UNIVERSITY OF CALIFORNIA
Santa Barbara
Department of Electrical and Computer Engineering
Problem Set No. 1
Issued:
October 1, 2008
Fall 2008
Due:
October 8, 2008
ECE 201A
1.
In this course we shall use many operator identities.
All of them are derivable from the following
relations of vector algebra.
(a)
In a triple scalar product
A BC
×
⋅
one may interchange the
⋅
and
×
without changing
the product.
An interchange of the order of the vector factors leaves the product unchanged, but changes
the sign if two adjacent factors are interchanged.
(b)
()
(
)
(
)
ABC A
C
B A
B
C
××=⋅
−⋅
.
(c)
The chain rule holds
f
gf
g
g
f
∇
⋅=
∇
+
∇
g
and
f
can be replaced by vectors and the multiplication by a vector multiplication.
Using these
facts, prove
2
(
)
AA
A
∇× ∇×
=∇ ∇⋅
−∇
(
)
EH
HE
∇⋅
×
= ∇×
⋅
− ∇×
⋅
.
Do not use decomposition into components in a coordinate system.
2.
(a)
Show that knowing the curl or divergence of a vector field alone is not enough to
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 Spring '09
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 Vector Calculus, Vector Space, Euclidean vector, Vector field, Santa Barbara Department of Electrical

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