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What is a good way to understand tensors?

From Quora:

When pipes freeze, the water in them expands and sometimes breaks the pipe.  But an interesting thing happens if you look at 100 broken pipes.  They almost all have the break running lengthwise along the pipe, never around its circumference.


The material of the pipe is about the same in both directions, so why does the pipe break in only one?

When ice starts to press against the inside of the pipe, the material of the pipe gets put under tension.  This means that if you were to cut a small slit in the pipe, you’d have to pull hard on both sides of the slit to keep it closed.  Further, the amount you pull depends on the length of the slit.  You’ll have to pull twice as hard to keep a 2cm slit closed compared to a 1 cm slit.  When the tension becomes so high that the metal’s own cohesive strength is too weak to hold it together, the pipe bursts.

The reason it bursts in one direction, though, is that the tension is higher in that direction.  The force needed to hold  a slit closed depends not only on how long the slit is, but also on the direction of the slit.  Ori Barbut showed an explicit calculation of this in his answer to Why is it more difficult to blow up a long balloon than a round one?  It turns out that the tension is exactly twice as high in the lengthwise direction as in the circumferential direction.

Once we know the tension in these two directions, we can figure out the force needed to hold any slit closed.  If the slit is 1cm at a 45-degree angle, wrapping diagonally like a barber pole, for example, the force needed is half way between that for a 1cm slit in each of the two original directions.

All of this holds only so long as the pipe is homogeneous.  In general, the tension could change from place to place, and if we have a long, crooked slit it would be more complicated to find the force to hold it closed.  As long as our slits are small, though, what we’ve just said holds.

This two-dimensional tension is an example of a tensor.  Specifically, it is a rank-two tensor.  You can think of it as a function.  The input to the function is a vector - in this case a vector that describes a slit in the pipe (the slit has magnitude and direction, making it a vector).  Its output is another vector - the force needed to hold the slit closed.  So a rank-two tensor, in general, is a linear function from vectors to vectors.  (The “linear” part is what we assumed by saying a slit twice as long requires twice the force, and that a slit at an angle can be calculated if you know what happens for lengthwise and circumference slits.)  The tensor is not the slit itself or the force itself.  It is a relationship between the possible slits and the forces that would be needed  to hold them closed.

Pressure is another tensor very similar to tension.  In fact, pressure is basically just negative tension.  Usually, we think of pressure as being just a single number.  But really, it is a linear function that maps flat surfaces onto forces (any flat surface has a force on it due to pressure).  When the pressure is the same in all directions, we can think of it as just a number, but when it pushes unevenly, like it might inside a crystal whose structure picks out special directions, we need the tensor description.

Many other familiar physical quantities can be represented by tensors, such as the mechanical strain of a solid, the moment of inertia of a mass distribution, the polarizability of a dielectric, the quadrupole moment of a charge distribution, and, in higher mathematics, even the Pythagorean theorem!  (The Pythagorean theorem as a tensor is called “the metric”.)

We can easily generalize the idea to  tensors of different rank.  A rank-three tensor would have two vectors as input and one vector as output.  A rank-four would involve four vectors.  A rank one tensor exists, too.  It takes in one vector and outputs “zero vectors”, meaning just a single number.  But a vector can do that, too.  Take a vector A, and for any input vector B, output the dot product of A and B.  Now the vector A becomes a rank-one tensor!  There are even rank-zero tensors - they are just numbers, although we usually refer to them as “scalars” instead.

When describing the curved geometry of general relativity, a rank-four tensor called the Riemann curvature tensor is used.  It has three inputs - a vector pointing in some direction you’re interested in, and two more vectors that create a parallelogram when put together.  You take the first vector and walk it around the edge of the parallelogram.  When it gets back to where it started, it’ll be slightly rotated if the spacetime is curved (this is the meaning of “curved spacetime”).  The change in the first vector due to this rotation is the output of the tensor.

Relativity is notorious for its heavy use of tensors.  Some other important ones there are the metric tensor, mentioned earlier, and the stress-energy tensor, which is the four-dimensional spacetime version of the pressure tensor, as well as various quantities that are derived doing operations on these tensors. 

