PHYS809Class1Notes
1. Vector Calculus
1.1. Gradient of a scalar field
A scalar field
( )
φ
r
is a scalar function defined at all positions
r
. The change in
φ
in moving from
r
to
r
+
d
r
is
.
d
d
φ
φ
=
∇
r
i
In Cartesian coordinates
.
x
y
z
φ
φ
φ
φ
∂
∂
∂
∇
=
+
+
∂
∂
∂
i
j
k
φ
∇
is a vector that lies in the direction of fastest increase in
φ
. It is normal to the surface
φ
= constant,
that passes through
r
.
1.2. Divergence of a vector field
In Cartesian coordinates
.
y
x
z
A
A
A
x
y
z
∂
∂
∂
∇
=
+
+
∂
∂
∂
A
i
If
0,
∇
=
A
i
A
is said to be
solenoidal
.
If
,
φ
= ∇
A
then
2
.
φ
φ
∇
= ∇ ∇
= ∇
A
i i
2
∇
is the
Laplacian
operator.
1.3. Curl of a vector field
In Cartesian coordinates,
curl
.
y
y
z
x
z
x
A
A
A
A
A
A
y
z
z
x
x
y
∂
∂
∂
∂
∂
∂
= ∇×
=

+

+

∂
∂
∂
∂
∂
∂
A
A
i
j
k
If
0,
∇×
=
A
A
is said to be
irrotational
.
In solid body rotation,
,
=
×
v
ω
r
where
ω
is fixed in magnitude and direction. Take
ω
=
ω
k
, so that
,
y
x
ω
ω
= 
+
v
i
j
and
2
2
.
ω
∇×
=
=
v
k
ω
Hence if there is no rotation,
0.
∇×
=
v
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1.4. Vector operator identities
(
)
(
)
2
0
0
.
φ
∇ ∇×
=
∇×∇
=
∇× ∇×
= ∇ ∇
∇
A
A
A
A
i
i
This last expression is useful for finding the Laplacian of a vector in nonCartesian coordinates.
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
,
,
,
,
.
ψ
ψ
ψ
ψ
ψ
ψ
∇
=
∇
+
∇
∇×
= ∇
×
+
∇×
∇
=
∇
+
∇
+
× ∇×
+
× ∇×
∇
×
=
∇×

∇×
∇×
×
=
∇

∇
+
∇

∇
A
A
A
A
A
A
A B
A
B
B
A
A
B
B
A
A
B
B
A
A
B
A
B
A
B
B
A
B
A
A
B
i
i
i
i
i
i
i
i
i
i
i
i
i
1.5. Grad, div and curl in general orthogonal curvilinear coordinates
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 Fall '11
 MacDonald
 Vector field, Gradient, Cartesian coordinates, h2 h3 A1

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