... It was a hoot none-the-less, got up real early and walked several km to a spot I'd picked out earlier on the beach, above the high tide line ( which arrived about an hour later post totality ). Totality is unreal - the wind dies, the birds go quiet, spookin' a bit etc ...
All highly tantalizing. Good shots there including the clouds!
We had cloud and drizzle for the 1999 solar eclipse that skimmed Cornwall here in the UK. Despite the cloud and rain, we clearly saw a 'sunset' all 360 degs around the horizon. The birds went quiet and a few even scurried off to roost.
Rather interestingly, as the shadow passed across us, it stopped raining. Then, as the light returned, the rain returned a few moments later...
There's a parallel comment to that in the BBC article for your eclipse:
Nice pictures Mike.....sounds like you had plenty of fun.
In all my years I have looked up in the sky day and night thousands and thousands of times but can't say it was quite the same as what you got to do.
I was usually out in my yard freezing by myself since I can't get my wife to do more than look out the window at something I show her if she is still awake.
I guess back in my younger days I may have had a Fosters at the same time
You also reminded me I still have an unplayed/unopened Mobile Fidelity remastered on virgin vinyl "Dark Side of the Moon" in my collection.
The other night while listening to a late night talk radio program while doing something else they once again mentioned people who believe that there are some form of life on that other side of the Moon who live under the surface in some sort of life supporting world of their own.......I guess they like those month long days there
@Martin : we were lucky indeed ! The cloud for all it's nuisance did add an artistic flavour to the shots. There was a moment when I realised that the autofocus/autoshutter on the camera was totally confused as to what distance/depth it should be using ( with all the cloud, sudden drop in light level and whatnot ) so I quickly took the filters off, went to manual mode and wound the telephoto out to infinity. Seems it worked, but I lost precious time for a few more shots though. A NASA crew down the beach a ways was ecstatic to have grabbed a break in cover at the right time. I met a chap on the beach - it's easy to socialise here, you just go up to someone who has some optical contrivance, introduce yourself and much can be discussed and learned - who had seen the Cornwall eclipse and he remembered much the same ( 'rain in Cornwall, how unusual' he said with a wink ). There were very many multi-eclipse addicts about! They come from absolutely everywhere over the globe, there were teams nearby me who came from India and Malaysia who were thoroughly enjoying themselves. I think most were thinking as I was : 'burn through the cloud baby, burn through' ... :-) :-)
@Magic : Fosters is still a classic here! Could do worse. I have a well played vinyl of the Pink Floyd album, with that cool graphic of light going through a prism on the front cover. My wife happily played photographer's assistant. We did carefully protect the Mark One Eyeball Optics using certified eclipse sunnies. I put two filters over my camera's aperture to protect the digital backplane as I knew it would simply lengthen the shutter time to account. Which meant I needed a really stable tripod so I bunged three coconuts in a shopping bag and hung it from the tripod's centre hook ! :-)
[ As you can tell from my gourmet comments - kangaroo again last nite with a 'Plush' cocktail - it's a great place even without astronomical events. I have definitely blown a wad of cash this week. But hey, what a way ... :-) ]
Now I've been thinking : what if I could turn up on Principe in 1919 with my camera. Mr Eddington would be delighted! While I have a bog standard 1990's vintage 35mm digital SLR it would have been way ahead of their delicate business with photographic plates. I could even now grab a star chart and overlay to suitable scale on that reverse color shot with the three stars. The two far away ( 10 and 4 o'clock ) serve as reference for any displacement of the nearby one at 10 o'clock. There's lots of niggle here though, the stars showing at eclipse are going to be the brighter ones, which hence will be closer to Earth than most and subject to parallax and proper motion etc. Anyhows it's do-able in theory. Einstein had made a factor of two error in his predictive calculations for that eclipse, but fortunately corrected that beforehand. Else it would have been less convincing. There was a minor kerfuffle at the time about whether the photographic plate measurements were swayed by wishful thinking, but even if so : a deflection was evident so the Sun did deflect light paths.
