Complex Analysis - Maths KU

166 Chapter 3 Analytic Functions
Level curve
f(x, y) = c 0
Tangent
(x 0 , y 0 )
∇f(x0
, y0 )
Figure 3.5 Gradient is perpendicular
to level curve at (x0, y0)
Note
☞
∇ is called a nabla or del), is defined to be the two-dimensional vector
∇f = ∂f ∂f
i + j.
∂x ∂y
(5)
As shown in color in Figure 3.5, the gradient vector ∇f(x0, y0) atapoint
(x0, y0) is perpendicular to the level curve of f(x, y) passing through that
point, that is, to the level curve f(x, y) =c0, where c0 = f(x0, y0). To
see this, suppose that x = g(t), y = h(t), where x0 = g(t0), y0 = h(t0)
are parametric equations for the curve f(x, y) =c0. Then the derivative of
f(x(t), y(t)) = c0 with respect to t is
∂f dx ∂f dy
+ =0. (6)
∂x dt ∂y dt
This last result is the dot product of (5) with the tangent vector r ′ (t) =x ′ (t)i+
y ′ (t)j. Specifically, at t = t0, (6) shows that if r ′ (t0) �= 0, then
∇f(x0,y0) · r ′ (t0) = 0. This means that ∇f is perpendicular to the level
curve at (x0, y0).
Gradient Fields As discussed in Section 2.7, in complex analysis
two-dimensional vector fields F(x, y) =P (x, y)i + Q(x, y)j, defined in some
domain D of the plane, are of interest to us because F can be represented
equivalently as a complex function f(z) =P (x, y)+iQ(x, y). Of particular
importance in science are vector fields that can be written as the gradient of
some scalar function φ with continuous second partial derivatives. In other
words, F(x, y) =P (x, y)i + Q(x, y)j is the same as
F(x, y) =∇φ = ∂φ ∂φ
i + j, (7)
∂x ∂y
where P (x, y) = ∂φ/∂x and Q(x, y) = ∂φ/∂y. Such a vector field F is
called a gradient field and φ is called a potential function or simply the
potential for F. Gradient fields occur naturally in the study of electricity
and magnetism, fluid flows, gravitation, and steady-state temperatures. In a
gradient force field, such as a gravitational field, the work done by the force
upon a particle moving from position A to position B is the same for all paths
between these points. Moreover, the work done by the force along a closed
path is zero; in other words, the law of conservation of mechanical energy
holds: kinetic energy + potential energy = constant. For this reason, gradient
fields are also known as conservative fields.
In the study of electrostatics the electric field intensity F due to a collection
of stationary charges in a region of the plane is given by F(x, y) =−∇φ,
where the real-valued function φ(x, y) is called the electrostatic potential.
Gauss’ law asserts that the divergence of the field F, that is, ∇·F, is proportional
to the charge density ρ, where ρ is a scalar function. If the region of the
plane is free of charges, then the divergence of F is zero ‡ : ∇·F = ∇·(−∇φ) =0
or ∇ 2 φ = 0. In other words: The potential function φ satisfies Laplace’s equation
and is therefore harmonic in some domain D.
‡ The electrostatic potential is then due to charges that are either outside the charge-free
region or on the boundary of the region.