Advanced Calculus fi..

126 Advanced Calculus, Fifth EditionFigure 2.13Gradient vector of F and surface F( r. y. z) = constEquation (2.99) for the tangent plane can now be written in the vector formgrad F . dr = 0 (r =.ri + yj + zk). (2.104)while the two Eqs. (2.102) for the tangent line becomegrad F . dr = 0. grad G . dr = 0. (2.105)Since (2.105) expresses the fact that dr = dxi + dyj + dzk is perpendicular to bothgrad F and grad G, one concludes thatdr x (grad F x grad G) = 0; (2.106)this equation again represents the tangent line. The vector grad F x grad G hascomponentsHence the tangent line can be written in the symmetric formX -XI- y - yl - Z - ZI;)F iJFJ ityac ,c il G ij G iJG i3G(2.108)or, in terms of Jacobians, in the formThis discussion and that of the preceding section show the significance of thedifferential. Taking differentials in an equation or system of equations correspondson the one hand to replacement of the equations by linear equations in the variablesdx, dy, . . . and on the other hand to replacement of curves and surfaces by tangentlines and planes.