**j.Vtfc -j- 



dx dy dz 



54. Def. If o> is a vector having continuously varying values in space, 



x, v 



(1) 



. da) -, da 



.u = .-j---]- .-T-, 

 dx J dy dz 



_ . d&) . . da) , 7 do) /e) \ 



and Vx^X++fcx- (2) 



.co is called the divergence of o> and Vxo> its curl. 

 If we set 



we obtain by substitution the equations 



_dX dY dZ 

 ~ + + 



./dZ dY\./dX dZ\/dY 



dZ\ 



-j-J 

 dx/ 



and yAra = ., v ___ ;T-/v __^ yT . vs --^_ 



which may also be regarded as defining V.w and Vxo>. 