You will sometimes see the basic definition of a tensor described differently.  You can also think of a rank two tensor as taking two vectors as input and having one number as output.  This is the same idea as before, but the input would be the slit you cut, and a unit vector in some direction.  The number would be the component of the force in that direction.  Which way of thinking about it is better depends on the precise situation.

If you’ve studied linear algebra, a rank-two tensor is a linear operator.  (N.B. I am glossing over the important difference between a vector space and its dual.)  If you pick a basis, it becomes no different from a matrix.  They are all basically the same idea.  The difference is that a matrix is defined as a bunch of numbers in an array with certain rules for how to do algebra on it.  A tensor is the actual relationship between physical quantities, and the matrix can represent that tensor.

Usually we don’t harp on this very much.  Physicists prefer a different way of keeping track of their tensors, called index notation.  It takes some getting used to, but it lets us manipulate tensors of rank three, four, or more, which would be impossible with matrices.  (A rank-three tensor would require a “matrix” that looks sort of like a cube.  Don’t confuse the rank of the tensor, which is how many vectors it takes as input, with the dimension you’re working in.  The ordinary pressure tensor is rank two, meaning two vectors are involved, but dimension three, meaning the vectors are in three dimensions.  The Riemann curvature tensor in GR is rank four and dimension four.)

One reason physicists like tensors is that the tensor does not depend on the coordinates you use.  For our pipe, we could define the x-direction to go down the length of the pipe and the y-direction to go around the pipe.  Then, if we wanted a formula for the force needed to hold closed a slit going in some arbitrary direction, it would look pretty sloppy.  It would have to tell us to take the x-part of the slit vector and multiply that by T1 times some unit vector, then take the y-part and multiply that by T2 times some unit vector, then add those.  Further, we could have taken the x-direction to be wrapping up diagonally around the pole clockwise, and the y-direction to be wrapping up and around the pole counterclockwise.  Then we’d have an even more complicated formula with different numbers in it.  With a tensor, though, we can simply write F = TS, meaning force is tension acting on the slit, or in index notation

F^a = T^a_bS^b

and this formula looks the same no matter what coordinate system we use.  (However, as we change coordinate systems, the numerical values of things like T^a_b will change.)

 After all, the coordinates are just some silly choice we made, some imaginary grid we pasted down on the surface.  The physics shouldn’t depend on that, and tensor notation is a way of making this concept explicit.  It’s so important, in fact, that some physics literature actually defines a tensor by what happens to it when you change coordinates.

So a tensor is a way of representing a linear relationship between vector quantities.  It is especially important in continuum mechanics, relativity, and some areas of higher math, and we like them because they show the coordinate-independence of physical laws.

Filed under physics math tensor

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Why do humans need goggles to see underwater?

From Quora:

The salient difference is that the speed of light in water is slower than the speed of light in air. 

When light crosses a boundary between two media, it bends, an effect called refraction.  The extent of refraction depends on the difference in the speed of light in the two media.  In fact, the mathematical law describing refraction, Snell’s Law, can be derived by assuming that light takes the fastest possible route between two points.  To minimize total travel time, light rays will bend so that less distance is traveled through the slow medium and more distance through the fast medium.

(For a geometric derivation of Snell’s Law following an argument I read in the Feynman Lectures, check out my blog post here: http://arcsecond.wordpress.com/2… )

If the two media have very different speeds of light, there’s a big effect.  If the speed of light is equal in the two media, the fastest route is simply a straight line, so no refraction occurs at all.

Now we need to look at a human eye.  Here’s the Wikipedia diagram:

from http://upload.wikimedia.org/wiki…
Your eye bends light with both your cornea and your lens.  However, according to Wikipedia (http://en.wikipedia.org/wiki/Eye), two-thirds of the refractive power of your eye (in air) is in your cornea. 

The cornea’s refractive index is about 1.376 (same article).  This number is just the speed of light in vacuum divided by the speed of light in the cornea, so light goes 1.376 times faster in vacuum than in the cornea.  

