dC
S
C
dS
n(x)
t(x) ˆ
ˆ
and therefore cancel. The segments of the line integrals cancel
between all the tiles except those on the outer border that have
no adjacent circulation to cancel them.
If the line integral is nonzero, the vector field has a net rotation along the curve, so the integral of its curl over the surface is
nonzero. The curl of a vector field shows where rotations arise.
A common application is describing the velocity field of a moving fluid. The upper portion of Fig. A.6-5 shows streamlines,
lines parallel to the velocity vector at any point, for a viscous
fluid flowing past a circular object. The velocity is zero at the
object, and increases with distance away from it. The flow is
symmetric on the bottom of the object. The lower portion of
the figure shows contours of the curl of the velocity field with
larger values, indicating greater rotations, close to the object.
Two useful identities, whose proofs are left for the problems,
are that the curl of a gradient and the divergence of a curl are
zero:
∇
∇ ∇
∇ ∇ · (∇ ∇ ∇
∇ ∇ × u) = 0
(18)
∇
∇ ∇
∇ ∇ × (∇ ∇ ∇
∇ ∇φ) = 0.
(19)
Equation 19 can be used with Stokes’ theorem to show that for
a vector field written as the gradient of a scalar, the curl, and
hence circulation around an arbitrary curve, are zero. This idea
is used in mechanics to prove that a conservative force (one that
can be written as the gradient of a potential) has a line integral
that is independent of path, because its circulation around any
path is zero. These relations give insight into seismic waves,
because P waves have no curl and S waves have no divergence
(Section 2.4.1).
A.6.5 Laplacian
The Laplacian operator is formed by taking the divergence of
the gradient of a scalar field, which yields a scalar field
∇
∇ ∇
∇ ∇ 2 φ = ∇
∇ ∇
∇ ∇ · ∇
∇ ∇
∇ ∇φ =
∂
∂
∂
∂
∂
∂
2
1
2
2
2
2
2
3
2
φ
φ
φ φ
x
x
x
i ,
,,
+
+
=
(20)
where the last form uses index notation and the summation
convention. By analogy, the Laplacian of a vector field is a vector field whose components in Cartesian coordinates are the
Laplacians of the original vector components,
∇
∇ ∇
∇ ∇
2 u = (∇ ∇ ∇
∇ ∇
2
u 1 , ∇
∇ ∇
∇ ∇
2 u 2 , ∇
∇ ∇
∇ ∇
2
u 3 ).
(21)
For example, the ê 1 component of ∇
∇ ∇
∇ ∇ 2 u is
∂
∂
∂
∂
∂
∂
2
1
1
2
2
1
2
2
2
1
3
2
u
x
u
x
u
x
.
+
+
(22)
In Cartesian coordinates, the Laplacian of a vector satisfies
∇
∇ ∇
∇ ∇ 2 u = ∇
∇ ∇
∇ ∇(∇ ∇ ∇
∇ ∇ · u) − ∇
∇ ∇
∇ ∇ × (∇ ∇ ∇
∇ ∇ × u),
(23)
Fig. A.6-4 Geometry for Stokes’ theorem: 4(x) is a unit vector pointing
outward at the point x from an element dS of the surface S. dC is an
element of the curve C bounding S, with tangent 6(x).
This can be written as a determinant
∇
∇ ∇
∇ ∇ × u = det
ê
ê
ê
1
2
3
1
2
3
1
2
3
∂
∂
∂
∂
∂
∂
x
x
x
u
u
u
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
,
(15)
or, using index notation, in a compact form as
∇
∇ ∇
∇ ∇ × u = ε ijk
∂
∂
u
x
k
j
= ε ijk u k, j .
(16)
Some physical insight into the curl comes from Stokes’
theorem, which relates the integral of the curl of a vector field
over a surface S to the line integral around a curve C bounding
S (Fig. A.6-4) as
Ύ
C
u · 6 dC = Ύ
S
(∇ ∇ ∇
∇ ∇ × u) · 4dS.
(17)
Here dS is an element of surface area with normal 4(x), and dC
is an element of the curve with tangent 6(x). Analogous to the
case of Gauss’s theorem applied to a volume, we can think of
the surface as composed of infinitesimal tiles, each with a line
integral of u · 6 around it. The border of each tile is shared with
another tile, but, because the line integral, or circulation, is
computed in a counterclockwise manner, the integrals along
this border are the same but of opposite sign for the two tiles,
A.6 Vector calculus 461
S
C
dS
n(x)
t(x) ˆ
ˆ
and therefore cancel. The segments of the line integrals cancel
between all the tiles except those on the outer border that have
no adjacent circulation to cancel them.
