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  1. E

    Why must VTOL engines be larger than normal engines?

    Apparently my idea actually works in theory. Or at least, aircraft designers at Georgia Tech have designed a STOL fixed-wing aircraft based on the same principle, which is that the apparent airspeed over the wing is what generates lift. They're calling it a "blown wing" design...
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    Why must VTOL engines be larger than normal engines?

    How does the foil know that the air which is moving over it is compressed air and not "free" air? It just sees air moving over it. We already agree that a foil in a stationary tunnel experiences lift. What's the difference here? From the frame of reference of the foil there is no difference...
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    Why must VTOL engines be larger than normal engines?

    I understand that my question leaves this interpretation open, but this is what I had in mind: https://docs.google.com/drawings/d/1WhC7r3491KIbeSNgewC9NnJqyDCJpwl46nwnKOSw6WY/edit?usp=sharing The air would be deflected out of the back of the craft, not touching the walls. The difference...
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    Why must VTOL engines be larger than normal engines?

    Could an aircraft achieve lift by blowing a fast stream of air over an internal fixed wing? I.e., by having an onboard wind tunnel with a wing suspended inside? Couldn't that be used to achieve fixed-wing VTOL without a T:W ratio greater than one?
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    Why must VTOL engines be larger than normal engines?

    Ok, this is clearly the source of my confusion. And, I suppose I still don't understand how that is true. It feels like violates a conservation law, but actually I see now that it doesn't. (Though I don't understand how it's true, I can see that it's not impossible). That's true! I didn't...
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    Why must VTOL engines be larger than normal engines?

    Let me put this another way. I don't understand how a plane can take off when its thrust to weight ratio is less than one. Yet, many planes do so. The 747-400 has a T/W of 0.27. Can someone explain how a plane can take off (and a fortiori stay aloft) when its T/W is less than one?
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    Why must VTOL engines be larger than normal engines?

    I understand that this is the assertion. I am hoping someone can come in here with a more convincing analysis. At the moment that liftoff occurs, the lift on the airfoils is 9.8x N. Thus the drag is > 9.8x N. If the engine is not producing 9.8x N of thrust, then the plane slows down and the...
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    Why must VTOL engines be larger than normal engines?

    I am studying to be a pilot and something I've read in the book confuses me. It is that Vertical Takeoff and Landing (VTOL) engines need to be heaver than their corresponding traditional takeoff/landing counterparts. Wikipedia says the same thing here. I am going to interpret "larger" and...
  9. E

    Choosing two numbers uniformly

    Funny story, but |A| is 60, not 55. Proof by enumeration: Prelude> length [ (i,j) | i <- [0..10], j <- [0..10], i /= j, (i + j) <= 10] 60 Prelude> *[ (i,j) | i <- [0..10], j <- [0..10], i /= j, (i + j) <= 10]...
  10. E

    Choosing two numbers uniformly

    Ohh, thanks. I see his thought process now in his answer. Anyway, we are right, and I just confirmed that in Haskell. So I'll email him with the revision. Haskell program: Prelude> length [ (i,j) | i <- [0..10], j <- [0..10], i /= j, (i <= 5 || j <= 5)] 90
  11. E

    Choosing two numbers uniformly

    This is a solved problem and I am having a hard time working through the answer. Question . Choose two numbers uniformly but without replacement in {0,1,...,10}. What is the probability that the sum is less than or equal to 10 given that the smallest is less than or equal to 5? Answer...
  12. E

    Help with some unfamiliar set notation

    Thanks for the link, Tim! That fully clears it up for me. One of the biggest frustrations with mathematics self-study is that it math isn't a linear progression of knowledge, but rather a web of knowledge, and whatever thread I pursue is constantly touching other threads which are still in the...
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    Help with some unfamiliar set notation

    Here's another. Now, what does 2^{\Omega} mean in this context? Omega is a set; how do you raise an integer by a set?
  14. E

    Help with some unfamiliar set notation

    Oh, I think I get it. It is fair to say that the big cup in \bigcup _{n}A_{n} should read like \sum_{n}A_{n} in big sigma notation?
  15. E

    Quotient Rule for Derivatives

    There are two different concepts. The quotient rule is how to find the derivative of a function by decomposing it into the quotient of two smaller functions, f and g, each which you independently know how to differentiate. The quotient rule yields a formula for a derivative. L'Hopital's rule...
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    Help with some unfamiliar set notation

