studiot

As the author of post#5, did you read the rules here? Why should I go rambling over the net to find out what should be summarised in a post?

In Newtonian mechanics, the total mechanical energy of an isolated system is conserved. Energy may be swapped between individual reservoirs of energy. Often these are just Kinetic Energy and Potential Energy. Further energy can be introduced into the system by outside agents, at the expense of removing the isolation constraint.

Yes mass is a better quantity than volume to work with, both for the mechanics of viscous flow and the chemistry of reaction rate kinetics. Each of this will lead to a differential equation. These differential equations will be coupled or simultaneous and you solve them as such. For the chemical kinetics of solution look at the Law of Mass Action https://www.google.co.uk/search?hl=en-GB&source=hp&biw=&bih=&q=law+of+mass+action&gbv=2&oq=law+of+mass+action&gs_l=heirloom-hp.3..0l10.1297.9234.0.9781.26.13.4.9.10.0.204.1436.5j7j1.13.0....0...1ac.1.34.heirloom-hp..0.26.1953.HxTKyxqO_ak

5 posts and I still have no idea what the debate about the Physics of time is about. There is, anyway, far too much Physics (of time) to fit into one thread here so can someone (since the OP is not listed as having returned since starting this thread last November) establish a simple discussion point please?

Just to make wtf's question a little more difficult How many are left if you only leave every 3rd number instead of every second? What about only leaving every 4th number? What happens if we only leave every nth number? And what happens if we let n tend to infinity?

Once again some preliminary work is needed. It is important to realise that transfinite numbers do not obey the same rules of arithmetic as finite numbers. In particular if you remove (subtract) a finite number or a lesser infinity of numbers from an infinite set you still have the original cardinality. So if you remove an infinite subset of R made up of all the rational numbers you are still left with larger infinity of irrational ones. It is far easier, though, to follow the route of the pioneers and construct the number system, starting with the simple counting numbers and adding new types of number as they become necessary and then exploring as fully as possible the properties of these new types of numbers. That way you prove the rationals have the same cardinality as the integers. http://math.stackexchange.com/questions/12167/the-set-of-rationals-has-the-same-cardinality-as-the-set-of-integers.

Of the choices listed I would suggest that Environmental Science offers the least 'learning load' for those with small backgrounds in the Life Sciences. The syllabus seems to allow someone with an analytical mind to discuss topics logically. The other syllabuses on offer seem to lead on to further work in their respective areas and require a good deal of learning of facts, terminology etc that will frankly be baggage to you.

First terminology again, sorry. An 'interval' is a technical term for a set of real numbers which contains every point between the end points. We don't get to pick and choose. For other technical reasons not of interest here we should only apply 'interval' to the real numbers. Another technical point of interest is that intervals come in two types. Closed intervals include their end points, open intervals do not. This may become useful and relevant if this discussion develops. So The set of real numbers [math]\left\{ {x:1 \le x \le 11,x \in R} \right\}[/math] Is an closed interval between 1 and 11 and the set [math]\left\{ {x:1 < x < 11,x \in R} \right\}[/math] Is a open interval between 1 and 11. However the sets [math]\left\{ {x:1 \le x \le 11,x \in N} \right\}[/math] and [math]\left\{ {x:1 \le x \le 11,x \in Q} \right\}[/math] are not intervals since they refer to integers (denoted N) and to rational numbers denoted Q). Both these sets are subsets of the real number interval. The set of integers is finite , the set of rationals is infinite, and has a lower cardinality than the interval set in R. This is just one way to approach the meaning of numbers, however. We realise that we need different types of number to satisfy different equations. For instance consider the equation 2x = 5 There is no integer that satisfies this equation. In fact there is an infinity of such equations, one for each odd number.

This is why I started with the smallest infinity, that of the whole numbers or integers. An infinite set, chosen from the reals, (ie an infinite subset of the reals) can be chosen to only include the whole numbers. This has already be shown to be of smaller size than the set of reals itself.

You may not have realised it but they are all over the place. Take a cube [math]\delta x,\delta y,\delta z[/math] in continuum mechanics. You can consider the electric/magnetic/fluid/heat flux through the cube and use the engineer's most popular equation Input = output plus accumulation to derive all sorts of useful stuff. Or you can look at each face and note that there is (could be) a shear and normal stress associated with each face. Flux or stress is something per unit area [math]\delta x\delta y[/math] etc and it makes no sense to have zero area in the denominator of a definition. But for engineering purposes you still need a table of the stress at a point, the flow at a point etc.

Since wtf has posted such an able discussion of numbers let us return to the other part of your question - that of infinitesimals. Infinitesimals are not numbers and are not really used these days by pure mathematicians. They are, however, of immense use in applied maths where numbers are given significance in some physical sense and called quantities. Infinitesimals are quantities that are finite but small compared to the main bulk of the property or quantity being considered. We can conceive of a sequence of these getting smaller and smaller and calculate what is known as a 'limit' for some compound property ( a quantity made up of more than one infinitesimal) which we regards as the 'value' of that property at a point. A good example is density which is the ratio of mass to volume. We call the density at a point the limit of this ratio as we shrink the infinitesimals of mass and volume. Obviously they can never actually be allowed to reach zero or we would be trying to divide by zero. At one time the differential calculus was predicated upon such a ratio but we adopt a different approach today.

