class 3 - vector calculus

7/27/2019 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


2/37

Unnumbered slides contain comments that I inserted and

are not part of Professors Devenports original

presentation.


3/37

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


5/37

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


6/37

7

J. KURIMA, N. KASAGI and M. HIRATA (1983)

Turbulence and Heat Transfer Laboratory, University of Tokyo

LOW REYNOLDS NUMBER AXISYMMETRIC JET


7/37

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


8/37

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


9/37

3. Vector Calculus

Fluid particle: Differentially Small Piece

of the Fluid Material


10/37


11/37

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


12/37

13

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


15/37

16

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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17

High Speed Flow Past an Axisymmetric Object

Shadowgraph picture is from An Album of Fluid Motionby Van Dyke


17/37

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


18/37

19

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


19/37

20Picture is from An Album of Fluid Motionby Van Dyke

Mach approximately 2.0

For closed loops, e.g. Circulation sV d.


20/37

21

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


21/37

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


22/37

23

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


23/37

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


24/37


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


25/37


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


26/37

27

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


27/37

28

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


28/37

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.


29/37

30

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


30/37

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


31/37

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 ) ( )


32/37

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


33/37

34

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


34/37

35

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


35/37

36

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


36/37

37

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


37/37

38

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

class 3 - vector calculus

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