studiot

Hi Seth, It is difficult to estimate the as it depends heavily on your model. Most folks just quote a figure worked out by somebody else (perhaps with a reference). Anyway the most baseline figure is what you get with the simplest heat balance of what would happen if the Earht's atmousphere suddenly disappeared, but everything else remained the same and is done in the Cambridge book for the whole solar system. It is called the equilibrium temperature. Thermodynamics of the Earth and Planets A P Douce Cambridge 2011 Pages 630 ff However there are wrinkles in this. The Earth's axis is tilted at about 23.5 degrees and also the distance from the Sun varies, both during one year and from year to year. So the surface (even without the effect of the atmousphere) receives uneven insolation. And of course half of it is is shaded by night at any one time. You might think that the Moon was therefore a better candidate as it has no atmousphere to speak of. But, whilst it is true that the Moon rotates so that all its surface gets insolation, the shading effect of the Earth on the Moon is far greater than the shading effect of the Moon on the Earth owing to their size difference. So the Moon receives less insolation than your calculations would suggest. Coming back to the Earth's atmousphere, it affects the fugure's in several ways and more sophisticated models need to take these into acount. Both the atmousphere and ocean distribute energy from wrmer zones to colder ones, thus modifying the average temperature difference for Stefan energy flows. The atmousphere absorbs/ reflects some of insolation (Beer Lambert) so reducing the actual values. The atmousphere also reflects some surface emission back (greenhouse effect) again modifying the surface temperatures. A more sophisticated model , therefore, splits the surface into zones and applies appropriate insolation and emission values, spread over time then integrated over time and averaged, and then finally all collected together and again averaged.

Not all, only those in the White House. 😉

You need to look at the maths here https://en.wikipedia.org/wiki/D'Alembert_operator https://en.wikipedia.org/wiki/Four-gradient

I realise I missed the reciprocal. Apologies. It should read c-2 = ε0μ0

One of the most fundamental principles in Physics is that of the isotropy and homogeneity of space / spacetime. In order to theorise the observation that it does not matter where or when we look, the physics of light and physics in general appears much the same. The principle of relativity follows directly from this, as does the principle of equivalence. In other words there is no preferred origin for empty space or time. It is a requirement of every cubic metre of space and every second of time be indistinguishable from every other. This leads directly to the constancy of c since c2 = ε0μ0 If you want to consider non constant c then you must have non isotropic space /spacetime then epsilon and mu become tensors to reflect the non isotropy / homegeneity and c2 becomes a tensor product. In short you have either a non vacuum or an aether. Any maths you use must also reflect this.

So you have done some Physics already. Great and good luck with the exam. 😀 A question for you to think about. What happens when you add water at 2oC to water at 3oC. Do you get water at 5oC ? If not why not ?

Seconded +1

How does that old saying go? There's none so blind as those that won't see, or so deaf as those that won't listen.

This is very clearly the case. This is not at all what I would expect from someone who understands what they are claiming. I would expect replacement equations for those of Maxwell. Then I would expect to see some math deriving the conditions for wave motion from the equations and finally ending with a wave equation. What I would not expect to see is the assumption of that which was to be derived, namely assume c(x, t). I would also expect the claimant to understand that the definitions and derivations of Maxwell work on the basis of a vacuum with no gravity fields or anything else. c does not even appear until the last line of the standard derivation. The onus is entirely on the claimant to derive the claim. So show us your maths please. Edit Gosh I keep x posting with swansont. Multiplying by c does not rescale time at all. In conventional (SI) units it transforms the dimensions so that the product has the dimensions of length. This is in much the same way as mass (which can be variable) transforms the dimensions of acceleration to yield Force, in Newtonian theory. As I understand matters the choice of c and other natural constants is because they are constant in a universe where the dimensional quantities are variable according to circumstance and observer. This then avoids the issue of having to refer back to standards of mass, length and time etc.

But in Natural units the speed of light is dimensionless and exactly 1. https://en.wikipedia.org/wiki/Natural_units So whatever units you measure distance and time or other constants in is irrelevant. c is still constant.

If c varies first and foremost you need to specify what it varies with. Secondly you then need to rewrite our system of mathematics since constancy is a requirement of Maxwell's equations, and the wave equation in general. Can you offer this ?

