Country Boy

I thought this had already been answered. x ranging from 0 to 4, and y, for each x, ranging from sqrt{x} to 2 is the same as y ranging from 0 to 2 and, for each y, x ranging from y^2 to 4. That is, this integral is the same as J= \int_0^2\int_{y^2}^4 (1+ y^2 cos(x\sqrt{y})dx dy To do that, first let u= x\sqrt{y} so that du= \sqrt{y}dx When x= y^2, u= y^2\sqrt{y}= y^{3/2} and when x= 4, u= 4\sqrt{y}= 4y^{1/2}. The integral becomes (1/sqrt(y))\int_0^2\int_{y^{3/2}}^{4y^{1/2}} 1+ y^2cos(u)du dy

That would imply that it is at least as hard to talk directly about death as it is to talk about sex. That is not my experience! I think you need to consider the enormous emotional impact of sex.

"Time slows down" for an object in motion, relative to another object. Acceleration is not necessary nor are "g- forces". Yes, a sub-atomic particle would experience "slower time" compared to the "laboratory" time, but as DrP says, that would be a negligible effect. The "aging of a radioactive isotope"" has been demonstrated by moving the isotope at high speeds in a cyclotron.

1200kg is (1200)(9.81)= 11772 N. The maximum force on the cable can be 16000 N which leaves 16000- 11772= 4228 N for the passengers. That corresponds to 4228/9.81= 431 kg. That's close enough to your 435 kg. We probably rounded differently. You are correct that the elevator moving at a constant speed does not change that. Accelerating upward at 1.4 m/s^2 adds an additional force of 1.4m where m is the mass. The tension on the cable due to the elevator car itself is (1200)(9.81+ 1.4)= (1200)(11.21)= 1200= 13452 N. That leaves 14000- 13452= 548 N for the passengers. That, including the acceleration, is 549/(9.81+ 1.4)= 11.21 kg.

If f is not a function of any variable, then it makes no sense to talk about its derivative. If this is a "deterministic problem', which it would be if you are using Newtonian physics as in you first post, x will have a specific value for any specific time. In that case, x is a function of the time. Perhaps you are not clear on what the word "function" means.

Do you remember the famous picture of "Washington Crossing the Delaware"? George Washington was probably not standing up in the boat but the ice floes were there. Earth, in the late 1700s was in the middle of the "little ice age" and has been warming ever since. How much pollution or human activity has influenced that, and how much we can do about it can be subject to debate but the fact of warming is not.

For God's sake, don't ask us! Ask your dissertation director. It is the people who will be reviewing your dissertation that count! Generally speaking, while a master's thesis can be a "review" of other work, a doctoral dissertation must be original work- but that original work can extend previous work by others.

Well, yes, 2600 years is "more than 500 years" but that seems a strange way of putting! Not a very good approximation!

The OP is assuming that you cannot have a time interval smaller than a Planck time: "there is limit in our universe to the smallest amount of time which is Plank Time, which is the smallest unit of time possible in the universe made from natural units.' That is, of course, not true. Planck units are simply the units in which all the basic physical constants are "1". Essentially a Planck unit of time is the time a photon would take to move the width of an electron. There is no claim that there cannot be smaller distances, or times, or masses.

b+ c+ d+ e+ ...= (a+ b+ c+ d+ e+ ...)- a. For m> 1, sum_{i= m}^infinity a_i is equal to \sum_{I= 1}^infiity a_i - \sum_{i= 1}^{m- 1} a_i. The second sum is just a finite sum so is easy. In particular. sum_{i= 0}^infinity 1/2^i= 2 so sum_{i=1}^\infinity= 2- 1= 1, sum_{i= 2}^infinity 1/2^i= 2- 1- 1/2= 1/2, etc.

Often people who do not understand mathematics look at a math text and it looks like non-sense. Unfortunately, a few then decide that if they write non-sense, it is mathematics!

