52 AOE 5104 Class 4 9408

AOE 5104 Class 4 9/4/08
• Online presentations for today’s class:
– Vector Algebra and Calculus 2 and 3

•
•
•
•

Vector Algebra and Calculus Crib
Homework 1
Homework 2 due 9/11
Study group assignments have been made and
are online.
• Recitations will be
– Mondays @ 5:30pm (with Nathan Alexander) in
Randolph 221
– Tuesdays @ 5pm (with Chris Rock) in Whitemore 349


I have added the slides without numbers. The numbered slides are the
original file.


Last Class
ez

• Changes in Unit
Vectors
• Calculus w.r.t. time
• Integral calculus w.r.t.

space
• Today: differential
calculus in 3D

P'
P e
r

z

eθ

r
θ

deθ = −dθe r
de r = dθeθ
de z = 0

dθ

∂ ( A + B ) ∂A ∂B
=
+
∂t
∂t ∂t
∂ ( A.B )
∂B ∂A
=A
+

B
∂t
∂t ∂t
∂( A × B )
∂B ∂A
= A×
+
×B
∂t
∂t ∂t

∫ ( A + B ) dt = ∫ Adt + ∫ Bdt


Oliver Heaviside
1850-1925


Shock in a CD Nozzle

Bourgoing & Benay (2005), ONERA, France

Schlieren visualization
Sensitive to in-plane index of ref. gradient


Differential Calculus w.r.t. Space
Definitions of div, grad and curl
In 1-D
df

 1

= lim  [ f ( x + ∆x) − f ( x) ] ÷
dx ∆x → 0  ∆x


In 3-D

 1

gradφ ≡ lim  Ñ
φ ndS ÷
δ τ→0 δ τ ∫
 ΔS


 1

divD ≡ lim  Ñ
D.ndS ÷
∫
δ τ →0 δ τ
 ΔS

 1

curlD ≡ − lim  Ñ
D × ndS ÷
δ τ →0 δ τ ∫
 ΔS



D=D(r), φ = φ (r)

n
dS

Elemental volume δτ
with surface ∆S


Gradient
φ=
h
hig

φ ndS
(medium)
φ=

n

φ ndS
(large)

low

Resulting φ ndS
dS

φ ndS
(small)
φ ndS
(medium)

 1

gradφ ≡ lim  Ñ
φ ndS ÷
∫
δ τ →0 δ τ
 ΔS


Elemental volume δτ
with surface ∆S

= magnitude and direction of the slope in the scalar field at a point


Review

Gradient
gradφ ≡ Limδτ →0

1
φndS
∫
δτ ΔS

Magnitude and direction
of the slope in the scalar
field at a point


Gradient

low

• Fourier´s Law of Heat Conduction

∂T
q = − k
= − k∇T .n
∂n

h
hig

φ=

∂φ
e s .∇φ =
∂s

s, e s

φ=

• Component of gradient is the partial derivative

in the direction of that component

∇φ


Differential form of the Gradient
Cartesian system
Evaluate integral by expanding the variation in
φ about a point P at the center of an elemental
Cartesian volume. Consider the two x faces:

∂φ dx 

φ
n
dS
≈
φ
−

(−i )dydz
∫
∂x 2 

Face 1
∂φ dx 

φ
n
dS

≈
φ
+

(+i )dydz
∫
∂x 2 

Face 2
∂φ
i
dxdydz
adding these gives
∂x

gradφ ≡ Limδτ→0

1
φndS
∫
δτ ΔS

φ = φ(x,y,z)

k
P

dz

i

j
Face 2

Proceeding in the same way for y and z
we get j

∂φ
∂φ
dxdydz, so
dxdydz and k
∂z
∂y

Face 1

dx

1
dy
φndS
∫
δτ ΔS
 ∂φ
1  ∂φ
∂φ
∂φ
∂φ
∂φ

= Limδτ→0  i

dxdydz + j dxdydz + k
dxdydz  = i
+ j +k
δτ  ∂x
∂y
∂z
∂x
∂y
∂z


gradφ ≡ Limδτ→0


Differential Forms of the Gradient
gradΦ = ∇Φ
Cartesian
 ∂Φ  ∂Φ  ∂Φ
 ∂  ∂  ∂ 
i
+j
+k
=  i
+ j + k Φ
∂x
∂y
∂z
∂y
∂z 
 ∂x