Good morning Angelo and welcome. Thank you for your comments +1 and go well in your studies. Feel free to ask questions here at SF, there is plenty of help availble for those who want to learn. But please start you own thread for them, one topic per thread. As regards the arithmetic of quantities with units. Quantities with units have two parts. The value or magnitude of that quantity and the unit of physical property it represents. Mathematically (arithmetically) you can manipulate magnitudes as much as you like. But physically the results of that manipulation must also make sense. In your studies you will find out that there are sometimesmore than one way to add or multiply them together. Further sometimes the addition makes sense, but not the multiplication and sometimes the other way round. And worse sometimes one can be positively misleading. Let me give you an example with the property of length. Let us consider four houses, A, B C, D connected together by electric cables. In fig1 the houses are connected in a straight line so the total length is simple addition. However multiplying two or more of the lengths doesn't lead to any useful calculation and indeed could be misleading since the area enclosed is not a loopor closed geometric figure. Fig 2 is a closed figure (rectangle) so adding the lengths again produces the total length. But this time the electrical loop encloses an area. This has physical implications for the magnetic field, Faraday's Laws etc. Fig3 is another configuration of the network where the total length is the sum of the individual spacings, but no area is enclosed. Furthermore no real meaning can be attached to the product of say the length AB x BC (which does not involve D at all). Life gets more complicated of you try to add 50 Hz to 10 Hz oscillations or even 50 Hz to 50 Hz. You do not get and oscilation at 60 Hz or 100 Hz in either case. But you can meaningfully multiply the two oscillations and calculate the product of the two sinusoids - (you get the sum and difference frequencies)

Hello MigL. Well the original poster doesn't appear to want to discuss his own topic, perhaps because it was posted in a haphazard way last Christmas day. However the question (correctly) referred to "the concept of randomness " not to the word 'random' which is the etymological root in this case. Randomness is "the quality or state of being random" - Oxford E D. Now randomness is the noun derived from random, which in turn is both an adverb and an adjective. As such it is meaningless without a noun or verb to qualify. Random is derived from the Old French randir = to run via the Middle English adverb randon which referred particular to riding but also to other activities to rush about in a headlong, haphazard or aimless manner. In modern (including technical) English, when combined with suitable verbs or nouns, this sense is in someway preserved. But this does not mean that every aspect of the noun or verb suffers the state of randomness. A good example is 'random stone walling'. Here a masonry wall is built of unsorted shapes and sizes of stones (as they come) but the 'at random' , but the overall wall shape and size will be conventional and predictable. Random numbers are more difficult to get a handle on. I have pointed out the Kolmogorov definition in a previous thread, a few years ago. This states that a random number is simply the shortest possible way of writing that number so your 8 character statement is random by that definition as compared to your 34 character 'formula' for it. This would accord with the standard way of finding numbers from random number tables. Another closely allied term is 'arbitrary'. This term is often used in simple mathematical proofs to indicate that any of a possibly infinite range of values will satisfy the current deterministic calculation. Random, as already noted, can apply to small systems as well.

Did you want an argument or a discussion ? 😀 Could not not stretch a filter fabric over the top of the sump, under the water ? As to the sump material. If it is bituminous I would avoid it, although in the past water tanks have been painted / lined internally with bituminous material, which is not in itself poisonous, in the past there is a risk of cancer from the bitumen.

Yes I have downloaded this pdf as the subject is interesting. However I wonder how you consider it restores Fermat's original proof ? You have quite a few pages of algebra and calculus to wade through, though they may indeed be correct I can't reconcile them with the simplicity Fermat claimed, not the algebra and other mathematics of his time.

Do you realise that each line of the table is independent of any other line ? So you fill it in line by line. There are 7 base quantities in SI units and, (by coincidence ?), 7 lines in the table. Look here https://sciencenotes.org/si-base-units/ 😀 So, starting with the first line how are you getting on with time ?

Calculating percentage 'accuracies' can be very misleading which is why it is a pity you did not answer my post. The questions I asked were designed to help us to help you since you have asked the same question twice now, this is the one I was am still concentrating on. The answer to this is that industry standards have grown up on the basis of calculating the error. But the analysis depends upon the circumstances and the formulae you are using which I why I asked about calculus. You really need a good understanding of the meaning of the words, error, significant figures, decimal places, accuracy, precision. Your example of the area of a circle is more difficult and less versatile than one using the circumference since it only contains multiplication. If you would like to work through the following question, which also contains addition, and its implications please let us know. It is a much more useful question to start with. A man who is 2 metres tall walks all the way round the Earth at the equator. Given that the radius of the Earth is 6378137.3472 metres, how much further does his head travel than his feet ? This question is about what is known as 'the difference of two large numbers' and the loss of accuracy that can arise if it is handled in the wrong way. The answer is also quite suprising. If you want to proceed further, please let us know.

