1. 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
5:30pm (with Nathan Alexander) in Randolph 221
5pm (with Chris Rock) in Whitemore 349

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

3. Last Class Changes in Unit Vectors Calculus w.r.t. timeez e
Changes in Unit Vectors
Calculus w.r.t. time
Integral calculus w.r.t. space
Today: differential calculus in 3D P' z P er r  d

4. Oliver Heaviside
Oliver Heaviside's parents were Rachel Elizabeth West and Thomas Heaviside. Thomas was a wood engraver and water colour artist. Oliver, the youngest of his parents four sons, was born at 55 King Street in Camden Town. He caught scarlet fever when he was a young child and this affected his hearing. This was to have a major effect on his life, making his childhood unhappy with relations between himself and other children difficult. However his school results were rather good and in 1865 he was placed fifth from 500 pupils.
Academic subjects seemed to hold little attraction for Heaviside however and at age 16 he left school. Perhaps he was more disillusioned with school than with learning since he continued to study after leaving school, in particular he learnt Morse code, studied electricity and studied further languages in particular Danish and German. He was aiming at a career as a telegrapher and in this he was advised and helped by his uncle Charles Wheatstone (the piece of electrical apparatus known as the Wheatstone bridge is named after him).
In 1868 Heaviside went to Denmark and became a telegrapher. He progressed quickly in his profession and returned to England in 1871 to take up a post in Newcastle upon Tyne in the office of Great Northern Telegraph Company which dealt with overseas traffic.
Heaviside became increasingly deaf but he worked on his own researches into electricity. While still working as chief operator in Newcastle he began to publish papers on electricity, the first in 1872 and then the second in 1873 was of sufficient interest to Maxwell that he mentioned the results in the second edition of his Treatise on Electricity and Magnetism. Maxwell's treatise fascinated Heaviside and he gave up his job as a telegrapher and devoted his time to the study of the work. He later wrote:-
I saw that it was great, greater, and greatest, with prodigious possibilities in its power. I was determined to master the book... It took me several years before I could understand as much as I possible could. Then I set Maxwell aside and followed my own course. And I progressed much more quickly.
Although his interest and understanding of this work was deep, Heaviside was not interested in rigour. His poorest subject at school had been the study of Euclid, a topic in which the emphasis was on rigorous proof, an idea strongly disliked by Heaviside who later wrote:-
It is shocking that young people should be addling their brains over mere logical subtleties, trying to understand the proof of one obvious fact in terms of something equally .. obvious.
Despite this hatred of rigour, Heaviside was able to greatly simplify Maxwell's 20 equations in 20 variables, replacing them by four equations in two variables. Today we call these 'Maxwell's equations' forgetting that they are in fact 'Heaviside's equations'. FitzGerald wrote:-
Maxwell's treatise is cumbered with the debris of his brilliant lines of assault, of his entrenched camps, of his battles. Oliver Heaviside has cleared these away, has opened up a direct route, has made a broad road, and has explored a considerable trace of country.
Heaviside results on electromagnetism, impressive as they were, were overshadowed by the important methods in vector analysis which he developed in his investigations. His operational calculus, developed between 1880 and 1887, caused much controversy however. He introduced his operational calculus to enable him to solve the ordinary differential equations which came out of the theory of electrical circuits. He replaced the differential operator d/dx by a variable p transforming a differential equation into an algebraic equation. The solution of the algebraic equation could be transformed back using conversion tables to give the solution of the original differential equation. Although highly successful in obtaining the answer, the correctness of Heaviside's calculus was not proved until Bromwich's work.
Burnside rejected one of Heaviside's papers on the operational calculus, which he had submitted to the Proceedings of the Royal Society , on the grounds that it:-
... contained errors of substance and had irredeemable inadequacies in proof.
Tait championed quaternions against the vector methods of Heaviside and Gibbs and sent frequent letters to Nature attacking Heaviside's methods.
Heaviside went on to achieved further advances in knowledge, again receiving less than his just deserts. In 1887 Preece, a GPO technical expert, wrote a paper on clear telephone circuits. His paper is in error and Heaviside pointed this out in Electromagnetic induction and its propagation published in the Electrician on 3 June Heaviside, never one to avoid controversy, wrote:-
Sir W Thomson's theory of the submarine cable is a splendid thing. .. Mr Preece is much to be congratulated upon having assisted at the experiments upon which (so he tells us) Sir W Thomson based his theory, he should therefore have an unusually complete knowledge of it. But the theory of the eminent scientist does not resemble that of the eminent scienticulist, save remotely.
In this paper Heaviside gave, for the first time, the conditions necessary to transmit a signal without distortion. His idea for an induction coil to increase induction was never likely to be taken up by the GPO while Preece was in charge of research proposals. Heaviside dropped the idea but it was patented in 1904 in the United States. Michael Pupin of Columbia University and George Campbell of ATT both read Heaviside's papers about using induction coils at intervals along the telephone line. Both Campbell and Pupin applied for a patent which was awarded to Pupin in 1904.
Not all went badly for Heaviside however. Thomson, giving his inaugural address in 1889 as President of the Institute of Electrical Engineers, described Heaviside as an authority. Lodge wrote to Nature describing Heaviside was a man:-
... whose profound researches into electro-magnetic waves have penetrated further than anyone yet understands.
Heaviside was elected a Fellow of the Royal Society in 1891, perhaps the greatest honour he received. Whittaker rated Heaviside's operational calculus as one of the three most important discoveries of the late 19th Century.
In 1902 Heaviside predicted that there was an conducting layer in the atmosphere which allowed radio waves to follow the Earth's curvature. This layer in the atmosphere, the Heaviside layer, is named after him. Its existence was proved in 1923 when radio pulses were transmitted vertically upward and the returning pulses from the reflecting layer were received.
It would be a mistake to think that the honours that Heaviside received gave him happiness in the last part of his life. On the contrary he seemed to become more and more bitter as the years went by. In 1909 Heaviside moved to Torquay where he showed increasing evidence of a persecution complex. His neighbours related stories of Heaviside as a strange and embittered hermit who replaced his furniture with:-
... granite blocks which stood about in the bare rooms like the furnishings of some Neolithic giant. Through those fantastic rooms he wandered, growing dirtier and dirtier, and more and more unkempt - with one exception. His nails were always exquisitely manicured, and painted a glistening cherry pink.
Perhaps Heaviside has become more widely known due to the Andrew Lloyd Webber song Journey to the Heaviside Layer in the musical Cats , based on the poems of T S Eliot:-
Up up up past the Russell hotel Up up up to the Heaviside layer
However it is doubtful if many people understand the greatness and significance of the achievements of this sad misunderstood genius. As stated in [3] Heaviside was:-
A mathematical thinker whose work long failed to secure the recognition its brilliance deserved ...
Article by: J J O'Connor and E F Robertson