Cylindrical  ∂Φ
∂
Φ
∂
Φ
∂


 ∂
e
e
θ
θ ∂
+
+ ez
= er +
+ e z Φ
er
∂r
r ∂θ
∂z
∂z 
 ∂r r ∂θ
Spherical





  ∂ eθ ∂
e φ ∂Φ
eφ ∂ 
 ∂Φ eθ ∂Φ
+
+
=  e r +
+
Φ
er
∂r
r ∂θ r sin θ ∂φ
 ∂r r ∂θ r sin θ ∂φ 

These differential forms define the vector operator ∇


Gradient of a vector , V: V = ui + vj + wk
V ( r + dr ) − V ( r ) ≡ dV = Grad V gdr where dr = dxi + dyj + dzk
Here Grad V maps a vector, dr, into another vector, dV, and plays the role of a derivative for the vectorfield
 ∂u
∂u
∂u   ∂v
∂v
∂v   ∂w
∂w
∂w 
dV = dui + dvj + dwk =  dx + dy + dz ÷i +  dx + dy + dz ÷j + 
dx +
dy +

dz ÷k
∂
x
∂
y
∂
y
∂
x
∂
y
∂
z
∂
x
∂
y
∂
z

 
 

 ∂u

∂x
du
  
   ∂v
dV =  dv  = 

 dw  ∂x
   ∂w
 ∂x


∂u
∂y
∂v
∂y
∂w
∂y

 ∂u
∂u 
÷

∂z ÷
 ∂x
dx
 
 ∂v
∂v ÷ 
÷dy  → Grad V ≡ 
∂z ÷ 
 ∂x
dz 

 ∂w
∂w ÷
÷

 ∂x
∂z 


∂u
∂y
∂v
∂y
∂w
∂y

∂u 
÷
∂z ÷
∂v ÷
÷
∂z ÷
∂w ÷
∂z ÷


Some people prefer to use "dyadic" notation, to call Grad V a dyad, and to write it as follows:
Grad V ≡

∂u
∂u
∂u
∂v
∂v
∂v

∂w
∂w
∂w
i ⊗ i + i ⊗ j+ i ⊗k + j⊗ i + j⊗ j+ j⊗k +
k ⊗i +
k ⊗ j+
k ⊗k
∂x
∂y
∂z
∂x
∂y
∂z
∂x
∂y
∂z

where ⊗ denotes the so-called dyadic product, which does not have a geometric interpretation. Often ⊗ is omitted. The
first base vector indicates the component of V being differentiated and the second indicates the direction of the derivative;
hence, Grad V has nine components: three derivatives of each of its three components.
continued


Gradient of a vector , V: V = ui + vj + wk ;

V ( r + dr ) − V ( r ) ≡ dV = Grad V gdr where dr = dxi + dyj + dzk

definition:
Grad V gdr ≡

∂u
∂u
∂u
∂v
∂v
∂v
∂w
∂w
∂w
i (i gdr ) + i ( jgdr ) + i (k gdr ) + j(i gdr ) + j( jgdr ) + j(k gdr ) +
k (i gdr ) +
k ( jgdr ) +
k (k gdr )
∂x
∂y
∂z
∂x
∂y
∂z
∂x
∂y
∂z