The refractive index of water (according to Google) is 1.33 (http://www.google.com/search?sou… ) .  Refraction depends on the fractional change in the speed of light, so in water your cornea bends light as much as a lens in air whose refractive index is 

\frac{1.376}{1.33} = 1.034

That means you’re losing about 90% of your cornea’s refractive power, or 60% of your total refractive power, when you enter the water. 

The question becomes whether your lens can compensate for that.

I didn’t find a direct quote on how much you can change the focal distance of your lens, but we can estimate that in water your cornea is doing essentially nothing, and ask whether your lens ought to be able to do all the focusing itself.

For a spherical lens with index of refraction n sitting in a medium with index of refraction n_0, the effective focal length is

f = \frac{nD}{4(n-n_0)}

This is the lensmaker’s equation - I simply looked it up. (http://www.edmundoptics.com/tech… )

The refractive index of your vitreous humor is about 1.33 (like water), and the refractive index of your lens, (Wikipedia again), varies between 1.386 and 1.406.  Let’s take 1.40 as an average.  Then, plugging in the numbers, the effective focal distance of a spherical eye lens would be five times its diameter.  

The Wikipedia picture of a human eye makes this look reasonable - a spherical lens might be able to do all the focusing a human eye needs, even without the cornea.

The problem is that your eye’s lens isn’t spherical.  From the same Wikipedia article

In many aquatic vertebrates, the lens is considerably thicker, almost spherical, to increase the refraction of light. This difference compensates for the smaller angle of refraction between the eye’s cornea and the watery medium, as they have similar refractive indices. Even among terrestrial animals, however, the lens of primates such as humans is unusually flat.

So, the reason you can’t see well underwater is that your eye’s lens is too flat.

If you wear goggles, the light is refracted much more as it enters the cornea - the same amount as normal. 

Googling for “underwater contact lens”, I found an article about contact lenses made with a layer of air ( http://sportsillustrated.cnn.com… ), allowing divers to see sharply underwater.  However, even with corrective lenses, things will not be totally normal.  For example, things underwater will appear closer to you than they actually are because your depth perception depends on detecting the angles of incoming light rays, and those angles change when going from water to air.

Technical note:
In physics “the speed of light” frequently refers to the constant c from relativity theory.  This constant doesn’t change as you go between media - c is built into the universe.  What changes is the effective propagation velocity of light on a macroscopic scale.  On a microscopic scale, photons still always travel at c.

 Reference:
adapted from an answer to a similar question at
http://physics.stackexchange.com…

Filed under physics math interesting quora

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En iyi fizikçi kim? (Cevap: John Bardeen)

Quora’daki bu soruya, Joshua Yoon’un verdiği cevap ufkumu açtı. Öyle bir insandan bahsetti ki tam aradığım kişi… Yani Transistör konusunu asla adamakıllı anlatamadılar… Hep merak ederdim. Kaynağına eriştim şimdi.

İşin ilginci, Bardeen çok enteresan bir adammış, yani bir dahiye göre normal birisi.

Daha da ilginci, üç çocuğundan ikisi de fizikçi; bir tanesinin danışmanı Richard Feynman.

Stand on the shoulders of giants.

Always learn from the masters.

İşte o cevap:

There are many awesome physicists out there, but I don’t think there’s enough love in this post for John Bardeen.

John Bardeen, if people look him up in the internet, was one of the guys that invented the transistor. The Information Age as we know it today would have not been possible without his contributions;  virtually anything that runs on electronics these days utilizes the power of transistors.

Because of his efforts, he was awarded the Nobel Prize in Physics at Stockholm. But when he had arrived at the ceremony, he had only brought one of his children since at the time his other children were studying at Harvard. King Gustav was pissed, even scolding Bardeen for doing such a thing. In response, Bardeen says that he’ll bring his children along….next time.

And what do you know, Bardeen and two others come up with the first microscopic theory on superconductivity and then goes back to Stockholm….againwith all his children this time.

Damn.

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Wiki

In October 1945, John Bardeen began work at Bell Labs. Bardeen was a member of a Solid State Physics Group, led by William Shockley and chemist Stanley Morgan.

