studiot

Your original question is too general to supply a specific answer and there is no universal answer available. So here are some possible lines of attack. Most functions defined by integrals have alternative definitions which may be more easily invertible. Every integral has a counterpart differential equation so using that may be fruitful in some cases. But there are pitfalls to inversion for instance the sine function maps the entire number line to the interval (-1, 1). So there will be no inverse at all for numbers outside this range. Bessel functions cannot be expressed in terms of simpler functions so must be inverted from tables or some other numeric procedure.

Hilbert Sequence Space (symbol l2) is an infinite dimensional vector space with a defined inner product. Rn and Cn are not infinite dimensional since n is defined as a number and therefore cannot be infinite. https://en.wikipedia.org/wiki/Hilbert_space Edit note you will need to scroll down to the section about Hilbert sequence space, the first part of the Wiki article is about finite dimensional Hilbert spaces.

OK thank you for that information. The teaching of electricity and magnetism nearly always runs along the following path these days. A force on a suitable body is noted and an electric/magnetic effect deduced to account for it. This effect is defined in terms of the force produced and the position in space and is a vector quantity. The work done as the body moves around is calculated and related to the position to define a potential field. Sometimes a second vector quantity is defined as a vector derived from the first one. Either E and B can be defined first, with D and H as the derived vectors or the other way round (which is the older method). But often the significance of the second vector is not presented, hence your question. So consider this statement in the light of the following. You need two vectors to have a dot (scalar) product and energy is a scalar. Starting with a mechanical example we have Stored Mechanical Energy per unit volume = 1/2 stress x strain Stored Electrical Energy per unit volume = 1/2 E x D Stored Magnetic Energy per unit volume = 1/2 B x H edit (note these are all dot products not cross products) Are you beginning to see a pattern ? There are differences in the nature of stress, E and B but scalar multiplied by another suitable vector they yield the energy stored in the system. There are actually many more examples that could be listed and they all spring from a common root in continuum mechanics. How are we doing?

How many Royal Colleges do you have in the US of A? A school leaver is someone who didn't die waiting for the bell.

Enjoy Life to mean life for GPs GPs will no longer be allowed to leave the profession, according to plans drawn up by the Department of Health. An imminent change of law will make it illegal to stop being a GP. The move is expected to eliminate the current shortage of 5000 GPs within a few years. The news follows a call from the Royal College of General Practitioners for urgent action to halt the exodus from general practice. Under the proposals expected to be announced by the government next week, the GP contract will become binding on both parties. A government spokesman said: “GPs have long enjoyed the protection of a contract for life, but they were free to end it at any time by quitting the profession, moving abroad or choosing to retire. These options will no longer be available. Life must mean life.” From now on, GPs will be expected to practice until they die, with an option to work on after death open to those who wish to take national clinical posts, leadership roles in professional bodies or jobs with the Care Quality Commission. Acknowledging that GPs would be unable to retire under the new rules, the government said that the benefits of stopping work in later life were “widely exaggerated”. “All the evidence suggests that people who retire are more like to die before those who are still in work. On that basis forcing GPs to work until they drop may actually improve their life expectancy,” the spokesman said. Ending GP retirements would have other benefits, including simplified funding and contractual arrangements for general practice, and a modest saving in pension contributions. GPs will be expected to surrender their passports to remove the temptation to move overseas to work. They would also no longer be able to take holidays abroad, an objection dismissed by the Department of Health as “largely academic”. The government accepts that not all GPs will welcome the changes, but said it had run out of options for ending the recruitment crisis. Incentive schemes giving GPs the opportunity to work longer hours, collaborate in more meaningful ways or meet helpful people with clipboards have had limited success. Initiatives to recruit overseas doctors have also suffered setbacks. One ended in chaos earlier this week after an undercover team of NHS England recruitment officers tried to snatch an American doctor from a United Airlines flight, provoking a storm of criticism on social media. The government says it is still considering other ways to “ease the supply” of new recruits, including a compulsory national service style scheme under which all school-leavers would work as a GP for two years. “Not only would this be character building for the young people concerned, those who discovered an aptitude for clinical practice might choose to go on to get proper qualifications and stay in the profession,” said the spokesman.

