1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

468 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS

that the line integral of the tangential component of V (x, y) along any simply
closed contour be identically zero. If we consider the rectangle in Figure 11.47,
then the tangential component is given by p on the bottom edge, q on the right
edge, - p on the top edge, and - q on the left edge. Integrating and equating the
resulting circulation integral to zero yield

1

y+l>y 1x+l>:z;

Y [q(x+.C:.x,t)-q(x,t)]dt- "' !P(t,y+.C:.y)- p(t,y)]dt= O.

As before, we apply the mean value theorem and divide through by .C:.x .C:.y, and

obtain the equation

1 1y+ti.y 1 rx+l>:r;

.C:.y Y q.,(x1,t)dt- .C:.x}., p 11 (t,yz)dt=O.

We can use the mean value for integrals with this equation to deduce that

q., (xi, yi) -Py (xz, yz) = O. Letting .C:.x -+ 0 and .C:.y -+ 0 yields

q., (x, y) -Pv (x, y) = 0.

Equation (11-33) and this equation show that the function f (z) = p (x, y) -

iq (x, y) satisfies the Cauchy-Riemann equations and is an analytic function. If

we let F ( z) denote the antiderivative of f ( z), then

F(z) = (x, y) +iefJ(x, y), (11-34)

which is the complex pote nt ial of the flow and has the property

F' (z) = x (x, y) -il/Jx (x, y) = p (x, y) + iq (x, y) = V (x, y).

Since <j>., = p and </>y = q, we also have

grad</> (x, y) = p(x, y) + iq (x, y) = V (x, y),

so</> (x, y) is the velocity potentia l for the flow, and the curves

<f>(x,y) =I<,

are called equipotent ials. The function efJ (x, y) is called the stream function.

The curves

efJ (x, y) = I<2

are called streamlines and describe the paths of the fluid particles. To demon-

strate this result, we implicitly differentiate efJ (x, y) = K 2 and find that the slope

of a vector tangent is given by

dy -1/J., (x, y)

dx = 1/Jy (x, y) ·