 ∂u
∂u
∂u   ∂v
∂v
∂v   ∂w
∂w
∂w 

Grad V gdr ≡  dx + dy + dz ÷i +  dx + dy + dz ÷j + 
dx +
dy +
dz ÷k ≡ dV
∂y
∂z   ∂x
∂y
∂z   ∂x
∂y
∂z 
 ∂x
similarly
dr gGrad V ≡

∂u
∂u
∂u
∂v
∂v
∂v
∂w
∂w
∂w
(dr gi )i + (dr gi ) j + (dr gi )k + (dr gj)i + (dr gj) j + (dr gj)k +
( dr gk )i +
(dr gk ) j +
(dr gk )k
∂x
∂y
∂z

∂x
∂y
∂z
∂x
∂y
∂z

 ∂u
 ∂v
 ∂w ∂w
∂u
∂u 
∂v
∂v 
∂w 
dr gGrad V ≡  i + dxj + k ÷dx +  i + j + k ÷dy +  i +
j+
k ÷dz ≠ Grad V gdr
∂
x
∂
y
∂
z
∂
x
∂
y
∂
z

∂
x
∂
y
∂
z






∂v
∂w   ∂u
∂v
∂w   ∂u
∂v
∂w 
 ∂u
≡  dx + dy +
dz ÷i +  dx + dy +
dz ÷j +  dx + dy +
dz ÷k ≠ dV
∂x
∂x   ∂y
∂y
∂y   ∂z
∂z
∂z 
 ∂x

Note: in general

 ∂u

 ∂x
 ∂v

 ∂x
 ∂w

 ∂x

∂u
∂y
∂v
∂y
∂w
∂y

 ∂u
∂u 
÷

∂z ÷
 ∂x
 dx 
 ∂v
∂v ÷ 
dy

≠
dx
dy
dz
{
}
÷ 

∂z ÷ 
 ∂x
dz 

 ∂w
∂w ÷
÷

∂z 
 ∂x

∂u
∂y
∂v
∂y
∂w
∂y

∂u 
÷
∂z ÷
∂v ÷

÷
∂z ÷
∂w ÷
÷
∂z 

continued


Addition of dyads is defined so that it is consistent with the parallelogram law of addition (PLA):
 Txx

A = TgC =  Tyx
T
 zx

Txy
Tyy
Tzy

 S xx
Txz  Cx 
÷ 

Tyz ÷C y  and B = SgC =  S yx
 
S
Tzz ÷
 C z 
 zx

S xy
S yy
S zy

S xz  C x 
÷ 
S yz ÷C y 
 
S zz ÷
 Cz 

R = A + B → according to the PLA Rx = Ax + Bx , etc.
→ Rx = ( TxxC x + Txy C y + Txz C z ) + ( S xxC x + S xy C y + S xz Cz )
→ Rx = ( Txx + S xx ) Cx + ( Txy + S xy ) C y + ( Txz + S xz ) Cz
→ R = A + B = TgC + S=
gC = ( T + S) gC
 Txx + S xx

where T + S =  Tyx + S yx
T + S
zx
 zx

Txy + S xy
Tyy + S yy
Tzy + S zy

Txz + S xz 
÷

Tyz + S yz ÷
Tzz + S zz ÷


To preserve the parallelogram law of addition, we need to add the corresponding
components of the dyads.

continued


Decomposition of GradV where V denotes velocity
 ∂u

 ∂x
 ∂v
GradV = 
 ∂x
 ∂w

 ∂x

∂u
∂u  

÷
∂x
∂z ÷ 
∂v ÷  1  ∂u ∂v 
+ ÷
÷=  

∂z ÷  2  ∂y ∂x 
∂w ÷  1  ∂u ∂w 
÷
+
∂z   2  ∂z ∂x ÷


∂u
∂y
∂v
∂y
∂w
∂y

1  ∂u ∂v  1  ∂u ∂w   
+ ÷
+
0


÷÷ 
2  ∂y ∂x  2  ∂z ∂x  ÷ 
∂v
1  ∂v ∂w  ÷  1  ∂v ∂u 
 +
÷÷ +   − ÷
∂y
2  ∂z ∂y  ÷  2  ∂x ∂y 
÷ 
1  ∂v ∂w 

