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  1. R

    Complex analysis antiderivative existence

    Homework Statement a) Does f(z)=1/z have an antiderivative over C/(0,0)? b) Does f(z)=(1/z)^n have an antiderivative over C/(0,0), n integer and not equal to 1. Homework Equations The Attempt at a Solution a) No. Integrating over C= the unit circle gives us 2*pi*i. So for at least...
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    Differential geometry acceleration as the sum of two vectors

    Basically the main question I have is if the only way to do this would be to explicitly calculate the expression for N(t).
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    Differential geometry acceleration as the sum of two vectors

    So I should able to write a(1) as a linear combination of T(1) and N(1), correct? But how do I get N(t)? Taking derivatives of T(t) is very messy and the professor said it didn't involve any long calculations. Furthermore we haven't learned about the binormal and normal in this section yet...
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    Differential geometry acceleration as the sum of two vectors

    Homework Statement a(t)=<1+t^2,4/t,8*(2-t)^(1/2)> Express the acceleration vector a''(1) as the sum of a vector parallel to a'(1) and a vector orthogonal to a'(1) Homework Equations The Attempt at a Solution I took the first two derivatives and calculated a'(t)=<2t, -4t^2...
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    Cauchy riemann equations and constant functions

    Ah yes, I meant unit circle. I was thinking the domain would need to have restrictions because I thought I proved it yet found a counterexample at the same time. But for the original problem: Let f=U + iV. We have |f|=c for some constant c. |z|^2=zz*, so we have |f|^2=c^2 which implies...
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    Cauchy riemann equations and constant functions

    But what about the function f(z)=z=x+iy? Isn't that function analytic on the say, unit disk with a constant modulus but f is not constant?
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    Cauchy riemann equations and constant functions

    Homework Statement Let f(z) be analytic on the set H. Let the modulus of f(z) be constant. Does f need be constant also? Explain. Homework Equations Cauchy riemann equations Hint: Prove If f and f* are both analytic on D, then f is constant. The Attempt at a Solution I think f need...
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    Linear equations, solution sets and inner products

    Homework Statement Let W be the subspace of R4 such that W is the solution set to the following system of equations: x1-4x2+2x3-x4=0 3x1-13x2+7x3-2x4=0 Let U be subspace of R4 such that U is the set of vectors in R4 such the inner product <u,w>=0 for every w in W. Find a 2 by 4...
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    Open and closed intervals and real numbers

    Yay! How exactly does uniqueness follow though? It seems like its trivial to prove.
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    Open and closed intervals and real numbers

    So far: Since S is bounded above and below, by Dedekind completeness there exists a supremum of S. Call it b. Again by dedekind completeness we can say there exists an infimum of S. Call it b. By definition of sup and inf, a<b. We are left to show a,b are unique and that S is exactly one...
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    Open and closed intervals and real numbers

    Good idea. Isn't it an axiom that if a nonempty subset of R has an upper bound, then it has a least upper bound/sup(S)?
  12. R

    Open and closed intervals and real numbers

    Homework Statement Show that: Let S be a subset of the real numbers such that S is bounded above and below and if some x and y are in S with x not equal to y, then all numbers between x and y are in S. then there exist unique numbers a and b in R with a<b such that S is one of the...
  13. R

    Additive functions, unions, and intersections.

    I am such an idiot. I tried it before representing the left hand side as a union of disjoint sets but for some reason I didn't bother manipulating the right hand side in the same way. Thanks a ton!
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    Additive functions, unions, and intersections.

    Homework Statement A function G:P--->R where R is the set of real numbers is additive provided G(X1 U X2)=G(X1)+G(X2) if X1, X2 are disjoint. Let S be a set, Let P be the power set of S. Suppose G is an additive function mapping P to R. Prove that if X1 and X2 are ARBITRARY(not necessarily...
  15. R

    Set theory and ordered pairs

    No. But why does it matter if it is in it or not?
  16. R

    Question about set theory and ordered pairs

    Hi I was reading through a textbook and I came across the set theoretic definition of an ordered pair (Kuratowski), where (x,y)={{x},{x,y}}, which apparently can be shortened to {x,{x,y}}. This seems to be the standard definition for an ordered pair in set theory so that we can determine...
  17. R

    Set theory and ordered pairs

    Homework Statement Determine whether or not the following definition of an ordered pair is set theoretic (i.e. you can distinguish between the "first" element and the "second" element using only set theory). (x,y)={x,{y}} Homework Equations The Attempt at a Solution I am inclined to...
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    Scaling of the vertical projectile problem nondimensionalization

    So O is just the order of the taylor approximation that replaces the right hand side?
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    Scaling of the vertical projectile problem nondimensionalization

    I actually redid the nondimensionalization part and it was correct, but not properly scaled which explains why I didn't get a term that was <<1. The correctly scaled nondimensional equation is d2Y/dT^2=-1/(1+eY)^2 where e<<1. It doesn't really change though, I still don't see where the...
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    Scaling of the vertical projectile problem nondimensionalization

    Homework Statement Restate the vertical projectile problem in a properly scaled form. (suppose x<<R). d2x/dt^2=-g(R^2)/(x+R)^2 Initial conditions: x(0)=0, dx(0)/dt=Vo Find the approximate solution accurate up to order O(1) and O(e), where r is a small dimensionless parameter. (i.e. the...
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    Intro to analysis proof first and second derivatives and mean value theorem

    If d>c then f'(xo)<0 and xo>c. f'(c)=0 and f'(xo)<0. But this contradicts that f'(x) is strictly increasing. So the statement must be true.
  22. R

    Intro to analysis proof first and second derivatives and mean value theorem

    Ah my mistake, the quantity is positive. I think I know where this is headed. So f'(xo)>f'(c)=0 and c>xo. But f'(x) must be strictly increasing on I since f''(x)>0, and thus the statement must be true by contradiction.
  23. R

    Intro to analysis proof first and second derivatives and mean value theorem

    It is negative. I'm still a bit puzzled as to how the mean value theorem relates to proving this problem.
  24. R

    Intro to analysis proof first and second derivatives and mean value theorem

    The MVT tells me that there exists some Xo in [d,c] such that f'(Xo) = [f(c)-f(d)]/(c-d).
  25. R

    Intro to analysis proof first and second derivatives and mean value theorem

    I still can't figure this one out - I get the feeling the proof is much more simple than I think it is.
  26. R

    Intro to analysis proof first and second derivatives and mean value theorem

    It implies that f(x) is concave up? It also implies c is a minimum by the second derivative test but we haven't covered that as of this section in our textbook.
  27. R

    Intro to analysis proof first and second derivatives and mean value theorem

    Homework Statement Let f(x) be a twice differentiable function on an interval I. Let f''(x)>0 for all x in I and let f'(c)=0 for some c in I. Prove f(x) is greater than or equal to f(c) for all x in I. Homework Equations Mean value theorem? The Attempt at a Solution f''(x)>0...
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    Linear algebra proof - Orthogonal complements

    Homework Statement Let V be an inner product space, and let W be a finite dimensional subspace of V. If x is not an element of W, prove that there exists y in V such that y is in the orthogonal complement of W, but the inner product of x and y is not equal to 0. Homework Equations The...
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