Class 3 - Vector Calculus
Transcript of Class 3 - Vector Calculus
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AOE 5104 Class 3 9/2/08
Online presentations for todays class: Vector Algebra and Calculus 1 and 2
Vector Algebra and Calculus Crib
Homework 1 due 9/4
Study group assignments have been made and areonline.
Recitations will be Mondays @ 5:30pm (with Nathan Alexander)
Tuesdays @ 5pm (with Chris Rock)
Locations TBA
Which recitation you attend depends on which study groupyou belong to and is listed with the study groupassignments
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Unnumbered slides contain comments that I inserted and
are not part of Professors Devenports original
presentation.
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4
Cylindrical Coordinates
R
er
eez
Coordinates r, , z
Unit vectors er, e, ez(indirections of increasing
coordinates)
Position vector
R= rer+ zez
Vector components
F= Frer+Fe+FzezComponents not constant,
even if vector is constant
r
z
F
x
y
z
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5
Spherical Coordinates
r
er
e
e
rF
Coordinates r, ,
Unit vectors er, e, e (indirections of increasing
coordinates)
Position vector
r= rer
Vector components
F= Frer+Fe+Fe
Errors on this slide in online presentation
y
x
z
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1 1
1
r r x y z
x r r
r r
r
r
F F F F F F
F F F F
x r y r r r
r r r
hh r r
hh r
sin cos , sin sin , cos
sin cos sin sin cos
sin cos sin sin cos
cos cos cos sin sin
F e e e i j k
F
i e i e i e i
r i j k
r re i j k
re i j k
1
x r
r
h rh
F F F F
sin cos sin
sin cos cos cos sin
r
r re i j
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7
J. KURIMA, N. KASAGI and M. HIRATA (1983)
Turbulence and Heat Transfer Laboratory, University of Tokyo
LOW REYNOLDS NUMBER AXISYMMETRIC JET
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8
Class ExerciseUsing cylindrical coordinates (r, , z)
Gravity exerts a force per unit mass of 9.8m/s2
on the flowwhich at (1,0,1) is in the radial direction. Write down thecomponent representation of this force at
a) (1,0,1) b) (1,,1) c) (1,/2,0) d) (0,/2,0)
R
er
eez
r
z
9.8m/s2
a) (9.8,0,0)
b) (-9.8,0,0)
c) (0,-9.8,0)
d) (9.8,0,0)
xy
z
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Vector Algebra in Components
1 1 2 2 3 3
1 2 3
1 2 3
1 2 3
2 3 3 2 1 3 1 1 3 2 1 2 2 1 3
cos
where is the smaller of the two angles
between and
sin where is determined by the
right-han
A B A B
A B
e e e
A B
e e e
A B A B n n
A B A B A B
A A A
B B B
A B A B A B AB AB A B
d rule
A
B
n
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3. Vector Calculus
Fluid particle: Differentially Small Piece
of the Fluid Material
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Concept of Differential Change In a
Vector. The Vector Field.
V
-2
-1
0
1
2
y/
L
-2
0
2
-T/ U
L0
1
2
z/L
V+dV
dV
V=V(r,t)
=(r,t)Scalar fieldVector field
Differential change in vector
Change in direction
Change in magnitude
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PP'
er
e
ez
d
r
z
Change in Unit Vectors
Cylindrical System
rdd ee
ee dd r
0zde
e+de
er+de
r
er
ede
der
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14
Change in Unit Vectors
Spherical System
sin
cos
sin cos
e e e
e e e
e e e
r
r
r
d d d
d d d
d d d
r
er
e
e
r
See Formulae for Vector
Algebra and Calculus
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15
Example
kjir zyx
dt
drV
zr zr eer
R=R(t)
Fluid particle
Differentially small
piece of the fluid
material
V=V(t) The position of fluid particle moving in a flowvaries with time. Working in different coordinate
systems write down expressions for the position
and, by differentiation, the velocity vectors.
O
... This is an example of the calculus of vectors with respect to time.
dtdrV
zrdt
dz
dt
dr
dt
dreee
Cartesian
System
Cylindrical
Systemz
rr
dtdz
dtdr
dtdr eee
kjidt
dz
dt
dy
dt
dx
ee dd r
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Vector Calculus w.r.t. Time
Since any vector may be decomposed into scalar
components, calculus w.r.t. time, only involves scalar
calculus of the components
dtdtdt
ttt
ttt
ttt
BABA
BAB
ABA
BAB
ABA
BABA
.