Good Morning Axel, I see you are new here so welcome. 😀 You are correct in thinking that the value used for π affects the end result in calculations. There is indeed considerable theory which is used in Science and Engineering to select the appropriate number of decimal places to use. In order to explain this further perhaps you would like to tell us a little about you maths background since calculus is needed for this. Do you know any calculus or anything about 'significant figures' ? In the meantime a couple of rules of thumb. Firstly it is often good enough to adopt the value that π = √10 Which can make a calculation easier as this will often cancel. [math]r = \sqrt {\frac{{78.5}}{\pi }} \simeq \sqrt {\frac{{78.5}}{{\sqrt {10} }}} = \sqrt {\frac{{7.85*10}}{{\sqrt {10} }}} = \sqrt {7.85*\sqrt {10} } = 4.98[/math] This example is trivial but shows the principle. Secondly it is common practice to emplou one or two 'guard digits'[ So if you are expecting the result to be accurate to 3 significant figures (3SF) you would work to 4 or 5 sf and round at the end. Better theory would give you a better answer.

Pull-backs are defined for smooth (differentiable) transformations only; for example the pull-back of a Jacobian (which acts on columns from the left) is another matrix of partial derivatives acting on rows from the right.

https://math.stackexchange.com/questions/1812391/relation-between-the-dual-space-transpose-matrices-and-rank-nullity-theorem No more time tonight sorry.

Good on you for your choice of subject. Perhaps your lecturers are trying to make their students think ? At third level you need self directed learning (reading around the subject) more than ever. You learn almost as much from your fellow students as from your lecturers, especially in a subject that devotees are passionate about. Also this subject requires a wide ranging scientific background, so much of the first year is spent bring students from disparate backgrounds up to a required common level. The pace and standard ramps up after that. From the times of your posting I deduce that you are probably in OZ or NZ. This book, from Oxford University Press has most of the examples set in this continent. He also wrote a smaller first year university textbook about the subject. Sadly Prof Selby passed in 2018. But remember ScienceForums when you want to discuss something. You can post more generally in the Earth Science section, the Homework section has special rules so we are not allowed to answer set work directly.

Hello Ed, You are clearly near the beginning of your studies here, but it would be useful if you were to indicate your interest in physical geography / physical geology / geomorphology is. Especially if you are going to post more questions for clarification. It really is a fascinating subject. The three diagrams you posted were from an older era and the subject has moved on since. In those days the view was rather narrow and static. Today we try to look at such questions in a wider sense, both in how the particular landforms developed and fit in both with their surrounding areas on the ground and our best theoretical work on the subject. I note that the three diagrams (which could easily have been drawn by Arthur Holmes a very famous pioneer) have been carefully edited to avoid showing any explanation, for the purpose of making them into a question. I take it you are aware that all the rocks on Earth can be classified into sedimentary or igneous (volcanic). All three diagrams could be formed from either types of rock by 'denudation processes'. Denudation is the overall term for the general process by which earth and rock is detached from the main body of material and transported away from the site to somewhere else. Swansont has suggested that the waterfall might be formed like the ones on the Colorado river in the Grand Canyon. These layers here are mostly volcanic and look like the ones in the Rio Grande picture. Here, as with the Grand Canyon there is a large scale upper plateaux or mesa, heavily dissected by a major river in an arid area. More can be seen here https://www.nps.gov/grca/learn/nature/grca-geology.htm The whole area has been uplifted and is a bit like the simplified diagram below. Note the timeline goes from right to left. The final stages of this process, which have not yet happened, look like this butte in Utah. Note here the pile of debris at the bottom, which has not yet been transported away. This could also be as the first diagram of your question. The waterfall diagram shows nor canyon sidewalls , more like Niagara, which is forming on sedimentary rock. In your middle diagram the resisant layers could be part of an anticline, with the top worn away, as with this section of the Welsh Borders.

Hello and welcome. We have another member with similar interests so look at their thread.

I think you are clear what is being eroded or weathered, but did you think about my question ? In each of the three pictures what agent is dong the eroding. Note it is probably different in each case.