    I am reading http://walrandpc.eecs.berkeley.edu/126notes.pdf on the theory of random processes. The authors are making use of some unfamiliar notation early on, and I don't want to move on without understanding their formalisms. First one is the := operator and a union operator that looks like...
  17. E

    Area of a circle by integration

    Above I posted what I believe is the correct derivation for the area of a circle, which was the question I posted here. And as for the original question I posed, I suppose I haven't quite put it together, but it's basically there. But I believe the answer is this: ##\left(\text{Area of the...
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    Area of a circle by integration

    Yes, thanks for pointing it out. The only help I need is someone else to check the work and say whether it looks right or not, since I think I have posted the answer :). I also wanted to record this posterity, in case anyone else wanted to find the answer to this. It's not really easy to google...
  19. E

    Area of a circle by integration

    Since the time I posted this and today, I've done maybe 150 more integrals, and I'm ready to tackle this. First the area of a circle centered on the origin is ##A = -4 \int_{0}^{r}\ \sqrt{r^{2}-x^{2}} dx \\ = -4 \int_{0}^{r} r \sqrt{1 - \frac{x^{2}}{r^{2}}} dx \\ = -4r\int_{0}^{r} \sqrt{1 -...
  20. E

    [Spivak Calculus, Ch. 5 P. 9] Showing equality of two limits

    Edit: actually, not quite sure what he's asking for here.
  21. E

    Area of a circle by integration

    Look below that, where I am back to algebra on ##f(x)##, no dx in sight.
  22. E

    Area of a circle by integration

    There was no dx in anything i wrote. I was simply concerning myself with writing true expressions for y. My ultimate plan was to integrate one of those expressions, but as you can see, I hadn't gotten that far. I have (now) seen the examples online of integrating my original function ##y = f(x)...
  23. E

    Area of a circle by integration

    I know what's wrong with my formula for ## f(\theta)##: ##f(\theta) = y##, but it should equal r. I need to find an expression for r in terms of theta.
  24. E

    Area of a circle by integration

    I have ##f(\theta)## and I wish to integrate with respect to ##\theta##. I'm sorry: I'm not saying that your input isn't useful, but I'm not sure how to use it.
  25. E

    Area of a circle by integration

    My curiosity was piqued by another poster who's trying to find the area of a lune using calculus. I wanted to do this now. So I don't highjack his thread, I'm making a new post about it. First, the geometric picture of a lune that constitutes my basic plan of attack. First I want to solve a...
  26. E

    Find the area of the lune formed (Using calc please)

    Area of circles by integration Whoops, meant to make a new thread.
  27. E

    Derivative of a difficult integral

    Though I am convinced by Kurtz's answer, I am having a hard time seeing where I went wrong in my own reasoning. Is my function composition wrong? Does ##F(x) ≠ (f \circ G \circ H)(x)##?
  28. E

    Derivative of a difficult integral

    Ok guys, I struggled through that one and I think I understand it now. To test my knowledge, I've attempted another one, which isn't solved. ##F(x) = \sin \left ( \int_{0}^{x} \sin \left ( \int_{0}^{y} \sin^{3} t dt \right ) dy \right )## So now I will try another composition. I will use...
  29. E

    Derivative of a difficult integral

    Thanks. I think I get it. Because we're integrating over ##dy##, that means we're plugging in ##x## for y when we integrate.
  30. E

    Derivative of a difficult integral

    Thanks for your reply. When you put it that way, we have F'(x) = f(y) = \int_{8}^{y} \frac{1}{1+t^{2}+sin^{2}t}dt But the given answer has \int_{8}^{x} \frac{1}{1+t^{2}+sin^{2}t}dt. How can we just change the integrand variable like that?
  31. E

    Derivative of a difficult integral

    Spivak, 14.1.iii: Derivative of F, where I have the FTC. The answer is also given, But I don't know how to find it. First I want to try a function decomposition: and from the Chain Rule: But now I'm stuck. For I do not know an expression for p' or q'. And worse, r' doesn't show up...
  32. E

    Closest approach of a parabola to a point, using lagrange multipliers

    thanks everyone. I took a few days off from math to clear my head, but I will revisit this result in the morning and try to convince myself of this result in linear algebra
  33. E