The assignment of the group was to seek a solid-state alternative to fragile glass vacuum tube amplifiers.
Their first attempts were based on Shockley’s ideas about using an external electrical field on a semiconductor to affect its conductivity. These experiments mysteriously failed every time in all sorts of configurations and materials.
The group was at a standstill until Bardeen suggested a theory that invoked surface states that prevented the field from penetrating the semiconductor. The group changed its focus to study these surface states, and they met almost daily to discuss the work. The rapport of the group was excellent, and ideas were freely exchanged.
[7] 
By the winter of 1946 they had enough results that Bardeen submitted a paper on the surface states to 
Physical Review.
Brattain started experiments to study the surface states through observations made while shining a bright light on the semiconductor’s surface. This led to several more papers (one of them co-authored with Shockley), which estimated the density of the surface states to be more than enough to account for their failed experiments. The pace of the work picked up significantly when they started to surround point contacts between the semiconductor and the conducting wires with 
electrolytes. Moore built a circuit that allowed them to vary the frequency of the input signal easily and suggested that they use glycol borate (gu), a viscous chemical that didn’t evaporate.
Finally they began to get some evidence of power amplification when Pearson, acting on a suggestion by Shockley,
[8] put a voltage on a droplet of gu placed across a P-N junction.

Magazin kısmı:

On December 23, 1947, Bardeen and Brattain—working without Shockley—succeeded in creating a point-contact transistor that achieved amplification. By the next month, Bell Labs' patent attorneys started to work on the patent applications.[9]

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Bell Labs’ attorneys soon discovered that Shockley’s field effect principle had been anticipated and patented in 1930 by Julius Lilienfeld, who filed his MESFET-like patent in Canada on October 22, 1925.[10]

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Shockley took the lion’s share of the credit in public for the invention of transistor, which led to a deterioration of Bardeen’s relationship with Shockley.[11] Bell Labs management, however, consistently presented all three inventors as a team. Shockley eventually infuriated and alienated Bardeen and Brattain, and he essentially blocked the two from working on the junction transistor.

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Bardeen began pursuing a theory for superconductivity and left Bell Labs in 1951.
Brattain refused to work with Shockley further and was assigned to another group. Neither Bardeen nor Brattain had much to do with the development of the transistor beyond the first year after its invention.[12]

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The “transistor” (a combination of “transfer” and “resistor”) was 1/50 as large as the vacuum tubes it replaced in televisions andradios and allowed electrical devices to become more compact.

Eğitim Hayatı:

By 1951, Bardeen was looking for a new job. Fred Seitz, a friend of Bardeen, convinced the University of Illinois at Urbana-Champaign to make Bardeen an offer of $10,000 a year. Bardeen accepted the offer and left Bell Labs.[9] He joined the engineering faculty and the physics faculty at the University of Illinois at Urbana-Champaign in 1951. He was Professor of Electrical Engineering and of Physics at Illinois. His first Ph.D. student was Nick Holonyak (1954), the inventor of the first LED in 1962.

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In 1957, John Bardeen, in collaboration with Leon Cooper and his doctoral student John Robert Schrieffer, proposed the standard theory of superconductivity known as the BCS theory (named for their initials

Özel Hayatı:

Bardeen married Jane Maxwell on July 18, 1938. While at Princeton, he met Jane during a visit to his old friends in Pittsburgh.

Bardeen was a man with a very unassuming personality. While he served as a professor for almost 40 years at the University of Illinois, he was best remembered by neighbors for hosting cookouts where he would cook for his friends, many of whom were unaware of his accomplishments at the university. He would always ask his guests if they liked the hamburger bun toasted (since he liked his that way). He enjoyed playing golf and going on picnics with his family.[13]

It has been said that Bardeen proves wrong the stereotype of the “crazy scientist.”[13] Lillian Hoddeson, a University of Illinois historian who wrote a book on Bardeen, said that because he “differed radically from the popular stereotype of genius and was uninterested in appearing other than ordinary, the public and the media often overlooked him.”[13]

Filed under physics person Author feynman john bardeen transistor important