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High Speed Flow Past an Axisymmetric Object
Shadowgraph picture is from An Album of Fluid Motionby Van Dyke
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1
1
lim where is evaluated at any
point on and the largest 0 as
similarly for other integrals
lim etc.
I s s
s s
V s V s
B i n
i i in
iA
i i
B i n
i in
iA
d
n
I d
Line integrals
divide the path intosmall segments:
is a typical onesi
n
si
Vi
A
B
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Integral Calculus With Respect to Space
D=D(r), = (r)
n
dS
Surface S
Volume R
D(r)
ds
O
r
D(r)
d
Line Integrals
s D s D sB B B
A A Ad d d
For closed loops, e.g. Circulation V sd
A
B
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20Picture is from An Album of Fluid Motionby Van Dyke
Mach approximately 2.0
For closed loops, e.g. Circulation sV d.
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Integral Calculus With Respect to
Space
D=D(r), = (r)
n
dS
Surface S
Volume R
D(r)
ds
O
r
D(r)
d
Surface Integrals
n D n D nS S SdS dS dS Se.g. Volumetric Flow Rate through surface S S dSnV.
For closed
surfaces
Volume Integrals
D
R R
d d
A
B
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22Picture is from An Album of Fluid Motionby Van Dyke
Mach approximately 2.0
n
dS
. n D n D nS S SdS dS dS
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In 1-D
0S
1grad lim dS
n
In 3-D
Differential Calculus w.r.t. Space
Definitions ofdiv, gradand curl
0S
1div lim dS
D D n
0S
1curl lim dS
D D n
Elemental volumewith surface S
n
dS
D=D(r), = (r)
0
1lim ( ) ( )
xdf
f x x f xdx x
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Alternative to the Integral Definition of Grad
We want the generalization of
0
0
1lim ( ) ( )
lim ( ) ( ) Grad
Grad
Grad
rr r r r
i j k
i j k
x
df df f x x f x df dx
dx x dx
df f f f d
f f fdf dx dy dz f dx dy dz
x y z
f f ffx y z
continued
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Alternative to the Integral Definition of Grad
Cylindrical coordinates
0
cos sin
cos sin sin cos
lim ( ) ( ) Grad
Grad
1Grad
r
r i j k e k
r i j i j k
e e k
r r r r
e e k
e e k
r
r
r
r
r r z r z
d dr rd z
dr rd dz
df f f f d
f f f
df dr d dz f dr rd dz r z
f f ff
r r zcontinued
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Alternative to the Integral Definition of Grad
Spherical coordinates
sin cos sin sin cos
1sin cos sin sin cos 1
1cos cos cos sin sin
1sin cos sin
sin cos sin
r i j k
r re i j k
r re i j k
r re i j
r i
r r
r
r
r r r
hh r r
h rh r
h rh
d dr
sin cos cos cos cos sin sin
sin sin cos sin
, , Grad sin
1 1
Grad sin
j k i j k
i j e e e
e e e
e e
r
r
r
rd
r d dr rd r d
F F FdF r dr d d F dr rd r d
r
F F F
F r r r e
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ndS
(large)
Gradient
0S
1grad lim dS
n
dS
n
Elemental volume
with surface S
ndS
(small)
ndS
(medium)
ndS
(medium)
= magnitude and direction of the slope in the scalar field at a point
ResultingndS
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Gradient
Component of gradient is the partial derivative
in the direction of that component
Fouriers Law of Heat Conduction
es s
n
Tq k k T
n
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The integral definition given on a previous slide
can also be used to obtain the formulas for the
gradient.
On the next four slides, the form of GradF inCartesian coordinates is worked out directly from
the integral definition.