∂w
÷  1  ∂w − ∂u 
 +
÷
÷  2  ∂x ∂z ÷
2  ∂z ∂y 
∂z

 

1  ∂u ∂v  1  ∂u ∂w  
− ÷

 −
÷÷
2  ∂y ∂x  2  ∂z ∂x  ÷
1  ∂v ∂w  ÷
0
 −
÷÷
2  ∂z ∂y  ÷
÷
1  ∂w ∂v 
÷
− ÷
0

÷
2  ∂y ∂z 


GradV = SttrainV + RotV
read strain of V and rotation of V
SttrainV is symmetric, Sij = S ji .
RotV is antisymmetric, Rij = R ji .
SttrainV has six distinct components.

RotV has only three distinct components.

Note:
i
∂
Curl V =
∂x
u

j
∂
∂y
v

k
 ∂w ∂v   ∂u ∂w   ∂v ∂u 
∂
=
− ÷i + 
−
÷j +  − ÷k ≡ Ω x i + Ω y j + Ω z k
∂z  ∂y ∂z   ∂z ∂x   ∂x ∂y 
w

i

Let A = Ax i + Ay j + Az k represent any vector; then Curl V × A = Ω x
Ax
 0
1
RotV gA =  Ω z
2
 −Ω y

−Ω z
0
Ωx

Ω y   Ax 
÷ ÷
−Ω x ÷ Ay ÷ =
 ÷
0 ÷
  Az 

j
Ωy
Ay

 0
1
and RotV =  Ω z
2
 −Ω y

−Ω z
0
Ωx

Ωy 
÷
−Ω x ÷
0 ÷


k

Ω z = ( Az Ω y − Ay Ω z ) i + ( Ax Ω z − Az Ω x ) j + ( Ay Ω x − Ax Ω y ) k
Az

 Az Ω y − Ay Ω z 
1
÷ 1
Ax Ω z − Az Ω x ÷ = Curl V × A

2
÷ 2
 Ay Ω x − Ax Ω y 
continued


vB
B

.

A and B are two points in the same fluid separated
by the infinitesimal position vector dr
v B = v A + dv = v A + Grad v gdr

dr

.

A

vA

= v A + Rot v gdr + Strain v gdr
1
= v A + Curl v × dr + Strain v gdr
2

we can identify the three components of the motion of a continuum:
1) translation at the velocity v A
1
2) rigid-body rotation at the angular velocity ω = Curl v
2
note: the vector Curl v × dr is simultaneously ⊥ to both Curl v and dr;
the fact that Curl v × dr is ⊥ to dr means that this term does not cause
dr to change length
3) deformation given by Strain v gdr
v B = vω
v r gd

A + r× d + Strain


to here
during Δt

fluid particle
moves from
here
B

v B ∆t

.

d

dr

.

A

time t

v A ∆t

.B’
dr′

. A’
time t + Δt

using Taylor series and the PLA, we can write
d = v A∆t + dr′ = dr + v B ∆t
v B = v A + dv =



Divergence
Fluid particle, coincident
with δτ at time t, after time
δt has elapsed.

