A channel dedicated to hard maths problems such as difficult integrals with some hard STEP questions along the way. If you have a specific question you would like me to solve in a video then message me on instagram @jagosprivatelife
I just need to write more to make this point. If I want to watch a video about math I can add my own music. Do you not comprehend that some people are incapable of hearing human speech while music is playing? That the music takes over their attention? Why TF do you think you can make such a unilateral choice for all potential viewers of your video? You unbelievable a**h**e and f*** you for wasting my time having to write this comment. F***ing idiots, why do you always just do the same thing. Music is soso so so so easy. I can choose my own music if I want to
You differentiate wrt a, x constant. At the same step, you integrate wrt x. But if x is constant, then dx = 0. Therefore, the integral is also 0, which is not how you continue the derivation. Have I misunderstood something?
_"Have I misunderstood something?"_ Yes. Probably you mean the operation at 6:54. Diffentiating with respect to "a" and integrating with respect to "x" are two different steps. This sequence is reversed at 6:54 ("we're going to bring the derivative inside of the integral"). This reversal is allowed in many cases, that is, if the function in question has a sufficiently benign behaviour. You can construct weird cases where this does not work, so we have to take care.
Is it possible to get an analytic solution for arbitrary limits of integration i.e.: other than - to + infinity? I'm aware of numerical methods that converge quickly.
Terrific -- Clearn and natural and benign and adult manner and tone -- This 'style' -- (which cannot be just 'put on' for the occasion -- It comes from being completely comfortable with one's subject matter) -- is so important for the simple quality of 'effective communication' -- i.e. necessary for 'communication', as such. Puts the pupil at ease -- but also enables complicated ideas to be easily absorbed by a beginning pupil -- whether the pupil is learning mathematics or the staff of a corporation or voters going into an election or the troops going into battle. (As David Hilbert said -- the leading research mathematician of the XX century, "You don't really understand a concept until you are able to explain it to the layman.")
São Paulo (Brazil), January 5, 2015. Alvaro C. G. Gemignani PROBABILITY AND DETERMINISM: A CONCEPTION OF ETERNAL LIFE ACCORDING TO A PURELY MATERIALIST PERSPECTIVE The advances in physics, chemistry, biology and other sciences seem to have favored a materialist conception of the world where the great hopes of Christianity, unfortunately, did not fit. However, it is becoming increasingly evident that, in science, if we can be sure that we are wrong, we can never be sure of the truth with regard to a total conception of the universe. And, perhaps, scientific development itself may provide elements that invert the atheistic materialist tendency that characterized science. In this brief essay, we want to demonstrate how two respectable hypotheses (one microcosmic and the other macrocosmic) can be combined in order to lead to the surprising conclusion that it would be acceptable to consider the possibility of resurrection and eternal life from a purely materialistic perspective. So let's see. Vladimir Kéler included in his book "L'univers des physiciens" (The universe of physicists) an interesting speculation about the probable existence of the true atom, really indivisible. Otherwise, let us admit that there are a number of different types of indivisible particles that would form the fundamental structure of all matter in the universe. I. Chklovski, in his book "Univers, vie, raison" (Universe, life, reason), presents another interesting speculation, according to which the entire universe would be an enormous amount of matter in eternal oscillation. So, for all eternity, the universe would have moments of maximum expansion, then it would start to recede (under the effect of the "gravitational spring") and reach moments of maximum concentration (forming a "corpuscle" of singular density), and then reassumes its expansion movement, by means of a new "Big Bang". It is possible that the universe is now expanding, having created environments favorable to the emergence of life in several points, such as the Earth. In any case, the images that come to us are images of the past, as the speed of light is limited. From the oscillating universe hypothesis, it can be assumed that the amount of existing matter is finite or limited. If matter is limited, it can be deduced that the amount of indivisible particles that make up the universe, although immense, is also limited. In short, the universe would be in eternal oscillation, being composed of an enormous, but limited amount, of indivisible elementary particles. And, if so, considering the mathematical principles of probability, in this infinite number of oscillations the universe would always end up reassuming combinations identical to those of certain past oscillations, as in a constantly repeated draw where the same number drawn before ends up to be drawn again. In this way, any possible configuration of the universe will have existed and will exist an infinity of times. The current configuration of the universe, to which we belong, will have been repeated and will be repeated an infinite number of times, exactly as it now presents itself. It is evident that there could not be a memory of these repetitions, but we can admit them by reasoning. On the other hand, if at any moment over time the present is a consequence of the past, in the universe everything happens according to an eternal plan. If the universe is in fact eternally repeatable, it can be said that each person's life is eternal, even if intermittently. Furthermore, if we consider that with death the sensation of time disappears, it would be, for those who die, as if there were a "resurrection" right after death, without the memory of past repetitions. If we consider from the moral point of view, each weight on the conscience would be a suffering that would be repeated for all eternity (hell). The peace of mind and inner joy, that result from a dignified existence, would represent a happiness that would never end (paradise). Therefore, a substantially Christian lifestyle would be the best option. As can be seen, from a purely materialistic point of view, Christianity's hopes of resurrection and eternal life may even seem quite plausible, especially if we consider that reality is always richer than any theory that science can construct to know it. BIBLIOGRAPHY KÉLER, Vladimir. L'univers des physiciens. 2nd edition. Moscow: Éditions Mir, 1967. CHKLOVSKI, I. Univers, vie, raison. Moscow: Éditions de la Paix (1960s?). ALVES, Rubem. Philosophy of science (introduction to the game and its rules). 3rd edition. São Paulo: Editora Brasiliense, 1982.