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0
S
0
1grad lim dS
1
lim
n
i j k i j k dxdydz dxdydz dxdydz x y z x y z
Face 2
Differential form of the Gradient
0S
1grad lim dS
n
Cartesian system
dy
dx
dz
j
ik
P
Evaluate integral by expanding the variation in
about a point Pat the center of an elementalCartesian volume. Consider the twoxfaces:
= (x,y,z)
Face 1
e1
dS ( )2
n i
Fac
dxdydz
x
e 2
dS ( )2
n i
Fac
dx dydzx
adding these gives
i dxdydzx
Proceeding in the same way foryand z
j dxdydzy
k dxdydzz
andwe get , so
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An element of volume with a local Cartesian coordinate system having its origin
at the centroid of the corners, O
O .
x
y
z
x
z
y
M
Point M is at the centroid of the face perpendicular to the y-
axis with coordinates (0,y/2, 0)
Other points in this face have the coordinates (x,y/2, z)
F y axis
F x y z
We introduce a local Cartesian coordinate system and then use
a Taylor series to express at a point in the face
in terms of its value and the values of its derivatives at the origin:
( , , ) 2
2
0 0 0 0 0 0 0 0 00 0 0
2
F F y F F x z O l
x y z
O l x y z
( , , ) ( , , ) ( , , )( , , ) ( )
where ( ) denotes the remainder which contains the factors , ,
in at least quadratic formcontinued
Gradient of a Differentiable Function, F
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2
4
0 0 0 0 0 0 0 0 00 0 0
2
0 0 0
2
y
y axis
F x y z dS
F F y F F x z O l dxdz
x y z
F y x z F x z O l
y
z x
2 2- z - x
2 2
the integral over the face ( ) :
( , , )
( , , ) ( , , ) ( , , )( , , ) ( )
( , , )( )
the integ
n j
n
j
j
40 0 00 0 0
2
0 0 0
y
y y
y axis
F y x z F x y z dS F x z O l
y
FF x y z dS F x y z dS y
y
ral over the face ( ) :
( , , ) ( )( , , ) ( , , ) ( )
the sum of the integrals over these two faces is given by
( , , )( , , ) ( , , ) (
n j
n j
n n 4x z O l ) ( )
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4
4
4
0 0 0
0 0 0
0 0 0
y y
x x
z z
FF x y z dS F x y z dS y x z O l
y
FF x y z dS F x y z dS y x z O l
x
FF x y z dS F x y z dS y x z O l z
y
( , , )( , , ) ( , , ) ( ) ( )
similarly
( , , )( , , ) ( , , ) ( ) ( )
( , , )( , , ) ( , , ) ( ) ( )
n n j
n n i
n n k
0 0
1
lallfac s
x z
F F FF F x y z dS O l
x y z
F F FF
x y z
e
volume of the element
Grad lim ( , , ) lim ( )
Grad
n i j k
i j k
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Differential Forms of the Gradient
These differential forms define the vector operator
sinsin r
e
+r
e
+rer
e
+r
e
+re
ze+
r
e+
re
ze+
r
e+
re
zk+
yj+
xi
zk+
yj+
xi
=grad
rr
zrzr
Cartesian
Cylindrical
Spherical
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0S
1lim dS
V V ndiv
dS
n
Divergence
Fluid particle, coincident
with at time t, after time
thas elapsed.
= proportionate rate of change of volume of a fluid particle
Elemental volume
with surface S
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Differential Forms of the
Divergence
A.re+
re+
reA
r1+A
r1+
rAr
r1
A.z
e+r
e+
re
z
A+
A
r
1+
r
rA
r
1
A.zk+yj+xiz
A+y
A+x
A
A.=Adiv
rr
2
2
zrzr
zyx
sinsinsin
sin
Cartesian
Cylindrical
Spherical
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Differential Forms of the Curl
ArrAA
r
erere
r
1=
ArAA
zr
eere
r
1=
AAA
zyx
kji
=A=Acurl
r
r
2
zr
zr
zyx
sin
sin
sin
S0 dS1
Lim nAAcurl
Cartesian Cylindrical Spherical
Curl of the veloci ty vecto rV =twice the circumferentially averaged
angular velocity of
-the flow around a point, or
-a fluid particle
=Vortici ty Pure rotation No rotation Rotation
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e
0S
0S
0S
0S
Ce
0 0
C
1lim dS
1lim . dS
1lim . dS
1
lim . d
1lim .d lim
V V n
e V e V n
e V V e n
e V V e n
e V V s
e
curl
curl
curlh
curl s hh
curl
c2
0 0
1lim 2 2 lim
V
vurl v a
a a
dS
n
Curl
e
nPerimeter Ce
ds
h
dS=dsh
Area
radius a
v avg. tangential
velocity= twice the avg. angular velocity
about e
Elemental volume
with surface S