    Counting multiplicities of a particle lattice

    This is from Molecular Driving Forces, 2nd Ed. 5.3: Calculating the entropy of mixing. Consider a lattice with N sites and n green particles, and another lattice with M sites and m red particles. These lattices cannot exchange particles. This is state A. (a) What is the total number of...
  34. E

    Closest approach of a parabola to a point, using lagrange multipliers

    Thanks, Vela. Taking your word on it that (73-k)(52-k) - 36*36 = 0, I was able to solve for k and then back substitute for x and y. Is this result true in general? If I have a matrix equation equal to zero, does the equation only have non-trivial solutions if the determinant is zero?
  35. E

    Closest approach of a parabola to a point, using lagrange multipliers

    I have another related question. I am not sure if I should put it here or in a new thread, so I'll put it here. II.4.6: The equation 73x^{2} + 72xy + 52^{2} = 100 defines an ellipse which is centered at the origin, but has been rotated about it. Find the semiaxes of this ellipse by maximizing...
  36. E

    Closest approach of a parabola to a point, using lagrange multipliers

    Thank you both for your generous attention to my questions! I read both of your posts last night before bed, and again this morning over coffee. Gradually, the insight that the gradients must be parallel at the extrema is sinking into me. I am convinced of the intuitive argument now, and I just...
  37. E

    Closest approach of a parabola to a point, using lagrange multipliers

    I just tried to do the next problem in the chapter, which was also a minimum distance problem, and encountered the same thing with an imaginary root for y. I really hope someone can come in here and show me where I'm going wrong :).
  38. E

    Closest approach of a parabola to a point, using lagrange multipliers

    My justification for y ≠ 0 is false. We have from 0 = 4x - y^{2} that if y = 0, then x = 0. (0,0) could be the solution?
  39. E

    Closest approach of a parabola to a point, using lagrange multipliers

    How do I simplify \nabla[d^{2} + \lambda g]? Is \nabla[d^{2} + \lambda g] = \nabla d^{2} + \lambda \nabla g? If so, and if \nabla d^{2} + \lambda \nabla g = 0 is true for some \lambda \in \Re, then \nabla d^{2} - \lambda \nabla g = 0 must be true for some other number \gamma \in \Re...
  40. E

    Closest approach of a parabola to a point, using lagrange multipliers

    I'm sorry, Zondrina. I'm not saying that your graph doesn't help me, but I don't see how it does.
  41. E

    Closest approach of a parabola to a point, using lagrange multipliers

    Advanced Calculus of Several Variables, Edwards, problem II.4.1: Find the shortest distance from the point (1, 0) to a point of the parabola y^{2} = 4x. This is the Lagrange multipliers chapter. There might be another way to solve this, but the only way I'm interested in right now is the...
  42. E

    Closest approach of two skew lines in R3

    I'm stuck again. If expand the dot product, I get these equations: f_{1}(s_{1}) - g_{1}(t_{1}) + 2( f_{2}(s_{2})-g_{2}(t_{2}) ) - f_{3}(s_{3}) - g_{3}(t_{3}) = 0 \\ f_{1}(s_{1}) - g_{1}(t_{1}) + f_{2}(s_{2}) - g_{2}(t_{2}) + 2( f_{3}(s_{3}) - g_{3}(t_{3})) = 0 \\ Which is six...
  43. E

    Closest approach of two skew lines in R3

    Alright, thanks guys. I'll take it from here and see if I can finish it off. It should be easy now...
  44. E

    Closest approach of two skew lines in R3

    Well, the thing is, he hasn't defined the inner product in the problem statement. In this book so far, <> has been a generalization of what he calls the "usual inner product", which I understand is what most people just call the inner product, defined as x \bullet y = x_{1}y_{1} + x_{2}y_{2} +...
  45. E

    Closest approach of two skew lines in R3

    Hello all, and thanks again to all the help I've been getting with this book. This is a two part problem in Advanced Calculus of Several Variables, C. H. Edwards Jr. I have the first part and the second part should be easy, but I find I'm stumped. Since the second part builds on the solution of...
  46. E

    Isomorphisms have the same dimensions

    Thanks Dick, your remark that "Any element of I am L can be written as L(v) for some v in V" was what I was missing to prove that L(v_{1}), ... , L(v_{n}) spans W if v_{1}, ... , v_{n} spans V. Because any vector in v can be written as a_{1}v_{1} + ... a_{n}v_{n}, then any vector in W can be...
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