Let the function f(z) = u(x,y) + iv(x,y) be analytic in a domain D, and consider the families of leve curves u(x,y) = c_1 and v(x,y) = c_2, where c_1 and c_2 are arbitrary real constants. Prove that these families are orthogonal. More precisely, show that if z_0=(x_0, y_0) is a point D which is common to two particular curves u(x,y)=c_1 and v(x,y)=c2 and if f'(z_0) not equal to 0, the the lines tangent to those curves at (x_0, y_0) are perpendicular.
That is the long winded question. I know i have to show that the two gradient vectors are orthogonal, i.e., their dot product as vectors in the plane is zero, at any point of intersection between a level curve of u
and a level curve of v. I know how Cauchy-Riemann equations work, but not sure how to represent the functions as vectors since they are both constants - wouldn't du/dx=0 and so on ?
2007-03-07
21:46:10
·
2 answers
·
asked by
hey mickey you're so fine
3
in
Mathematics