Multidimensional Calculus: Differential Theory
Chapter & Page: 9–13
9.3
The Classic Gradient, Divergence and Curl
Basic Definitions in Euclidean Space
For expediency, we will first define the classical differential operators for scalar and vector fields
in a Euclidean space
E
using the “del operator”. Moreover, we will assume that our coordinate
system
{
x
1
,
x
2
,...,
x
N
}
is
Cartesian
.
The
del operator
is the “vector differential operator” given in our Cartesian system by
−→
∇
=
∇
=
N
summationdisplay
k
=
1
∂
∂
x
k
e
k
=
N
summationdisplay
k
=
1
e
k
∂
∂
x
k
=
N
summationdisplay
k
=
1
∂
x
∂
x
k
∂
∂
x
k
.
For any sufficiently differentiable scalar field
Ψ
and vector field
F
with corresponding coordi
nate formulas
Ψ(
x
)
=
ψ
(
x
1
,
x
2
,...,
x
N
)
and
F
(
x
)
=
N
summationdisplay
k
=
1
F
k
(
x
1
,
x
2
,...,
x
N
)
e
k
,
we define
the gradient of
Ψ
=
grad
(Ψ)
=
∇
Ψ
,
the divergence of
F
=
div
(
F
)
=
∇
·
F
,
and
the curl of
F
=
curl
(
F
)
=
∇
×
F
by the coordinate formulas
grad
(Ψ)
=
∇
Ψ
=
N
summationdisplay
k
=
1
e
k
∂
∂
x
k
ψ
(
x
1
,
x
2
,...,
x
N
)
=
N
summationdisplay
k
=
1
e
k
∂ψ
∂
x
k
=
N
summationdisplay
k
=
1
∂ψ
∂
x
k
e
k
,
div
(
F
)
=
∇
·
F
=
bracketleftBigg
N
summationdisplay
k
=
1
∂
∂
x
k
e
k
bracketrightBigg
·
N
summationdisplay
j
=
1
F
j
(
x
1
,
x
2
,...,
x
N
)
e
j
=
N
summationdisplay
k
=
1
∂
F
k
∂
x
k
,
and
curl
(
F
)
=
∇
×
F
=
bracketleftBigg
N
summationdisplay
k
=
1
∂
∂
x
k
e
k
bracketrightBigg
×
N
summationdisplay
j
=
1
F
j
(
x
1
,
x
2
,...,
x
N
)
e
j
=
vextendsingle
vextendsingle
vextendsingle
vextendsingle
vextendsingle
vextendsingle
vextendsingle
vextendsingle
vextendsingle
e
1
e
2
e
3
∂
∂
x
1
∂
∂
x
2
∂
∂
x
3
F
1
F
2
F
3
vextendsingle
vextendsingle
vextendsingle
vextendsingle
vextendsingle
vextendsingle
vextendsingle
vextendsingle
vextendsingle
=
bracketleftbigg
∂
F
3
∂
x
2
−
∂
F
2
∂
x
3
bracketrightbigg
e
1
−
bracketleftbigg
∂
F
3
∂
x
1
−
∂
F
1
∂
x
3
bracketrightbigg
e
2
+
bracketleftbigg
∂
F
2
∂
x
1
−
∂
F
1
∂
x
2
bracketrightbigg
e
3
.
Both
grad
(Ψ)
and div
(
F
)
can be defined assuming our space is of any dimension. However,
curl
(
F
)
requires that our space be threedimensional (in which case,we usually use the traditional
{
(
x
,
y
,
z
)
}
and
{
i
,
j
,
k
}
notation).
version: 10/31/2011
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Multidimensional Calculus: Differential Theory
Chapter & Page: 9–14
Observe that
grad
(Ψ)
and
curl
(
F
)
are vector fields, while div
(
F
)
is a scalar field.
One thing not obvious from the above definitions is why anyone would be interested in these
things. There are good reasons arising from both basic mathematics and from physics. The
“geometric/physical significance” of
grad
(Ψ)
will be discussed in a little bit. The importance
of the divergence and curl of a vector field will be discussed later, in conjunction with some
classic integral theorems. In anticipation of that discussion, let me introduce some terminology
that you may encounter in homework:
“
F
is solenoidal”
⇐⇒
“
F
is divergence free”
⇐⇒
∇
·
F
=
0
and
“
F
is irrotational”
⇐⇒
“
F
is curl free”
⇐⇒
∇
×
F
=
0
.
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 Summer '11
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 Math, Calculus, Derivative, Scalar, Cartesian Coordinate System, dt, Polar coordinate system

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