So now plonk yourself on the close companion of some really compact star like a neutron star or black hole. Ignoring the non-tenable aspects for beings such as us, you could see a whole range of distortions of the far star field close to the limb of such an object. I remember an earlier Windows screensaver which simulated the same by rolling a 'black hole' over the desktop. What a weird experience it would be for us who live in almost 'flat' spacetime. Conversely if you were born in a deep gravity well how unusual our circumstances would seem. You'd probably bump into and trip over everything as your depth perception couldn't adapt! Who can say? Could be a sci-fi novel in that. :-)
I've been thinking about tensors this week too. Here's my humble take on that ( beware, errors! ) :
Take a whole bunch of points, like those in the room you're in. In your mind, attach to each point a box into which you can place numbers. It could be simple like one number per box and thus maybe describe the temperature levels throughout the room. A 'scalar field'. Or it could be three numbers per box representing the movement of air, as three numbers can define a vector showing the direction of the 'wind' in the room generated by a fan, say. A 'vector field'.
But you can be more ambitious, as you can define any arrangement of numbers in the boxes. As per the task at hand. So it's a sort of accounting device to keep track of some process of interest. Imagine the interior of a solid. At each point within you can define an infinitesimal cube centred on the point. Infinitesimal means as small as you like but not zero. For each face of that cube you could look at force vectors with components thereof resolved with respect to that face. Resolve in the normal or perpendicular direction and you will have that force component which wants to compress or expand the cube. Resolve to the cube face ( it's a plane ) and you have the force components that want to shear or slide. So the totality of all the vectors resolved at all faces of the cube with each vector being a list of three numbers then becomes a record of the force state of that cube surrounding a single point. Throughout the whole solid we have boxes containing numbers ( suitably indexed by which face, which resolved direction, and which coordinate component per vector ) to then summarise the state of that body with regard to internal forces that want to distort it. One box of numbers per point in the solid.
Now here's the kicker : all those numbers have to be offsets from an origin in a coordinate system of some orientation & scale. Not just the positions of each point in the solid now, but also the numbers that I put into the accounting box that describe those vector directions. You want the description you have built to be independent of the choice of coordinate system. If you change your coordinate system the numbers in each box will in detail change for sure. But the relationship between the numbers in one box attached to a certain point should not alter with respect to those in a box attached to an ( infinitesimally ) nearby point. The change in observer viewpoint ( coordinate system ) brought about by a transformation in said viewpoint ought not lead to different physics. So a given lattice ion, say Bob, is going to want to move ( due to nett forces ) in a relative way to a nearby ion, say Carol. Bob and Carol are going to do what they do regardless of any accounting description used.
What I've discussed here is the idea of a 'covariant tensor' setup. Einstein's gravity equations use such things to ensure that there is a certain 'equality' of viewpoints. Not all observers will agree in detail for sure, as they are sitting at different places in spacetime after all, but they are going to be describing the same Universe. So the Bob's and Carol's will maintain their mutual behaviours.
Cheers, Mike ( FNQ Correspondent )
( edit ) I should have said more explicitly : the box of numbers attached to each point is the tensor. Scalars and vectors are a simpler type of tensor, or if you like tensors are a generalisation of vectors. A scalar is a tensor with only one number. You have to distinguish b/w the dimensionality of the coordinates of the physical system you describe, say three for space and four for spacetime versus that which applies to the boxes. The boxes can have any dimension at all, these are not 'physical' per se but represent the number of independent indices ( counters ) that you need to cover all the types of possibilities within the number box 'structure' you are using. So a tensor field, if of covariant character, thus by definition transforms in the way described keeping the physics consistent. There are contravariant tensors ( plus others no doubt ) that I haven't got around to.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Tensor calculus was introduced to Einstein by the mathematicians of the Padua University like Luigi Bianchi and Gregorio Ricci-Curbastro,who gave name to the Ricci tensor.
Tullio
@Mike : Now you have me tempted to maybe have a bottle of Fosters on football sunday......not the big can version which was called the "Fosters Oil" can here just because of the size and shape of the famous 25oz
Now the closest I have ever been to a kangaroo was at a zoo and that has been almost 30 years so you have been a bit closer than I have!
Yes I also have an old copy of Dark Side of the Moon next to my unplayed version and the wife likes the exact opposite of what I have so they may never make it to a turntable again (I did R&D for the Carver Corp. in the 80's)
Ribbons and magnetic fields.
Just imagine trying to travel the Earth in 1919 hoping to set up camera's on a clear night sky to try to prove or disprove gravitational deflection of light by the sun's mass, gravitational lensing,experimental verification of Albert Einstein's theory of general relativity.
Now those were some vintage cameras!
I haven't had a screensaver since early 1999 but the screen does seem to be my own black hole with the keyboard near the event horizon that draws you in for hours and some how I make it back out through the worm hole.
