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...
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...
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...
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...
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...
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...
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...
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...
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!
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...
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...
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...
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...
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...
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.
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.
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...
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...