Paradoxical Euler or Integration by Differentiation
Transcript of Paradoxical Euler or Integration by Differentiation
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Paradoxical Euler
or
Integration by Differentiation:
Andrew Fabian and Hieu Nguyen
Rowan University
A Synopsis of E236
The Euler Society 2008 Annual Conference
July 21, 2008
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Exposition de quelques paradoxes dans le calcul integral
(Explanation of Certain Paradoxes in Integral Calculus)
E236
Originally published in Memoires de l'academie des sciences de
Berlin 12, 1758, pp. 300-321; Opera Omnia: Series 1, Volume 22,
pp. 214 – 236 (Available at The Euler Archive: http://www.math.dartmouth.edu/~euler/ )
English translation by Andrew Fabian (2007).
Student Translations of Euler’s Mathematics (STEM) Project
- 7 students
- 3 faculty members
- 9 translations
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The First Paradox
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PROBLEM I (E236)Given point A, find the curve EM such that the perpendicular
AV, derived from point A onto some tangent of the curve MV,
is the same size everywhere.
dx Pp M
dy m
PS M
PM Mm
PR m
AP Mm
x AP
y PM
2 2ds Mm dx dy
PM M ydxPS
Mm ds
AP m xdyPR
Mm ds
a
,, PMSMm and are similarAPR
aAV
x
y
( “Tangential” distance)
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a AV PS PR
ydx xdy
ds ds
dy dsy x a
dx dx
ydx xdy ads
a
Note: If the solution curve is assumed to lie below x-axis, then
Euler’s DE becomes
dx
dydxa
dx
dyxy
22 )()()(
Euler’s Differential Equation:
2 2dx dydyy x a
dx dx
2 2dx dydyy x a
dx dx
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Ordinary Method of Integration
2 2ydx xdy a dx dy
Euler’s differential equation (differential form):
22222222 2 dyadxadyxxydxdydxy
1. Square both sides:
2 2 2
2 2
xydx a dx x y ady
a x
2 2 2 2 2a dy x dy xydx adx x y a
2. Solve for dy (Euler ignores negative solution):
3. Rewrite:
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2 2 2 2 2a dy x dy xydx adx x y a
22
2222 ;xa
uxdxxadudyxauy
4. u-substitution:
22
22
22222222 )()()( xadxuxxa
uxdxxaxaduxadxxydyxa
222222 23
)( xadxuxxauxdxxadu
2 2 2 2 2 2( )( 1)adx x y a adx a x u
322 2 2 2 2( ) ( )( 1)du a x adx a x u
LHS:
RHS:
Simplifies to:
2 2 2( ) 1a x du adx u
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2 2 2( ) 1 (true)a x du adx u
2 1u Case I:
222 ayx (Circle)
2 2 2 2y u a x a x
Thus:
2 22 1
du dxa
a xu
Case II:2 1u
1
2
2log 1 loga x
u u na x
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2 1a x
u u na x
2 22 ( 1) 1a x
n u u ua x
1
2
2 2 1a x a x
n una x a x
1 1
2 21
2 2
n a x a xu
a x n a x
1 12 21
2 2( )( )
n a x y a x
a x n a xa x a x
1( ) ( )
2 2
na x y a x
n
2 2( 1) ( 1)
2 2
n ny x a
n n
(Line)
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21y xp a p
:dy
pdx
dy
dx
21
apx
p
21
dp ap dpp p x
dx dxp
2
21
1
ay a p xp
p
Integrating by Differentiating
Case I: 0dp
dx
1dp
p xdx
21
ap dp
dxp
Euler’s Differential Equation:
2. Differentiate:
1. Rewrite in terms of
2 2ydx xdy a dx dy
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2 2x y
Solution to Case I:
2 2 2x y a (Circle)
2 2 2
21
a p a
p
2a
A
E
VMa
Eliminate the parameter:
2 2
2 21 1
ap a
p p
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Case II: 0dp
dx
(Tangent line)
Thus p = constant = n
2 21 1y xp a p y nx a n
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3 33ydx xdy a dx dy
2
3 23 (1 )
apx
p
3 23 (1 )
ay
p
Higher-Order Examples
Case I: 0dp
dx
Cubic:
33 1dy
y xp a p pdx
Case II: 0dp
dx
(Tangent line)
Thus p = constant = n
33 33 1 1y xp a p y nx a n
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( .)n n nnydx xdy a dx dx dy dx dy etc
1 1
1
.