It has been many years since I even had to develop any film but for some reason I carry a camera and collect these digital pictures.
Do you ever take pictures using a telescope?
The first time I was in my cardiologists office waiting to see him I was surprised to see the walls covered with photographs of planets and famous galaxies that were like Nasa Hubble photo's and when he comes in the first thing I ask is where he got them and he told me he had his own expensive telescope and camera and showed me a picture of it set up at his place.....but as a typical cardiologist like him he didn't have any time to discuss that so we went right back to why I was there and he seemed like a busy doctor so I imagine his telescope and camera is his peaceful place.
I've wanted to, but lack the time and in any case we get only 3/10 clear night skies where I am, essentially summertime only, plus a high local horizon on one side and the glow of a big city ( Melbourne ) on the other. I like pictures but would probably do photometry if given the chance.
NB. Forgot to mention : I never had any hope of seeing diffraction bands, not at sea level! Those go past at ~ 15,000 km/hr so you need a large area of landscape in view to perceive that.
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Solar astronomy and astrophotography is a nice hobby (a bit dangerous for kids of course). You can have quite OK results even with very, very modest equipment. The following is something that was more of a practical astronomy joke of mine, but it actually works: Celestron Firstscope, Baader solar filter foil, cardboard, Velcro tape, Baader zoom eye-piece (simpler would do as well), T2-adapter ring, and a simple, lightweight Olympus E420 DSLR
Nice for observing sunspots.
I encourage everyone to see their local astronomy club if you are interested in solar observations, chances are that you can try out some equipment there and then decide if that's something you would enjoy as a hobby.
... For years, tourists, astronomers and eclipse chasers had been anticipating a fantastic show over the Coral Sea on Nov. 14, 2012. Just after daybreak, the Moon would pass directly in front of the low-hanging sun, producing a total eclipse in plain view of many resort towns along the coast. More than a hundred thousand people (me and my family included) converged to witness the event.
@Martin : Yep, he's captured the mood of the event for sure! The clouds did add ambiance. He was to the south of myself, where they started the marathon. It was a hot day for them, many succeeded, but I don't remember seeing a guy with a rubber chicken! :-)
There were plenty about, here's a shot afterwards of a short stretch of the beach - it was really packed at totality, high tide and all.
@Bikeman : Nice sunspots! It is quite neat how you can get pretty good shots with a modest outlay and a bit of fiddling.
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Actually I've just found a great book teaching about transitioning from vector to tensor thinking : A Student's Guide To Vectors and Tensors by Daniel Fleisch. That's already borne fruit for me eg. it is a tensor's components that are contravariant/covariant and NOT the tensor per se. So a tensor 'is what it is', and thus co/contra-variance is a matter of choice within a given coordinate representation of said tensor. Note that Bernard F. Schutz ( a director at AEI/MPG, Potsdam ) has commented upon confusions arising when using contra/co-variant language and thus it's deprecation. The essential issue, I gather, is that expressing a given vector/tensor in terms of the basis vectors of different coordinate systems is NOT the same intellectual process as transforming a vector to a different vector ( components expressed in either system ).
That's rather self evident when it's put like that, really. So for instance if I have a vector pointing to the Andromeda Nebula then it will continue to do that even if I tilt my head and compare the different components of that single vector expressed in 'unrotated' and 'rotated' coordinate systems. However that ain't the same type of cow as ( but has much in common with! ) rotating the vector to point to the Lesser Magellanic Cloud and not the Andromeda Nebula. Confusion arises especially when one 'transforms' the basis vectors of one system to those of another ie. are you actually rotating ( say ) the original basis vectors to co-incide with a new set of basis vectors OR are you expressing each original basis vector in terms of components with regard to the new basis vector set. For me that has substantially lifted some fog/mist! :-)
I'll write a review when I'm done. :-)
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
RE: ... It was a hoot
)
All highly tantalizing. Good shots there including the clouds!
We had cloud and drizzle for the 1999 solar eclipse that skimmed Cornwall here in the UK. Despite the cloud and rain, we clearly saw a 'sunset' all 360 degs around the horizon. The birds went quiet and a few even scurried off to roost.
Rather interestingly, as the shadow passed across us, it stopped raining. Then, as the light returned, the rain returned a few moments later...