5. Shock in a CD Nozzle Bourgoing & Benay (2005), ONERA, France
M1=1.81, rho2/rho1=2.38, p2/p1=3.66, M2=0.61, U1~628m/s (if T=300K).
If b.l. 1mm thick, and shear accomodated by simple rotation of the fluid particles then rotation rate = times per second
Bourgoing & Benay (2005), ONERA, France
Schlieren visualization
Sensitive to in-plane index of ref. gradient

6. Differential Calculus w.r.t. Space Definitions of div, grad and curl
In 1-D
Elemental volume 
with surface S n dS
D=D(r), = (r)
In 3-D

7. Gradient  = high ndS (medium) ndS (large) n  = low Resulting ndS
(small)
Elemental volume 
with surface S
ndS
(medium)
= magnitude and direction of the slope in the scalar field at a point

8. Review
Magnitude and direction of the slope in the scalar field at a point
Gradient

9. Gradient
Component of gradient is the partial derivative in the direction of that component
Fourier´s Law of Heat Conduction
 = high
 = low

10. Differential form of the Gradient
Cartesian system
Evaluate integral by expanding the variation in  about a point P at the center of an elemental Cartesian volume. Consider the two x faces:
 = (x,y,z) k dz i P j
adding these gives
Face 2
Proceeding in the same way for y and z
we get
and
, so
Face 1 dx dy

