VECTOR ANALYSIS 95 



ds - an element of arc of a curve regarded as a vector whose direction is 

 at of the positive tangent to the curve. 



4.31 Gauss's Theorem: 



fff div AdV = 



4.32 Green's Theorem 



i. yyy</>v 2 ^F + 

 2. 



4.33 Stokes's Theorem: 



yy 



4.40 A polar vector is one whose components, referred to a rectangular system 

 of axes, all change in sign when the three axes are reversed. 



t.401 An axial vector is one whose components are unchanged when the axes 

 are reversed. 



4.402 The vector product of two polar or of two axial vectors is an axial vector. 



4.403 The vector product of a polar and an axial vector is a polar vector. 



4.404 The curl of a polar vector is an axial vector and the curl of an axial vector 

 is a polar vector. 



4.405 The scalar product of two polar or of two axial vectors is a true scalar, 

 i.e., it keeps its sign if the axes to which the vectors are referred are reversed. 



4.406 The scalar product of an axial vector and a polar vector is a pseudo-scalar, 

 i.e., it changes in sign when the" axes of reference are reversed. 



4.407 The product or quotient of a polar vector and .a true scalar is a polar 

 vector; of an axial vector and a true scalar an axial vector; of a polar vector 

 and a pseudo-scalar an axial vector; of an axial vector and a pseudo-scalar a 

 polar vector. 