At 19:57, you say f(t) = 0. Obviously as t goes to infinity, e^-(t^2) goes to zero. It appears to me that makes f(t) = improper integral of 0dx which is C, not zero. If I am missing something, please explain. Loved the video and subscribed.
Of course you usually cannot draw the "lim" under the integral, so lim [t→∞] ∫ f(t, x)dx will not be equal to ∫ lim [t→∞] f(t, x)dx in many cases. f(t, x) has to converge _uniformly_ to zero for all x to make this work. And even this might not be sufficient if the integral does not converge absolutely. However, absolute and uniform convergence can easily be confirmed in our present case.
One thing I never liked about this integration technique is that it is not obvious how to choose the auxiliary function. So it is difficult to teach to students, as they would just have to guess at the auxiliary function (or memorize examples) and hope for the best! Feynman has written that he learned the method from a 1926 math book by Frederick Woods (Advanced Calculus: A Course Arranged with Special Reference to the Needs of Students of Applied Mathematics) that was given to him by his high school physics teacher. Perhaps there were enough examples in that text that Feynman knew a whole whack of sample integrals to solve with this method.
_"One thing I never liked about this integration technique is that it is not obvious how to choose the auxiliary function."_ Lol. One thing which is inherent to mathematics is the fact that there is no known technique or algorithm to find a proof, or, appropriately, a disproof, of any mathematical statement. Moreover, it was shown (by Goedel, Turing and others) that such an algorithm can by no means exist.
Interesting, though seems long winded as presented here. It is understandable. Makes me wonder how this approach was discovered. I'll check out Leibnitz and other methods.
I found this to be easier by using the Laplace transform. First, I distributed the exp(-x). = ∫(0, ∞) exp(-x)/x - exp(-x(a+1))/x dx I then just took the coefficients of -x to be the value of s and did both Laplace transforms. This yielded the answer of ln(a+1).
This method is NOT called "Feynman Integration" , IT'S CALLED *Leibniz Integral Rule* . Gottfried Leibniz DISCOVERED THE RULE, Feynman POPULARISED IT. THIS IS Leibniz's technique, NOT FEYNMAN'S. GIVE THE CREDIT TO THE RIGHT PERSON FOR GOODNESS SAKE.
@@LactationMan Thanks -- (Mr "Uphold justice" may be 'autistic' -- They lack restraint -- Everything that departs from their habitual way of doing things is an affront.)
Hello, Thank you for this interesting solution. It is possible also to do the following: For a=0, the integral I(a)=0. Now, if you derive I(a) you get a very simple integral that gives you I'(a)=1/(1+a). This integration is also very simple and leads to your solution I(a)=ln(1+a). Thank you again for your very stimulating videos!
Right but it became popular due to Feynman using it, if I remember correctly he discovered it in a class textbook and couldn’t understand why no one was using it as it is very powerful for certain problems
that's how cults work. the cult identity is either attached to individuals being credited for mundane shit, or attached to complete nonsense. this is an example of the former in mathematics. an example of the latter in mathematics would be the claim that 1+1=2 is universally true despite the fact that fraction addition, polynomials, unit conversions, etc. all exist.
@@sumdumbmick (If you de-press the 'Shift' key -- as you depress an alphabetic character on your keyboard -- you will be able to display a 'block capital'. If you put capitals a the beginning of your sentences -- the man reading your post will be more likely to read the entirety of what you have posted -- because they will be less likely to assume that you are not a naive adolescent without a clue about life.)