Does this question have anything to do with the original?

+1 for good help with this homework/coursework question. @the etc Please note the rules for the special homework section. http://www.scienceforums.net/topic/75772-read-this-before-posting-in-homework-help/

The first was by Heaviside, the originator of the operational calculus and Laplace transform. However he was not a pure mathematician and could not prove his new methods in pure maths so he was ridiculed by pure mathematicians. His answer was basically it works when other methods don't and I need answers. The quote was to the Royal Society at a dinner where he was a guest speaker. The second quote was from Lord Samuel essays in Physics. Pearls of wisdom can also be found in the most unlikely of places. There is no reason why Poet's should not create some. Coleridge wrote this to his brother Some scientific wisdom can even be found in the Bible The Book of Ecclesiastes offers this and the Book of Kings offers a respectable (for the time) value for pi.

That refers to by far the most rational, although by no means perfect, post I have seen from madmac so I have given +1. Such a welcome development deserves encouragement.

Both those quotations had deeper implications, did you recognise either?

Have you studied electric fields? The relationship between B and H is very similar to the relationship between E and D. In many situations one is a constant times the other but in non-isotropic media they may point in different directions (they are all vectors). If you let me know whether you understand about E and D I will explain further.

I quite like the practice of heading chapters with a pithy and perhaps witty quotation. One of my favorites is Another good one is

Two thoughts about the replies here. The OP is talking about A levels, which are intended for 16 - 18 year olds. Many of the replies are about maths for a much younger age group. Several have offered problem solving. I like problem solving and thinking for myself, but many do not. They prefer to be told what to do. What do you think they would make of being forced to problem solve? Of course there is the question "Is the mantra maths, maths and more maths is good education for everyone" actually true? Or are we boring many with subject matter they don't appreciate and will never need?

In 1967 the A level exam was a privilege. Approximately 15% of pupils in any one year took A levels at that time. I don't have reliable data on the % of these who took maths, but it was one of about 15 popular subjects, but by no means the most popular so the % would have been around 1%. About 3% of pupils in those days went on to become students at University.

Welcome Martin, but I completely failed to understand the point of your OP. No one is forced to take any A level subject, let alone A level Maths. It is elective and what's more the number of A levels a student can take during 6th form is limited. So why did you take A level Maths?

Do we not already have an alternative and dare I say better classification. Subjective and Objective? Properties including gravity would fall into the subjective category. The great steop forward due to Einstein was to show that many so called properties depend upin the observer and no one observer's view is preferable. Maths would fall into the objective category, but Russell, Godel and others showed that there are limits to this. Maths is certainly not 'absolute', partly as imatfaal showed and partly due to Godels theorems.

Yes we have evidence that energy can do but also evidence that it can come in any value, depending upon circumstances. Does this affect you question? Professor Majid of London University edited a recent Cambridge University book on this very subject, with many famous contributors. On Space and Time https://www.amazon.co.uk/d/Books/Space-Time-Canto-Classics-Shahn-Majid/1107641683

To address the question part of your OP, here is a paraphrasing of a long section by William Berkson on this very subject. Up until Einstein physical substances (bodies) were regarded as having the following characteristics. 1) Definite, unique location in space and time. 2) Identifiable properties, uniquely specifiable that are always present with the body. Generally these properties are conserved in during changes in space and/or time. The notion of substance was incorporated into the scientific approach; scientific development was aimed at deducing all laws in terms of the motion of the substance. These Laws were regarded as the basic laws of the Universe. Einstein denied the existence of physical substance in this role or meaning. He thought that we cannot uniquely assign to any portion of matter or the field the properties that were traditionally regarded as substantial. In different frames they will have different values, according to the Lorenz transformations and Einstein asserted that we cannot regard any of these different values as the real one. All are equally real. For instance the mass of an object cannot be uniquely specified as in different frames an object will 'have' different masses and none can be singled out as the real mass. Similarly the same applies to the space occupied by that body; the dimensions of a body are different in different frames. The times associated with different parts of an extensive body will also be different so we cannot even view extension as a substantial property. Furthermore these arguments also apply to any aether we propose, dispelling the validy of an aether. Properties formerly and subsequently regarded as insubstantial such as the speed of light are the same in all coordinate systems. I would recommend Berkson's book as offering some clear and well researched and thought out insights, both philosophical and scientific, although there is some wading through to be done as well. There are many extracts from published writings, papers and lectures of Faraday, Maxwell, Lorenz, Einstein and a host of others. Fields of force A development of a World View from Faraday to Einstein. (actually he goes a bit back before Faraday and forward beyond Einstein)