n
dS

Elemental volume δτ
with surface ∆S

 1

div V ≡ lim  Ñ
V gndS ÷
∫
δ τ →0 δ τ
 ΔS

= proportionate rate of change of volume of a fluid particle

Review

Gradient

Divergence

 1

gradφ ≡ lim  Ñ
φ ndS ÷
δ τ →0 δ τ ∫
 ΔS


 1

div V ≡ lim  Ñ
V.ndS ÷
δ τ →0 δ τ ∫
 ΔS


Magnitude and direction
of the slope in the scalar
field at a point

For velocity: proportionate
rate of change of volume
of a fluid particle

Differential Forms of the
Divergence
r
divA
Cartesian
Cylindrical
Spherical

∂ Ax ∂ A y ∂ Az
+
+
∂x
∂y
∂z
1 ∂ rAr 1 ∂ Aθ ∂ A z
+
+
r ∂r
r ∂θ
∂z

1 ∂ r 2 Ar
1 ∂ Aθ sin θ
1 ∂ Aφ
+
+
2
r sin θ

∂θ
r sin θ ∂φ
r ∂r

=

r
∇gA

r ∂ r ∂ r ∂  r
= i
+ j + k ÷.A
∂y
∂z 
 ∂x
r
 r ∂ eθ ∂ r ∂  r
= er +
+ e z ÷.A
∂z 
 ∂r r ∂θ
r
 r ∂ erθ ∂
eφ ∂  r
= er +
+
÷.A
 ∂r r ∂θ r sin θ ∂φ 

Differential Forms of the Curl
r
 1 r r 
curl A ≡ − lim  Ñ
A × ndS ÷
δ τ →0 δ τ ∫
 ΔS

r
i
r
r
∂
curlA = ∇ × A =
∂x

r
j
∂
∂y

r
k
∂
1
=
∂z r

Ax

Ay

Az

Cartesian

r
r
e r reθ
∂
∂
∂r ∂θ
Ar rAθ

r
ez
∂
1
= 2
∂z r sin θ
Az

Cylindrical

Curl of the velocity vector ∇ × V =
twice the circumferentially averaged angular
velocity of
-the flow around a point, or
-a fluid particle
=Vorticity Ω

Pure rotation

r
er
∂
∂r

r erθ
∂
∂θ

Ar

rAθ

r sin θ erφ
∂
∂φ
r sin θ Aφ

Spherical

No rotation

Rotation


 1


curl V ≡ − lim  Ñ
V
×
n
dS
÷
δ τ →0 δ τ ∫
 ΔS

1
e.curl V = − lim
∫ΔS e.V × ndS
δ τ →0 δ τ Ñ

Curl
e
Perimeter Ce
Area δσ

 1

e.curl V = lim 
V.e × ndS ÷
∫
δ τ → 0 δσδ h Ñ
ΔS


 1


e.curl V = lim 
V.e × ndsδ h ÷
∫
δ τ → 0 δσδ h Ñ
ΔS


 1
e.curl V = lim 
δ τ → 0  δσ


= twice the avg. angular velocity
about e

n

dS

ds

δh
n


 Γ Ce 
V
.d
s
=

lim
÷
Ñ
∫ ÷ δ τ →0  δσ ÷
Ce


 1

v 
e.curl V = lim  2 vθ 2π a ÷ = 2lim  θ ÷
a →0 π a
a →0


a

Elemental volume δτ
with surface ∆S

dS=dsδh
radius a
vθ avg. tangential
velocity


Review

Gradient

Divergence

 1

gradφ ≡ lim  Ñ
φ ndS ÷
δτ → 0 δ τ ∫
 ΔS


 1

div V ≡ lim  Ñ
V.ndS ÷
δτ → 0 δ τ ∫
 ΔS


Magnitude and direction
of the slope in the scalar
field at a point

For velocity: proportionate
rate of change of volume
of a fluid particle

Curl
 1

curl V ≡ − lim  Ñ

V × ndS ÷
δτ → 0 δ τ ∫
 ΔS

For velocity: twice the
circumferentially averaged
angular velocity of a fluid
particle = Vorticity Ω


Oliver Heaviside
1850-1925

Writes Electromagnetic induction and its propagation over the
course of two years, re-expressing Maxwell's results in 3
(complex) vector form, giving it much of its modern form and
collecting together the basic set of equations from which
electromagnetic theory may be derived (often called "Maxwell's
equations"). In the process, He invents the modern vector
calculus notation, including the gradient, divergence and curl of a
vector.