If the line integral is nonzero, the vector field has a net rotation along the curve, so the integral of its curl over the surface is
nonzero. The curl of a vector field shows where rotations arise.
A common application is describing the velocity field of a moving fluid. The upper portion of Fig. A.6-5 shows streamlines,
lines parallel to the velocity vector at any point, for a viscous
fluid flowing past a circular object. The velocity is zero at the
object, and increases with distance away from it. The flow is
symmetric on the bottom of the object. The lower portion of
the figure shows contours of the curl of the velocity field with
larger values, indicating greater rotations, close to the object.
Two useful identities, whose proofs are left for the problems,
are that the curl of a gradient and the divergence of a curl are
zero:
∇
∇ ∇
∇ ∇ · (∇ ∇ ∇
∇ ∇ × u) = 0
(18)
∇
∇ ∇
∇ ∇ × (∇ ∇ ∇
∇ ∇φ) = 0.
(19)
Equation 19 can be used with Stokes’ theorem to show that for
a vector field written as the gradient of a scalar, the curl, and
hence circulation around an arbitrary curve, are zero. This idea
is used in mechanics to prove that a conservative force (one that
can be written as the gradient of a potential) has a line integral
that is independent of path, because its circulation around any
path is zero. These relations give insight into seismic waves,
because P waves have no curl and S waves have no divergence
(Section 2.4.1).
A.6.5 Laplacian
The Laplacian operator is formed by taking the divergence of
the gradient of a scalar field, which yields a scalar field
∇
∇ ∇
∇ ∇ 2 φ = ∇
∇ ∇
∇ ∇ · ∇
∇ ∇
∇ ∇φ =
∂
∂
∂
∂
∂
∂
2
1
2
2
2
2
2
3
2
φ
φ
φ φ
x
x
x
i ,
,,
+
+
=
(20)
where the last form uses index notation and the summation
convention. By analogy, the Laplacian of a vector field is a vector field whose components in Cartesian coordinates are the
Laplacians of the original vector components,
∇
∇ ∇
∇ ∇
2 u = (∇ ∇ ∇
∇ ∇
2
u 1 , ∇
∇ ∇
∇ ∇
2 u 2 , ∇
∇ ∇
∇ ∇
2
u 3 ).
(21)
For example, the ê 1 component of ∇
∇ ∇
∇ ∇ 2 u is
∂
∂
∂
∂
∂
∂
2
1
1
2
2
1
2
2
2
1
3
2
u
x
u
x
u
x
.
+
+
(22)
In Cartesian coordinates, the Laplacian of a vector satisfies
∇
∇ ∇
∇ ∇ 2 u = ∇
∇ ∇
∇ ∇(∇ ∇ ∇
∇ ∇ · u) − ∇
∇ ∇
∇ ∇ × (∇ ∇ ∇
∇ ∇ × u),
(23)
Fig. A.6-4 Geometry for Stokes’ theorem: 4(x) is a unit vector pointing
outward at the point x from an element dS of the surface S. dC is an
element of the curve C bounding S, with tangent 6(x).
This can be written as a determinant
∇
∇ ∇
∇ ∇ × u = det
ê
ê
ê
1
2
3
1
2
3
1
2
3
∂
∂
∂
∂
∂
∂
x
x
x
u
u
u
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
,
(15)
or, using index notation, in a compact form as
∇
∇ ∇
∇ ∇ × u = ε ijk
∂
∂
u
x
k
j
= ε ijk u k, j .
(16)
Some physical insight into the curl comes from Stokes’
theorem, which relates the integral of the curl of a vector field
over a surface S to the line integral around a curve C bounding
S (Fig. A.6-4) as
Ύ
C
u · 6 dC = Ύ
S
(∇ ∇ ∇
∇ ∇ × u) · 4dS.
(17)
Here dS is an element of surface area with normal 4(x), and dC
is an element of the curve with tangent 6(x). Analogous to the
case of Gauss’s theorem applied to a volume, we can think of
the surface as composed of infinitesimal tiles, each with a line
integral of u · 6 around it. The border of each tile is shared with
another tile, but, because the line integral, or circulation, is
computed in a counterclockwise manner, the integrals along
this border are the same but of opposite sign for the two tiles,
A.6 Vector calculus 461