112 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



Now let us consider the curves 5 as the one set of coordinate-curves. The deriva- 

 tive of a scalar a with respect to 5 will then be the one partial derivative of the 

 dependent variable a. It is a special case that the curves 5 are parallel and equi- 

 distant straight lines. If we use two such systems of lines which are normal to each 

 other, and call the length of arc along the one set x, and along the other set y, the 

 two partial derivatives will be 



(a) F* = p y = 



?x 2y 



The fields of these partial derivatives can be determined by use of the divided sheet 

 of fig. 8i. 



The two partial derivatives are the rectangular components of the ascendant 

 F of the scalar. As we have shown already (Statics, section 17), this vector is 

 directed along the normal n to the equiscalar curves a = const., and is numerically 

 equal to the derivative of a with respect to the length of arc n measured along the 

 normal curves 



(b) F = ^ 



an 



In order to abbreviate we shall introduce here a useful notation. The fact that 

 the vector F is in the defined relation to the scalar a will be expressed by the single 

 vector-equation 



(c) F = Va 



This equation is by definition equivalent to the two scalar equations (a), and in 

 the case of the three-dimensional field it will be equivalent to three such equations. 

 A vector G of the opposite direction 



(d) G = -Va 

 represents the gradient of the scalar a. 



The field of the ascendant or of the gradient can be found by algebraic methods 

 (section 157) from the fields of the two rectangular components; but it can also be 

 derived directly from the field of the given scalar a. This direct method will involve 

 separate determinations of the direction and of the magnitude of the vector. 



The vector-lines can be drawn at once as normal curves to the equiscalar curves 

 a = const. If we wish to have the direction represented by isogons, we have to use 

 the directional differentiation described in section 163, and to give the isogons such 

 numbers that they represent the direction of the normal curves n, not of the equi- 

 scalar curves a = const. 



The intensity-field of ascendant or gradient are found by use of the differentiat- 

 ing sheet of fig. 81 in accordance with formula (b). The field will contain no zero- 

 curve. It will only have zero-points at the points of maximum, minimum, and 

 maximum-minimum of the scalar a. These zero-points will be singular points of 

 intersection of the vector-lines as well as of the isogons. As points for absolute 

 minimum of the scalar value of the vector they will be surrounded by closed curves 

 of equal intensity. The drawing of the field is therefore very much facilitated by 

 the circumstance that these zero points are given beforehand. 



