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...
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...
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...
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...
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...
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...
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...
Homework Statement
a) In the autocollimation experiment of Sec. 3.5, a thin convex lens is placed 25 cm from a pinhole that acts as a point source of light. A plane mirror placed 20 cm behind the lens reflects the light back through the lens, and the reflected light forms a sharp spot beside...
Homework Statement
A pair of students measure the refraction of light in passing from acetone into air for several angles. When the angle of incidence is 30, the angle of refraction is 42.
b) What is the angle of incidence (in degrees) when the refracted angle is the critical angle...
Homework Statement
In the experiment, you will study an oscillator called a "torsion pendulum." In this case, the restoring "force" is the torsion constant of the wire that suspends the weight X and the inertial term is the rotational inertia of the suspended mass. You will compare the periods...
Homework Statement
Suppose that gcd(a, b) = 1 with a, b > 0, and let x0, y0 be any integer solution to the equation ax + by = c. Find a necessary and sufficient condition, possibly depending on a, b, c, x0, y0 that the equation have a solution with x > 0 and y > 0.
Homework Equations...
Homework Statement
A Block of mass M is attached to a cable of length L. It is initially held at the horizontal position at rest. It is then released in the motion of a pendulum and collides with a bullet of mass m, traveling at a speed of Vi, at the lowest point of its trajectory. The...
Homework Statement
I have to use stokes' theorem and calculate the surface integral, where the function F = <xy,2yz,3zx> and the surface is the cube bounded by the points (2,0,0), (0,2,0),(0,0,2),(0,2,2),(2,0,2),(2,2,0),(2,2,2). The back side of the cube is open.
[/B]
Homework...
Homework Statement
Water in an irrigation ditch of width w = 3.0 m and depth d = 2.0 m
flows with a speed of 0.40 m/s. For each case, sketch the situation,
then find the mass flux through the surface: (a) a surface of area wd,
entirely in the water, perpendicular to the flow; (b) a surface...
Homework Statement
Imagine two scenarios. The system in both is the object and ruler.
In case 1, a rectangular object undergoes a collision with a ruler, which is initially at rest. It strikes the ruler on its center of mass.
In case 2, it is the same situation except the object...
Homework Statement
3. The height of a certain hill (in feet) is given by
h(x, y) = 10(2xy − 3x2 − 4y2 − 18x + 28y + 12)
where y is the distance (in miles) north, x is the distance (in miles)
east of the village.
(a) Where is the top of the hill located?
(b) How high is the hill?
(c)...
Homework Statement
A Carnot freezer in a kitchen has constant temperature of 260k, while the air in the kitchen has a constant temperature of 300K. Suppose the insulation for the freezer is not perfect and energy is conducted into the freezer at a rate of .15 Watts. Determine the average...
Prove that if s1 and s2 are subsets of a vectorspaceV such that....
Homework Statement
Prove that if s1 and s2 are subsets of a vector space V such that S1 is a subset of S2,then span(S1) is a subset of span(s2). In particular, if s1 is a subset of s2 and span(s1)=V, deduce that span(s2)=V...
Homework Statement
If you invest a dollar at 6% interest compounded monthly, it amounts to (1.005)^n, where n=# of months. If you invest 10$ at the beginning of each month for 10 years (120 months), how much do you have at the end of the 10 years?
Homework Equations
sum of a geometric...
Homework Statement
A stone is swirled in a horizontal circle 1.00m above the ground by means of a string 2.00m long. The string breaks, and the stone files off horizontally and strikes the ground 15.0m away. What was the centripetal acceleration of the stone before the string breaks...
Homework Statement
I reduced a much harder problem to the following:
Prove that if abs(a-b) is divisible by k, and if abs(b-c) is divisible by k, then abs(a-c) is divisible by k.
Homework Equations
none really.
The Attempt at a Solution
I tried setting abs(a-b)/k = n and abs(b-c)/k = m...
Homework Statement
On an exam we were given that the \Sigma 1/n^2 was (pi^2)/6. We were asked to prove that the value of \Sigma ((-1)^(n+1))/(n^2)) = (pi^2)/12.
I'm sorry about the lack of latex; this is the first time I've ever *tried* to use it.
Homework Equations
Basic...