There's a parallel comment to that in the BBC article for your eclipse:
Australia solar eclipse delights thousands
... There had been fears that cloudy weather would obscure the eclipse but the clouds parted just as it began, ...
... "And right after it finished, the clouds came back again. It really adds to the drama of it."...
All coincidence? Or a localized trick of solar energy?
That would be an interesting one to model in detail...
Thanks for the pics,
Cheers,
Martin
See new freedom: Mageia Linux
Take a look for yourself: Linux Format
The Future is what We all make IT (GPLv3)
Nice pictures Mike.....sounds
)
Nice pictures Mike.....sounds like you had plenty of fun.
In all my years I have looked up in the sky day and night thousands and thousands of times but can't say it was quite the same as what you got to do.
I was usually out in my yard freezing by myself since I can't get my wife to do more than look out the window at something I show her if she is still awake.
I guess back in my younger days I may have had a Fosters at the same time
You also reminded me I still have an unplayed/unopened Mobile Fidelity remastered on virgin vinyl "Dark Side of the Moon" in my collection.
The other night while listening to a late night talk radio program while doing something else they once again mentioned people who believe that there are some form of life on that other side of the Moon who live under the surface in some sort of life supporting world of their own.......I guess they like those month long days there
Thanks for showing us those nice pictures.
@Martin : we were lucky
)
@Martin : we were lucky indeed ! The cloud for all it's nuisance did add an artistic flavour to the shots. There was a moment when I realised that the autofocus/autoshutter on the camera was totally confused as to what distance/depth it should be using ( with all the cloud, sudden drop in light level and whatnot ) so I quickly took the filters off, went to manual mode and wound the telephoto out to infinity. Seems it worked, but I lost precious time for a few more shots though. A NASA crew down the beach a ways was ecstatic to have grabbed a break in cover at the right time. I met a chap on the beach - it's easy to socialise here, you just go up to someone who has some optical contrivance, introduce yourself and much can be discussed and learned - who had seen the Cornwall eclipse and he remembered much the same ( 'rain in Cornwall, how unusual' he said with a wink ). There were very many multi-eclipse addicts about! They come from absolutely everywhere over the globe, there were teams nearby me who came from India and Malaysia who were thoroughly enjoying themselves. I think most were thinking as I was : 'burn through the cloud baby, burn through' ... :-) :-)
@Magic : Fosters is still a classic here! Could do worse. I have a well played vinyl of the Pink Floyd album, with that cool graphic of light going through a prism on the front cover. My wife happily played photographer's assistant. We did carefully protect the Mark One Eyeball Optics using certified eclipse sunnies. I put two filters over my camera's aperture to protect the digital backplane as I knew it would simply lengthen the shutter time to account. Which meant I needed a really stable tripod so I bunged three coconuts in a shopping bag and hung it from the tripod's centre hook ! :-)
[ As you can tell from my gourmet comments - kangaroo again last nite with a 'Plush' cocktail - it's a great place even without astronomical events. I have definitely blown a wad of cash this week. But hey, what a way ... :-) ]
Now I've been thinking : what if I could turn up on Principe in 1919 with my camera. Mr Eddington would be delighted! While I have a bog standard 1990's vintage 35mm digital SLR it would have been way ahead of their delicate business with photographic plates. I could even now grab a star chart and overlay to suitable scale on that reverse color shot with the three stars. The two far away ( 10 and 4 o'clock ) serve as reference for any displacement of the nearby one at 10 o'clock. There's lots of niggle here though, the stars showing at eclipse are going to be the brighter ones, which hence will be closer to Earth than most and subject to parallax and proper motion etc. Anyhows it's do-able in theory. Einstein had made a factor of two error in his predictive calculations for that eclipse, but fortunately corrected that beforehand. Else it would have been less convincing. There was a minor kerfuffle at the time about whether the photographic plate measurements were swayed by wishful thinking, but even if so : a deflection was evident so the Sun did deflect light paths.
So now plonk yourself on the close companion of some really compact star like a neutron star or black hole. Ignoring the non-tenable aspects for beings such as us, you could see a whole range of distortions of the far star field close to the limb of such an object. I remember an earlier Windows screensaver which simulated the same by rolling a 'black hole' over the desktop. What a weird experience it would be for us who live in almost 'flat' spacetime. Conversely if you were born in a deep gravity well how unusual our circumstances would seem. You'd probably bump into and trip over everything as your depth perception couldn't adapt! Who can say? Could be a sci-fi novel in that. :-)
I've been thinking about tensors this week too. Here's my humble take on that ( beware, errors! ) :
Take a whole bunch of points, like those in the room you're in. In your mind, attach to each point a box into which you can place numbers. It could be simple like one number per box and thus maybe describe the temperature levels throughout the room. A 'scalar field'. Or it could be three numbers per box representing the movement of air, as three numbers can define a vector showing the direction of the 'wind' in the room generated by a fan, say. A 'vector field'.
