Download Note 5. Surface Integrals • Parametric equations of surfaces A

Note 5. Surface Integrals
• Parametric equations of surfaces
A surface in space can be represented by parametric equations
x = x(u, v), y = y(u, v), z = z(u, v),
(u, v) ∈ Q,
or equivalently, by the vector equation
r(u, v) = x(u, v) i + y(u, v) j + z(u, v) k,
(u, v) ∈ Q,
where Q is a region in the uv-plane. A unit normal vector of the surface is given by
∂y
∂y
∂x
∂z
∂z
n = ru × rv /|ru × rv |, where ru = ∂u
i + ∂u
j + ∂u
k and rv = ∂x
∂v i + ∂v j + ∂v k.
• Surface integrals with respect to surface area
If f is a continuous function on S, then
ZZ
ZZ
f (x, y, z) dS =
f x(u, v), y(u, v), z(u, v) ru × rv dA.
S
Q
In particular, when f = 1, the above surface integral gives the area of the surface.
• Surface integrals of vector fields
If F is a continuous vector field on an oriented surface S with unit normal vector n,
then the surface integral of F over S, or the flux of F across S, is
ZZ
ZZ
F·n dS =
F·(ru × rv ) dA.
S
Q
• The Divergence Theorem
The divergence of a vector field F = P i + Q j + R k is defined by
∂P
∂Q ∂R
+
+
,
div F = ∇ · F =
∂x
∂y
∂z
∂
∂
∂
where ∇ denotes the operator ∂x
i + ∂y
j + ∂z
k. Let E be a solid region whose
boundary surface S has outward orientation. Let F be a vector field whose component
functions have continuous partial derivatives on an open region that contains E. Then
ZZ
ZZZ
F·n dS =
div F dV.
S
E
• The Stokes Theorem
The curl of a vector field F = P i + Q j + R k is defined by
∂P
∂R
∂Q ∂P
∂R ∂Q
−
i+
−
j+
−
k.
curl F = ∇ × F =
∂y
∂z
∂z
∂x
∂x
∂y
Let S be an oriented piecewise smooth surface that is bounded by a simple, closed,
piecewise smooth boundary curve C with the orientation induced by the orientation
of S. Then
Z
ZZ
F·dr =
(curl F)·n dS.
C
S