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School of Electrical and Computer Engineering, Cornell University
ECE 303: Electromagnetic Fields and Waves
Fall 2007
Homework 1
Due on Aug. 31, 2007 by 5:00 PM
Reading Assignments:
i) Review the material on cartesian, cylindrical, and spherical coordinate systems from your favorite
freshman calculus book. Make sure you are comfortable in using these coordinate systems.
ii) Relevant sections of the online
Haus and Melcher
book for this week are 2.02.6, 3.2, 3.3. Note that
the book contains more material than you are responsible for in this course. Determine relevance by what
is covered in the lectures and the recitations.
Problem 1.1: (warm up)
Consider a scalar quantity
φ
given by the expression,
()
( )
2
2
2
,
,
z
y
x
a
z
y
x
+
+
=
Where
a
is some constant.
a) Find the gradient
∇
r
of the scalar
in the Cartesian coordinate system.
b) Express the scalar
given above in variables of the cylindrical coordinate system and then find the
gradient
∇
r
in the cylinderical coordinate system.
c) Express the scalar
given above in variables of the spherical coordinate system and then find the
gradient
∇
r
in the spherical coordinate system.
d) Find the divergence of the gradient for the scalar
(i.e. first find the gradient vector and then find the
divergence of the gradient vector). Note: The divergence of the gradient, written as
( )
∇
•
∇
r
r
, is also more
commonly denoted by the Laplacian operator
2
∇
, i.e.
( )
2
∇
=
∇
•
∇
r
r
.
Problem 1.2: (basic vector calculus review)
Consider a vector field
F
r
given by the expression:
()
z
z
y
y
x
x
z
y
x
F
ˆ
ˆ
ˆ
,
,
2
+
+
=
r
a) Find the total flux associated with the vector
F
r
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 Fall '06
 RANA
 Vector Calculus, Electromagnet, Line integral, Vector field, Stokes' theorem

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