But you can be more ambitious, as you can define any arrangement of numbers in the boxes. As per the task at hand. So it's a sort of accounting device to keep track of some process of interest. Imagine the interior of a solid. At each point within you can define an infinitesimal cube centred on the point. Infinitesimal means as small as you like but not zero. For each face of that cube you could look at force vectors with components thereof resolved with respect to that face. Resolve in the normal or perpendicular direction and you will have that force component which wants to compress or expand the cube. Resolve to the cube face ( it's a plane ) and you have the force components that want to shear or slide. So the totality of all the vectors resolved at all faces of the cube with each vector being a list of three numbers then becomes a record of the force state of that cube surrounding a single point. Throughout the whole solid we have boxes containing numbers ( suitably indexed by which face, which resolved direction, and which coordinate component per vector ) to then summarise the state of that body with regard to internal forces that want to distort it. One box of numbers per point in the solid.
Now here's the kicker : all those numbers have to be offsets from an origin in a coordinate system of some orientation & scale. Not just the positions of each point in the solid now, but also the numbers that I put into the accounting box that describe those vector directions. You want the description you have built to be independent of the choice of coordinate system. If you change your coordinate system the numbers in each box will in detail change for sure. But the relationship between the numbers in one box attached to a certain point should not alter with respect to those in a box attached to an ( infinitesimally ) nearby point. The change in observer viewpoint ( coordinate system ) brought about by a transformation in said viewpoint ought not lead to different physics. So a given lattice ion, say Bob, is going to want to move ( due to nett forces ) in a relative way to a nearby ion, say Carol. Bob and Carol are going to do what they do regardless of any accounting description used.
What I've discussed here is the idea of a 'covariant tensor' setup. Einstein's gravity equations use such things to ensure that there is a certain 'equality' of viewpoints. Not all observers will agree in detail for sure, as they are sitting at different places in spacetime after all, but they are going to be describing the same Universe. So the Bob's and Carol's will maintain their mutual behaviours.
Cheers, Mike ( FNQ Correspondent )
( edit ) I should have said more explicitly : the box of numbers attached to each point is the tensor. Scalars and vectors are a simpler type of tensor, or if you like tensors are a generalisation of vectors. A scalar is a tensor with only one number. You have to distinguish b/w the dimensionality of the coordinates of the physical system you describe, say three for space and four for spacetime versus that which applies to the boxes. The boxes can have any dimension at all, these are not 'physical' per se but represent the number of independent indices ( counters ) that you need to cover all the types of possibilities within the number box 'structure' you are using. So a tensor field, if of covariant character, thus by definition transforms in the way described keeping the physics consistent. There are contravariant tensors ( plus others no doubt ) that I haven't got around to.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Tensor calculus was
)
Tensor calculus was introduced to Einstein by the mathematicians of the Padua University like Luigi Bianchi and Gregorio Ricci-Curbastro,who gave name to the Ricci tensor.
Tullio
@Mike : Now you have me
)
@Mike : Now you have me tempted to maybe have a bottle of Fosters on football sunday......not the big can version which was called the "Fosters Oil" can here just because of the size and shape of the famous 25oz
Now the closest I have ever been to a kangaroo was at a zoo and that has been almost 30 years so you have been a bit closer than I have!
Yes I also have an old copy of Dark Side of the Moon next to my unplayed version and the wife likes the exact opposite of what I have so they may never make it to a turntable again (I did R&D for the Carver Corp. in the 80's)
Ribbons and magnetic fields.
Just imagine trying to travel the Earth in 1919 hoping to set up camera's on a clear night sky to try to prove or disprove gravitational deflection of light by the sun's mass, gravitational lensing,experimental verification of Albert Einstein's theory of general relativity.
Now those were some vintage cameras!
I haven't had a screensaver since early 1999 but the screen does seem to be my own black hole with the keyboard near the event horizon that draws you in for hours and some how I make it back out through the worm hole.