Sorry, but this still makes no sense. You say, in your first post, that "Definitions abound on the net, and i think i have a clear picture of them" but what you write doesn't show that. A "field" (in the mathematical sense) is a collection of objects, together with two operations, which we can think of as "+" (addition) and "*" (multiplication) such that Addition is associative and commutative, there is an "additive identity", "0", and every member has an "additive inverse". Multiplication is associative and commutative, distributes with addition, there is a "multiplicative identity", 1, and every member, except 0, has a multiplicative inverse. The set of all rational numbers, the set of all real numbers, and the set of all complex numbers are fields. A single polynomial is clearly NOT a "field" because it is just one object. Nor do the coefficients of a polynomial form a field- the sum or product of coefficients of a given polynomial are not necessarily coefficients of that polynomial. Of course if you are referring to a polynomial with real coefficients, those coefficients are from a field. The set of all polynomials also does not form a field because division of polynomials is not generally defined. Not all non-zero polynomials have "multiplicative inverses".

Will there be pink unicorns in these universes?

Good point. I should have said "USA" rather than just "American". On another point, I had a very good friend who objected to a person saying "American" when they meant "of the United States". I argued that "American" is a perfectly valid form because "The United States of America" is the only country that has the word "America" in its name.

I wasn't aware the Vicente Fox was a native-born American citizen!

Depending on the functions, one way might converge faster than the other but, assuming they converge, they will converge to the same result.

"dx/dθ * dx" is meaningless. Did you intend "dx/dθ * dθ"?

There is information left out of "problem z". Are we to assume that each leg, A to B and B to C, starts initial speed 0 or does A to B start with initial speed 0 and B to C with the speed with which we arrive at B? If we start a leg with speed [tex]v_0[/tex] and have constant (or average) acceleration a then in time t we will have gone a distance [tex](a/2)t^2+ v_0t[/tex]. Assuming that we started from A to B at initial speed 0 and it takes 1.5 hours at constant (average) acceleration [tex]11 m/s^2[/tex] then the distance from A to B is (5.5)(5400)+ 0(5400)= 29700 meters or 29.7 kilometers. The final speed would be 11(5400)= 59400 m/s. Now, do we start the leg from B to C at that speed or do we stop and start over again?

That's very odd. I googled "distance from heart to brain" and immediately got "In most people the brain is only 12 to 14 inches away from the heart, depending upon how tall they are". (There was also a very strange thing about the distance from the town of "Heart" in the United States to the town of "Brain" in France!)

To do a "Runge- Kutta" solver with more than function, do simultaneous R-K solvers on each function, at each step using all the function values that you have determined at h/2 and h.

We've even seen it happen! In July, 1992, comet Shoemaker-Levy 9 came sufficiently close to the sun that "tidal forces" (essentially what you call "warping space") broke it apart.

First he goes forward to 5 m and then back to 2m. That takes 8 seconds. Then he goes forward to 7m and back to 4m. That also takes 8 s so a total of 16 seconds. Then he goes forward to 9m and back to 6 m. That is now a total of 24 seconds. Finally, he goes forward to 11m and falls into the hole. That is a total of 29 seconds.

As strange said "Most objects don't have a resonance frequency." A resonance frequency is typically connected to a crystalline structure so that every part has the same frequency. What makes you think that a planet, made of many different substances, has a single resonance frequency.

The entire train has mass 8000+ 5(2000)= 18000 kg. Since it is accelerating at 2 m/s^2, the force on the entire train is 2(18000)= 36000 N. The force necessary to accelerate the engine alone would be 2(8000)= 16000 N so the cars must be exerting a retarding force on the engine of 36000- 16000= 20000 N. That, of course, means that the engine is pulling the cars with a force of 20000N. The force necessary to pull only the first car at that speed is 2(2000)= 4000 N so the first car must be pulling the second car with force 20000- 4000= 16000 N. Similarly, that second car must be pulling the third car with force 16000- 4000= 12000 N, the third car must be pulling the fourth car with force 12000- 8000= 4000 N and the fourth car must be pulling the fifth car with force 8000- 4000= 4000 N.

Actually, there are two bounded regions "between the circle x^2+ y^2= 1 and the graph y= |x|", one above the absolute value graph and below the circle, the other above the circle and below the absolute value graph. As Bender said, the first area is a quarter the area of the entire circle, pi/4, the second is the other three quarters the area of the circle, 3\pi/4.