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Classical Minimal Surfaces
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Classical Minimal Surfaces
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2372.pdf
Description
Identifier
Thesis
2573
Author
Nimchek, Michael T. (Michael Thomas), 1973
Title
Classical
Minimal
Surfaces
Publisher
Central Connecticut State University
Date of Publication
2016
Resource Type
Master's Thesis
Abstract
Minimal
surfaces
are
surfaces
that
minimize
area
locally
. They have
captured
the
imagination
of
mathematicians
since
they were
first
discovered
in the
late
eighteenth
century
. The
theory
of
minimal
surfaces
remains
an
active
area
of
research
. This
manuscript
explores
classical
minimal
surfaces
,
i.e.
,
minimal
surfaces
that were
discovered
during
the
eighteenth
and
nineteenth
centuries
. The
minimal
surface
equation
is
derived
using
methods
from the
Calculus
of
Variations
and
Gauss
'
Divergence
Theorem
. This
equation
is
then
converted
into a
useful
partial
differential
equation
in
both
Cartesian
and
polar
coordinates
. The
minimal
surface
equation
is
shown
to be
consistent
with
vanishing
mean
curvature
. The
catenoid
is
proven
to be the
only
nontrivial
surface
of
revolution
that
is
a
minimal
surface
(a
result
first
proved
by
Muesnier
in
1776)
and the
helicoid
is
proven
to be the
only
nontrivial
ruled
minimal
surface
(a
result
first
proved
by
Catalan
in
1842)
.
Scherk's
surface
is
proved
to be the
only
nontrivial
minimal
translation
surface
based
on
profile
curves
that
exist
in
orthogonal
planes
.
Separation
of
variables
is
then
considered
,
i.e.
,
surfaces
of the
form
u
=
f(x)g(y)
,
leading
to a
Cartesian
expression
for the
helicoid
. A
search
for
other
minimal
surfaces
based
on
separation
of
variables
is
undertaken
.
Though
no
new
minimal
surfaces
are
found
, a
method
for
identifying
possible
candidates
is
explored
.
Finally
,
theWeierstrass
Representation
Formula
is
introduced
and
used
to
generate
the
graphs
of
many
beautiful
minimal
surfaces
.
Notes
"
Submitted
in
Partial
Fulfillment
of the
Requirements
for the
Degree
of
Master
of
Arts
in
Mathematics.
";
Thesis
advisor
:
Nelson
Castenada.
;
M.A.,Central
Connecticut
State
University,,2016.
;
Includes
bibliographical
references
(leaf
84)
.
Subject
Minimal surfaces.
Department
Department of Mathematics
Advisor
Castañeda, Nelson
Type
Text
Software
System requirements: PC and World Wide Web browser.
Language
eng
OCLC number
969903159
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