It has been many years since I even had to develop any film but for some reason I carry a camera and collect these digital pictures.
Do you ever take pictures using a telescope?
The first time I was in my cardiologists office waiting to see him I was surprised to see the walls covered with photographs of planets and famous galaxies that were like Nasa Hubble photo's and when he comes in the first thing I ask is where he got them and he told me he had his own expensive telescope and camera and showed me a picture of it set up at his place.....but as a typical cardiologist like him he didn't have any time to discuss that so we went right back to why I was there and he seemed like a busy doctor so I imagine his telescope and camera is his peaceful place.
RE: Do you ever take
)
I've wanted to, but lack the time and in any case we get only 3/10 clear night skies where I am, essentially summertime only, plus a high local horizon on one side and the glow of a big city ( Melbourne ) on the other. I like pictures but would probably do photometry if given the chance.
NB. Forgot to mention : I never had any hope of seeing diffraction bands, not at sea level! Those go past at ~ 15,000 km/hr so you need a large area of landscape in view to perceive that.
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Hi! Solar astronomy and
)
Hi!
Solar astronomy and astrophotography is a nice hobby (a bit dangerous for kids of course). You can have quite OK results even with very, very modest equipment. The following is something that was more of a practical astronomy joke of mine, but it actually works: Celestron Firstscope, Baader solar filter foil, cardboard, Velcro tape, Baader zoom eye-piece (simpler would do as well), T2-adapter ring, and a simple, lightweight Olympus E420 DSLR
Nice for observing sunspots.
I encourage everyone to see their local astronomy club if you are interested in solar observations, chances are that you can try out some equipment there and then decide if that's something you would enjoy as a hobby.
Clear Skies
HB
For a fun comparison, here's
)
For a fun comparison, here's a NASA review of the Auzzie spectacle:
The Surprising Appeal of a Cloudy Eclipse
... For years, tourists, astronomers and eclipse chasers had been anticipating a fantastic show over the Coral Sea on Nov. 14, 2012. Just after daybreak, the Moon would pass directly in front of the low-hanging sun, producing a total eclipse in plain view of many resort towns along the coast. More than a hundred thousand people (me and my family included) converged to witness the event.
The night before the eclipse was crystal clear...
Suitably enthusiastic ;-)
Keep searchin',
Martin
See new freedom: Mageia Linux
Take a look for yourself: Linux Format
The Future is what We all make IT (GPLv3)
@Martin : Yep, he's captured
)
@Martin : Yep, he's captured the mood of the event for sure! The clouds did add ambiance. He was to the south of myself, where they started the marathon. It was a hot day for them, many succeeded, but I don't remember seeing a guy with a rubber chicken! :-)
There were plenty about, here's a shot afterwards of a short stretch of the beach - it was really packed at totality, high tide and all.
@Bikeman : Nice sunspots! It is quite neat how you can get pretty good shots with a modest outlay and a bit of fiddling.
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Actually I've just found a
)
Actually I've just found a great book teaching about transitioning from vector to tensor thinking : A Student's Guide To Vectors and Tensors by Daniel Fleisch. That's already borne fruit for me eg. it is a tensor's components that are contravariant/covariant and NOT the tensor per se. So a tensor 'is what it is', and thus co/contra-variance is a matter of choice within a given coordinate representation of said tensor. Note that Bernard F. Schutz ( a director at AEI/MPG, Potsdam ) has commented upon confusions arising when using contra/co-variant language and thus it's deprecation. The essential issue, I gather, is that expressing a given vector/tensor in terms of the basis vectors of different coordinate systems is NOT the same intellectual process as transforming a vector to a different vector ( components expressed in either system ).
That's rather self evident when it's put like that, really. So for instance if I have a vector pointing to the Andromeda Nebula then it will continue to do that even if I tilt my head and compare the different components of that single vector expressed in 'unrotated' and 'rotated' coordinate systems. However that ain't the same type of cow as ( but has much in common with! ) rotating the vector to point to the Lesser Magellanic Cloud and not the Andromeda Nebula. Confusion arises especially when one 'transforms' the basis vectors of one system to those of another ie. are you actually rotating ( say ) the original basis vectors to co-incide with a new set of basis vectors OR are you expressing each original basis vector in terms of components with regard to the new basis vector set. For me that has substantially lifted some fog/mist! :-)
I'll write a review when I'm done. :-)
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal