ECE 329
Homework 5
Due: Thursday, Feb 24, 2009, 5PM
1. Verifying
vector calculus identities
,
∇ ×
(
∇
Φ
) = 0
and
∇
·
(
∇ ×
A
) = 0
:
a) The
gradient
of a scalar field
Φ
is defined as
∇
Φ
≡
∂
Φ
∂
x
ˆ
x
+
∂
Φ
∂
y
ˆ
y
+
∂
Φ
∂
z
ˆ
z
.
Assuming that the order of di
ff
erentiation can be switched, show that
∇ ×
(
∇
Φ
) = 0
.
•
Consequently, any
curlfree
vector field can be expressed as the
gradient of some scalar
field — important in the definition of electrostatic potential studied in Chapter 6.
b) Given any di
ff
erentiable vector field
A
= ˆ
xA
x
+ ˆ
yA
y
+ ˆ
zA
z
, show that
∇
·
(
∇ ×
A
) = 0
by first
expanding
∇ ×
A
in terms of partial derivatives (e.g.,
∂
A
x
∂
x
,
∂
A
y
∂
x
etc.) of the components of
A
.
•
Consequently, any
divergencefree
vector field can be expressed as the
curl of some
other vector
field — important in the definition of vector potential studied in Chapter 6.
2. Another important vector identity is
∇ ×
(
∇ ×
A
) =
∇
(
∇
·
A
)
 ∇
2
A
,
where
∇
2
A
≡
(
∂
2
∂
x
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Kim
 Vector Calculus, Electromagnet, Fundamental physics concepts, Eo cos, divergencefree vector field, Verifying vector calculus, curlfree vector ﬁeld

Click to edit the document details