Math problem sets about Gaussian integrals, bosons and fermions and DIfferential forms on R - Mathematics
Please answer all of the question and provide the details of each step. Also please label the answer clearly
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Homework
Week
,
I
part
,
-
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.
Recall
the
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Find
formulas for polar
=
r
=
r
the
cos
sin
matrix
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Use
polar
this
t
I
the
to
=
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of
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that
coordinates
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on
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derivatives
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3
.
that
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be
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agree
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=
x
the
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Ex
and
,
formulas
J
( )
=
be
.
class notes
matrix
vector
column
a
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nxn
-
the
in
positive definite symmetric
a
let
and
x
4x4
these
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e
find
to
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formula
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answer
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have
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k
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.
spareof
the
of
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t
c
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I
3
2
3
3
general
t
deg
dimension
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=
k
n
4
degree exactly
tbs 4
-
So
variables
polynomials
to
equal
le
?
=3
n
0
ke
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dimension
the
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satisfy
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ply
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have
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L
formula
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city
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t
c
,
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t
d 41424
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a)
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B
(
=
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2
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4
=
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I
-
( ¥÷)
3
3
O
and
)
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the
to
answer
expand
out
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Compute
Check
det B
that
the
the
#
e
B
-
.
BY
polynomial
dy d
y
dysdy
1=7
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=
the
in
e
y
dy
correct
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with
sign
i
.
,
.
d
4
4434
formula
agrees
is
f
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antisymmetric
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a
compute
directly
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as
Use
that
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part
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La )
or
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.
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Homework
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Part
,
f
2
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6
.
Go
of
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forms
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in
the
make
that
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(
=
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be
can
in
old
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that
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v
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is
smooth
a
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vector
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if
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it
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different
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on
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field
Ext
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1123
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f
is
smooth
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off
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=
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a
3
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( divergence of
And
x
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form )
function
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smooth
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(
Vs
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)
V
=
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it
form
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calc
variable
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y
,
.
:
d
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t
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or
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j
differential
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class
in
multi
vector
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ur
+
out
school
-
a
lx.y.at
,
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ways
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v.
encoded
stretched
that
notice
,
derivatives
exterior
was
with
connection
familiar
more
computation of
the
through
form )
)
7
.
a)
Let
=
w
dw
Show
dy
x
O
=
w=
x
Compute
Is
it
w
O
-
encodes
implies
farm )
f
w
.
curl
a
-
free vector
field )
df for
-
-
( the
vector
be
the
.
of
dy
d
I
w
must
f
Find
Let
(
Lemma
function
b)
.
.
Poincare
The
f
ydx
t
ydx
-
a
some
fell
gradient
function )
.
.
find
to
possible
w
-
df
a
?
function f
such
that
3
\
8
coordinates
Changing
.
Suppose
change
we
IR ?
on
(
x
y
,
)
(
~
u
transform forms
need
,
.
coordinates
new
To
)
v
,
find
to
x
y
the
differentials transform
de
dy
and
a)
Consider
=
2¥
du
LAW
du
the
u
,
)
C
-
forms
dx
.
dy
and
x
,
v
.
I
°
.
)
y
rule
du
usual
as
coordinates
e
a
as
ZZ
t
y
Write
Lu
y
-
FTW
t
x
•
Luv )
x
chain
a
wedge together
polar
-
V
,
functions of
as
Then
=
u
Cr
←
r
cos
r
sin
O
r
O
)
iffy
:
O
polar
in
O
,
=
ten
( ¥)
coordinates
.
I
b)
write
dandy
words
polar
in
Should
.
from
c)
Show
dxndy
that
,
=
ingenue
det
5
given
du
.
du
Cx
,
y )
F-
rs
Luis
be
familiar
Problem
I !
,
( aziz
Zzz)
Jacobian
matrix
!
4g
9
.
Integration
Recall
that
on
curves
.
parameterized
a
y
curare
IR
in
Ele
is
i.
e
function
a
for
.
y
t
every
La
E
La
:
,
b
,
2
Jlt )
s ?
is
✓
radius
of
circle
:
(
!
O
,
2
I
it
or
ie
IR
Lt
x
.
)
cost
2
-
-
y
:
x
Given
we
to
an
1-
a
form
W
f
can
rewrite
ordinary
all
IR
,
2
4)
.
.
rn Lt ) )
,
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in
.
)
sint
)
Zsmt
-
1122
→
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l
3
(
O
t
-
,
)
l
,
4
-
t )
integrate
integral
.
.
and
,
by
differentials
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,
t
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3
Et
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t
4
W
y
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.
r
on
point
a
get )
,
Lo
Yi
)
it
?
L2cost
↳
( g.
=
1122
in
→
t
(
2
IR
→
in
a
using
terms
of
parameterized
the
dt
chair
,
then
,
convey
rule
doing
.
→
,
Example
Lets
Let
!
w
xdy
=
the
integrate along
ydx
-
§
(
boundary of
t
forman IRD
-
the
.
square
y
ri
×
terms
In
t
of
o
,
et
It
2
3
fyw
so
Actual
a)
problem
Let
w
si
w
te
etc
It
2
W
3
.
→
w
foot
=
Ldlt
→
W
4
I
tdlo
-
)
D
-
G- Hdt
→
Odt
-
It
-
Ida
-
w
-
-
fidttfzdt
-
t)
It
=
tf !
xdytydx
f
,
:
10,43
s
IR
0=2
boundary
fgw
xdy
ydx
-
,
y
square
fyw
of
.
.
=
circle
of
radius
a
Find
=
.
Find
Let
dt
Ddu
)
-
O
the
b)
O
=
constant
-
.
Nota
variable
cord
rule
.
( for the
OPTIMAL
10
Let
.
w
xdyndz
=
§
intrepid )
.
.
.
you
can
put
O
X
=
sin
y
=
sin
O
cos
O
z
=
find
(
Note
dy
dz
e
coordinates
O
C
cosy
fsw
:
you
which
It
will
need
means
sphere
,
y
S
on
:
everywhere except at
the poles
thats ok )
-
song
terms
in
unit
.
,
By rewriting
the
1123
in
Recall
be
S
let
and
,
of
O
.
y
@
4)
E
(O
,
it
)
x
fo
,
277
,
.
to
choose
an
determine
order
orientation
an
of the
the
I
of
coordinates
sign
in
the
S
to do
O
,
y
answer
.
.)
this
,
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