This is where you need to be precise. You are asking if your set B contains more numbers than your set A. In order to approach this question you need to precisely define. What you mean by more than? Which in turn begs the question what do you mean by "How many numbers does either set have"? Edit Nice simple answer wtf.+1

Well I had no idea that English was not your first language, yours is very good. So perhaps that is why you misread my purpose? Nor do I have any idea of your level of knowledge in mathematics. All I have to go on is your use of words which (no offence) is imprecise. Precision is very important in mathematics and I would have thought you might be interested in knowing the correct words. I have put a lot of effort into avoiding the imprecise statements that are often made in answering questions about this subject. But I did make a start on answering your question, I just did not finish it since there is a lot to take in. So my question at the end was designed to find out if you had any problems with the necessary background. So I'm glad to hear that you have followed it all.

Infinitesimals are not numbers.

Please use the word set for your collection of numbers - this is the accepted correct term. Older names are aggregate or collection or sometimes class. A 'group' is a very important particular type of set in mathematics and only some collections (sets) of numbers form groups. OK so the set of whole numbers has nothing between each member of the set. However the set is 'open ended' (has no beginning or end) as we can always add or subtract another 1 from any proposed first or last number. We say that the set runs from negative infinity to positive infinity, although infinity itself is not a member of the set. That is infinity is not a whole number. We use the positive whole numbers for counting (posh math word - enumeration) things. In particular we can (try to) count the number of members of any given set. 1,2,3,4.. etc We use this 'count' to measure the size of a set and compare the size of one set with another. This works just fine for the number of members in a finite set. So the set {1,3,5,7} which has 4 members is bigger than the set {1.3.5} which has only 3. But we have already noted that there is no end to the process of adding 1 to the count. We never actually reach 'infinity' So the number of members in the set of positive whole numbers is not a positive whole number. It is in fact a (the first) transfinite number or 'infinity'. Another way of looking at counting is the idea that we are putting the members of the counted set into one-to-one correspondence with the positive whole numbers. 1 2 3 4 W X Y Z So in considering just the set of positive whole numbers we have found an infinity. But we haven't included any of the fractional numbers in between, let alone those that can't be expressed as fractions. So we are forced to the conclusion that more transfinite numbers are needed to place infinite sets into one-to-one correspondence. How are you doing so far?

Am still waiting for the aforementioned comestibles to emerge from the roast spud server, so Dave please adjust the third one. Here is a scream shot. Oh dear can't upload this one, error 500 received.

Both are right! The Biot-Savart page is simply more complete in that it takes into account the medium via [math]{\mu _r}[/math] For air and many purposes such as the space between sub atomic particles [math]{\mu _r}[/math] is so close to 1 that we often ignore it. That is what is done in your second link from hyperphysics. A more satisfactory way to deal with this is to use [math]\mu = {\mu _0}{\mu _r}[/math] Then there is no ambiguity. Incidentally your hyperphysics link has some good pictures of the circular magnetic field lines around straight conductors carrying current. Look carefully at these and think about your other thread on crossed wires.

Since you posted this whilst I was busy polishing my own post, please see the last version.

I don't see a correct question asked either. So I thought sensei's answer pretty imaginative and thoughtful. Since we don't know where A, B, C or D are or which way L1 or L2 are flowing, and our answer options don't include a no effect option, suck it and see is the only sensible answer.

Said it all, Sensei +1

We want our inner product to be a map from a vector space to the Real numbers ie a scalar. Consider any complex number a+ib (a + ib)2 = (a + ib) (a + ib) = a2 + 2iab - b2 Which is another complex number ie another vector. To get a real number we must multiply by the complex conjugate (a + ib) (a - ib) = a2 + iab - iab + b2 = a2 + b2 This will always be a positive number we can take the real square root of. Don't forget that vectors here are Euclidian vectors and the inner product represents the Euclidian norm with is defined as the square root of the sum of the products of the coordinates.

Since you clearly wish to simply mess about without making a serious proposition I bid you and your thread adieu.

Does something require embodiment to be real and part of reality? Consider this example. I pick up something and pull on it. It stretches some. Then I let go. It returns to its original size. I pull on it some more (a bit harder this time). It stretches further and again returns to original condition upon release. This is called elasticity. So does elasticity exist? It elasticity real? So Can I weigh it? Can I see it? Can I smell, taste, feel, etc it? Well actually none of these. Yet I maintain that elasticity is real and that any system of definition that is unable to cope with this simple example of abstract existence is seriously deficient. We deal with many far more subtle effects in our encounters with reality than this so we need a sophisticated definition to cope with with all the vagaries and ramifications.

You that you have not yet answered the question I asked several times in this thread, most recently in post#45. Can I remind you that it is an express rule in the Speculations section that you do this> So before rushing onwards, taking nobody with you, please pause and give me a satisfactory answer. What does this thread have to do with its title about Newtonian Physics?

Do you also consider phlogiston real, just because famous people for at least 1500 years proclaimed it so?