Paul, could explain to me a non-mathematician why you say "Calculus
is Dead"? Feel free to over simplify. I quoted you and was
challenged to back up the statement. I said I thought it might have
something to do with "computer finite analysis" but that I did not
know the reasoning behind your comment. Be forewarned, your
comments may find their way to my friends old calculus professor at
Indiana University. :)
I explain further: while I was slapping shoes on a mean little
arab. we were talking about how the computer empowers non-
mathematicians/non engineers to attempt complex designs using CAD.
Another example is how (in my opinion) CAD has eliminated the need
for Trigonometry in the machine shop. Jerry
I am no mathematician either but as I understand it integral calculus
is all about computing the area or volume under certain mathematical
curves. It does this by slicing the area or volume into infinitely
thin slices,
calculating the area or volume of those slices and adding them up
to determine the total area or volume under the curve(s).
A computer can do this real easy and the curve(s) does/do not
have to have a mathematical definition such as y = x^2
to use a simple example. The curve(s) can have ANY shape.
All the computer needs is a set of coordinates for points
on the curve(s).
Way back before personal computers and electronic calculators
I was charged with calculating the speed attained as a function
of distance traveled for a land speed record car. Not being sharp
at all in calculus I opted to do it the long way.
I calculated the increase in speed and acceleration every .1
second knowing the driving wheel torque as a function of gear
ratio, engine RPM, car weight, aerodynamic down force, aerodynamic
drag, tire rolling resistance, coefficient of friction, the phase
of the moon and the distance the car traveled in that .1 second
and added them all up.
It took me a week to get to 400 MPH. I used a Friden mechanical
calculator that had a hundred buttons or more. See the attached
picture of the electro mechanical version. The one I used had a big
hand crank on the side of it. Second picture.
"Inventor: Frank Stephen Baldwin 1839-1925."
"Baldwin patented a pinwheel device that enabled carries in a compact
device. In 1900 he invented a method that allowed digits to be
entered by a single stroke."
"Each complete forward turn of the large crank on the right will add
the value set into the 8 x 10 keys into the bottom register of the
carriage. The main register on the carriage shows the result. The
smaller register above it counts the turns. Subtraction is achieved
by turning the crank in reverse. To multiply by more than one digit
the Repeat button is pressed and the crank turned as often as needed
for the low-order digit. Then the carriage is moved to the right
with the handle in front, so the next digit of the factor can be
cranked in."
http://www-db.stanford.edu/pub/voy/museum/pictures/display/1-
Calculators.htm
A computer can do that in about two or three ms after the
program is written. A few years later, after I had a computer
and the program, I used to astound my car magazine journalist
friends by predicting what their test car would do in the 1/4 mile
to the nearest .01 seconds. No need to actually test the car as long
as the manufacturer was not lying about the engine HP.
Paul Lamar ...No rotor no motor.
The main problem with integral calculus is that there are many
situations (curves, functions, etc.) that have no known solution. And
there is no "one size fits all" general solution. It used to be that
an engineer or physicist or mathematician had on his desk one or more
volumes of functions and transformations. Because no one I ever met
could remember them all. (I always thought that you had to have a
warped mind to be really proficient at it. I wasn't. :) )
The main alternative to calculus is Newton's method.
Newton's method of finding the area under a curve is to evaluate the
mathematical function that generates the curve in very small
increments and add them all up. "Small increments" has to be
experimentally arrived at: try one, then try a smaller one and if they
produce the same result within a negligibly small difference, you
probably have an acceptable answer. Paul's "long way" is an
application of Newton's method. Solutions by hand (i.e., no
electronics) using Newton's method were seldom employed because of the
labor involved, and a distinct probability of a non-trivial error
creeping in.
The electronic digital calculator, and to an even larger extent the
electronic digital computer, can be programmed to use Newton's method
to produce very accurate results in very short times for virtually any
situation.
Finite Element Analysis (FEA) is a process of taking a computer model
of something and dividing it into very small pieces (the finite
element), usually in 3 dimensions. The very small pieces are then
evaluated using Newton's method. The results of evaluating all of the
very small pieces are summed to produce the final result. FEA can be
used in a number of disciplines: mechanical design, heat transfer,
etc. Modern FEA software packages provide a great deal of help in
dividing whatever is to be analyzed into very small pieces. The
evaluation of each very small piece and the summation is essentially
automatic.
FEA and other processes that make use of Newton's method require no
calculus; no extensive knowledge of functions and transformations. And
it works for all situations that can be modeled in a computer.
The market for persons trained in calculus for calculus' sake is
vanishingly small, at least in the engineering community. Calculus is
still taught because it is useful in explaining scientific and
engineering concepts and principles.
As a day-to-day engineering tool, it is almost extinct. Which leads to
the statement it is obsolete. At least for the engineering disciplines
which is what Paul was referring to.
The above is a great simplification. Caveat emptor!
Doc Custer
Retired software engineer and sometime Audiologist
Building RV9-A. Fuselage
Thanks to Paul, Doc and Dennis. This is a most interesting topic
about which I now know a little. I wish I could get a second go
around as an engineer. Jerry
The Rotary Engine NewsLetter. Powered by Linux.
ACRE NL web site.
http://www.rotaryeng.net
Copyright 1998-2006 All world wide rights reserved.