11. Differential Forms of the Gradient
Cartesian
Cylindrical
Spherical
These differential forms define the vector operator 

12. continued

13. continued

14. continued

15. continued

16. . A B

17. . to here fluid particle during Δt moves from here B’ B d A’ A
time t
time t + Δt A A’ B’ B d
fluid particle
moves from
here
to here
during Δt

19. Divergence Fluid particle, coincident with  at time t, after time
t has elapsed. n dS
Elemental volume 
with surface S
= proportionate rate of change of volume of a fluid particle

20. Review Gradient Divergence
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
Gradient
Divergence

21. Differential Forms of the Divergence
Cartesian
Cylindrical
Spherical

22. Differential Forms of the Curl
Cartesian
Cylindrical
Spherical
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
No rotation
Rotation

23. Curl Elemental volume  with surface S e n dS Perimeter Ce Area 
dS=dsh
radius a
v avg. tangential velocity
= twice the avg. angular velocity
about e

24. Review Gradient Divergence Curl
For velocity: twice the circumferentially averaged angular velocity of a fluid particle = Vorticity Ω
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
Gradient
Divergence
Curl

25. Oliver Heaviside
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.

26. Integral Theorems and Second Order Operators

27. 1st Order Integral Theorems
Volume R
with Surface S
Gradient theorem
Divergence theorem
Curl theorem
Stokes’ theorem
ndS d
Open Surface S
with Perimeter C
ndS

28. The Gradient Theorem Finite Volume R Surface S
Begin with the definition of grad:
Sum over all the d in R: d
We note that contributions to the RHS from internal surfaces between elements cancel, and so:
nidS
di+1
Recognizing that the summations are actually infinite:
ni+1dS
di

29. Assumptions in Gradient Theorem
A pure math result, applies to all flows
However, S must be chosen so that  is defined throughout R S
Submarine
surface 

30. Flow over a finite wing S1 S1 S2
S = S1 + S2
R is the volume of fluid enclosed between S1 and S2
p is not defined inside the wing so the wing itself must be excluded from the integral

31. 1st Order Integral Theorems
Volume R
with Surface S
Gradient theorem
Divergence theorem
Curl theorem
Stokes’ theorem
ndS d
Open Surface S
with Perimeter C
ndS

32. Alternative Definition of the Curl
Perimeter Ce
Area  ds

33. Stokes’ Theorem Finite Surface S With Perimeter C
Begin with the alternative definition of curl, choosing the direction e to be the outward normal to the surface n: n
Sum over all the d in S: d
Note that contributions to the RHS from internal boundaries between elements cancel, and so:
dsi+1
di+1
Since the summations are actually infinite, and replacing  with the more normal area symbol S:
dsi
di

34. Stokes´ Theorem and Velocity
Apply Stokes´ Theorem to a velocity field
Or, in terms of vorticity and circulation
What about a closed surface?

35. Assumptions of Stokes´ Theorem
A pure math result, applies to all flows
However, C must be chosen so that A is defined over all S C
The vorticity doesn’t imply anything about the circulation around C
2D flow over airfoil with =0

36. Flow over a finite wing C S
Wing with circulation must trail vorticity. Always.

37. Vector Operators of Vector Products

38. Convective Operator
= change in density in direction of V, multiplied by magnitude of V

39. Second Order Operators
The Laplacian, may also be applied to a vector field.
So, any vector differential equation of the form B=0 can be solved identically by writing B=.
We say B is irrotational.
We refer to  as the scalar potential.
So, any vector differential equation of the form .B=0 can be solved identically by writing B=A.
We say B is solenoidal or incompressible.
We refer to A as the vector potential.

40. Class Exercise
Make up the most complex irrotational 3D velocity field you can. ?
We can generate an irrotational field by taking the gradient of any scalar field, since
I got this one by randomly choosing
And computing

41. 2nd Order Integral Theorems
Green’s theorem (1st form)
Green’s theorem (2nd form)
Volume R
with Surface S
ndS d
These are both re-expressions of the divergence theorem.