( .)nn
a p a p etcx
n p p etc
1
( ) ( ) .
( .)nn
na n a p n a p etcy
n p p etc
Case I: 0dp
dx
Arbitrary order:
( .)ny px a p p etc
Case II: 0dp
dx Thus p = constant = m
( .)ny mx a m m etc
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dy
y xp F p pdx
dy
dx
'( )dp dp
p p x F pdx dx
General Solution of Euler’s DE
1dp
p xdx
'( )
dpF p
dx
Integrating by Differentiating:
'( )x F p
y F p xp '( )F p pF p
Case I: 0dp
dx
(Parametric solution!)
Case II: 0dp
dx
y xp F p
Thus p = constant
(Tangent lines to solution in Case I)
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PROBLEM II (E236)On the axis AB, find the curve AMB such that having derived from one
of its points M the tangent TMV, it intersects the two lines AE and BF,
derived perpendicularly to the axis AB at the two given points A and B,
so that the “rectangle” formed by the lines AT and BV is the same size
everywhere.
2cBVAT
aAB 2dx Pp M
dy m
x AP
y PM
2 2ds Mm dx dy
,, PMSMm APRand are similar
dx
dyx
M
mTRRM
dx
dyxa
M
mMSSV
)2(
x
y
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(“Tangential area”)
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Euler’s Differential Equation
dx
dyxyRMPMAT
dx
dyxaySVBSBV
)2(
22c
dx
ady
dx
dyxy
dx
xdyyBVAT
2 2 2 ( )dy
y xp ap c a p F p pdx
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2
2 2 2
a px a
c a p
2
2 2 2
cy
c a p
0dp
dxCase I:
Integrating by Differentiating
Eliminating the parameter:
2 2 2
2 2 2 2
( )x a a p
a c a p
2 2
2 2 2 2
y c
c c a p
1)(
222
222
2
2
2
2
pac
pac
c
y
a
ax
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Solution to Case I:
2 2
2 2
( )1
x a y
a c
(Ellipse)
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Case II: 0dp
dx
222)( ancxany
222 ancnaAT
222 ancnaBV
2cBVAT
(Tangent line)
Thus p = constant = n
222 pacapxpy
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PROBLEM III (E236)Given two points A and C, find the curve EM such that if we
derive some tangent MV, which the perpendicular AV is directed
towards from the first point A, and we join the straight line CV to
V from the other point C, this line CV is the same size
everywhere.
a
b
x=AP
y=PM
CV=a
AC=b
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PROBLEM IV (E236)Given two points A and B, find the curve EM such that having
derived some tangent VMX, if the perpendiculars AV and BX are
directed towards it from the points A and B, the rectangle of these
lines is the same size everywhere.
x=AP
y=PM
AB = 2b
2cBXAV
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Generalization to 3-D
2-D: (Distance from curve to point)
2 2ydx xdy a dx dy
3-D: (Distance from surface to point)
2 2 2 2 2 2zdxdy xdydz ydxdz a dx dy dx dz dy dz
2
1dy dy
y x adx dx
22
1z z z z
z x y ax y x y
2 2
,,
dy dxx y a
dx dy
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The Second Paradox
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Student Translations of Euler’s Mathematics (STEM) Project Website:
http://www.rowan.edu/colleges/las/departments/math/facultystaff/nguyen/euler/index.html
References
Originally published in Memoires de l'academie des sciences de Berlin 12, 1758, pp.
300-321; Opera Omnia: Series 1, Volume 22, pp. 214 – 236. (Available at The Euler
Archive: http://www.math.dartmouth.edu/~euler/ )