We have had several students reports heavy Easter workloads It's all a dastardly plot by the establishment to keep them from enjoying that wonderful weather we are currently having. Rest assured there are plenty of competent responders here for your field. It's good to try to develop a relationship as folks are more willing to help when they know you a bit. A tip Wherever you go for help and advice (the web, college, books...) always establish the engineering conventions in use at the outset. There are a number of contradictory conventions so find out quickly to avoid cross purposes. There used to be a non profit teaching site and forum called allaboutcircuits.com They were very friendly and I had many good conversations there. But they have been taken over by commercial interests so there is always a hidden agenda there these days advertising certain products. About half of the excellent specialists left when the takeover happened, and I don't visit often now myself.

You really need to explain more about your question. Are you studying curve fitting? Do you have any expectation as to the form of the curve, I see that it goes up and then goes down again? Have you any thoughts on the curvature or the endpoints? When we try to fit a curve to data points, we have to decide in advance what is the maximum power of a collocating polynomial we are looking for and also whether we want this polynomial to match curvature as well as data values. Sometimes we use special curves such as splines to fit at the ends because we can't calculate the curvature at the endpoints as we don't know what the actual data is beyond our endpoints. A polynomial may fit quite well over the range but go wildly astray outside this. This is especially true of high order polynomials. Achieving curvature fit as well as best fit at the data points means that the data at some points has to be employed to obtain curvatures. So there is always a trade off between the use of the data points to achieve a closer fit at the data points and to get the right curvature. Of course there is also the statistical approach. That is to calculate the best average fit for a specific nominated curve. So which one is it to be here?

Have you heard of volumetric alarms? http://www.foxspyoutlet.com/security-equipment/smart-sensor-volumetric-alarm/

There is no such thing as absolute time. So there is no such thing as absolute 'running' of a clock. This is another way of saying or a consequence of the fact that every observer will see any clock running differently. You will not progress until you accept this basic fact.

The outward and return journeys cannot be symmetrical. Yes the travelling clock appears to run more slowly to the stay-at-home clock on both legs, but, On the outward leg travel is away from the base so light (or any other) signals take longer and longer to traverse the expanding distance between the two. On the return leg it is the other way round the separating distance is diminishing so signals arrive more and more quickly. This fact has nothing to do with Einstinian relativity, but must be factored in to any analysis.

Hello Mondie, it is a long time since we shared a thread. Anyway I am somewhat confused as to what you are seeking. Are you saying you understand the electrical/electronic engineering aspect from a point of view of connecting appropriate devices together, but not how the devices work and would like to understand how they work. Or do you need help with auto electrics as well? The main difference between auto electricity and domestic wiring is that domestic wiring is AC or alternating whilst autos run on DC or Direct current. It is true that modern autos use an alternating generator (alternator) rather than a direct generator (dynamo) but this is converted in a module mounted on or within the alternator. But yes the voltage is lower and the current higher for the same power requirement, but not otherwise. Modern vehicles also have a plethora of sensors (transducers) which measure/monitor all sorts of activity in the vehicle its mechanics and surroundings. Do you wish to know how these work, for example the lambda sensor in the cat converter? This is the province of control engineering. So tell us a bit more about your intended project so we can offer focused help and references.

The use of extremal quantifiers or qualifiers such as; none, all, never, always, perfect etc often leads to a paradox or other difficulty. Some examples I always lie. Every statement I make is false. Every statement containing extremals as noted can be disproved. I just wondered if you really meant to include something like these which work just as well with a finite set as an infinite one. For instance 'always' is a sort of infinity, but not for me since I have a finite life and can only make a finite number of statements in it. Otherwise If you really meant that all paradoxes are based on infinity there have already been plenty of counterexamples that disprove the claim.