I have to evaluate
$$\int^1_{-1} \int^{ \sqrt {1-x^2}}_{ - \sqrt {1-x^2}} \int^1_{-\sqrt{x^2+y^2}}dzdydx$$
using spherical coordinates.
This is what I have come up with
$$\int^1_{0} \int^{ 2\pi}_0 \int^{3\pi/4}_{0}r^2\sin\theta d\phi d\theta dr$$
by a combination of sketching and...
A satellite is in a circular orbit a distance $h$ above the surface of the earth with speed $v_0$. It suffers a head-on collision with some debris which reduces its speed to $kv_0$, where $k$ is a constant in the range $0<k<1$, but does not change its direction. Calculate the eccentricity of the...
Homework Statement
a satellite is in a circular orbit a distance $h$ above the surface of the earth with speed $v_0$, booster rockets are fired which double the speed of the satellite without changing the direction. Find the subsequent orbit.
Homework Equations
The Attempt at a Solution
Before...
I have found via integration that the y coordinate is $$y =h/2 = 120 mm$$. The x coordinate is $$x = \frac{-4r}{3\pi} = -51.9mm$$ and the z coordinate is $$z = r - \frac{4r}{3\pi} = 69.1 mm$$. I have no answers in my textbook so cant confirm whether i am correct or not.
Homework Statement
Prove that is ##A## is lower triangular and ##B_{ij}## is the matrix that results when the ith row and jth column of A are deleted, then ##B_{ij}## is lower triangular if i > j.
Homework Equations
The Attempt at a Solution
I know that a square matrix is lower triangular...
Homework Statement
Let X be a finite set and let x be an object which is not an element of X. Then X U {x} is finite and #(X U {x}) = #(X) + 1
The Attempt at a Solution
Let X be a finite set such that X has cardinality n, denoted by #X.
Suppose that ## x \notin X##, then the set X U {x} has...
Homework Statement
A catapult projects a stone in the normal direction to a playground which slopes at a angle of 10 degrees to the horizontal.
The initial speed of the stone is 18 m/s. calculate the range parallel to the playground.
Homework Equations
the usual ones!
The Attempt at a...
Homework Statement
Let X be a set. Then the set
{Y:Y is a subset of X}
prove this is a set.
Where do i start?
Really unsure, i know that i have to use the power set?
I have written down;
{0,1}^X
Homework Statement
Find the Geometric image of;
1. ## | z - 2 | - | z + 2| < 2; ##
2. ## 0 < Re(iz) < 1 ##
Homework Equations
The Attempt at a Solution
In both cases i really am struggling to begin these questions, complex numbers are not my best field.
There are problems before this one...
Homework Statement
t:P_3 -----> P_3
p(x) |---> p(x) + p(2)
Determine whether or not this function is linear transformation or not.
Homework Equations
For a function to be a linear transformation then t(0) = 0 , there are other axioms that must be satisfied, but that is not the problem...
Homework Statement
let z' = (a,b), find z in C such that z^2 = z'
Homework Equations
The Attempt at a Solution
let z = (x,y) then z^2 = (x^2-y^2, 2xy)
since z^2 = z', we have,
(x^2-y^2, 2xy) = (a,b)
comparing real and imaginary components we have;
x^2-y^2 = a,
2xy = b.
Now...
Homework Statement
Let n,m be natural numbers. Then n x m = m x n.
Prove this.
Homework Equations
In order to prove this i am asked to prove 2 Lemma that will be useful.
In my solution i will (attempt to) prove these first.
Definition of multiplication;
for all m in N
0 x m = 0,
(n++) x m...
Homework Statement
Let a be a positive number. Then there exists exactly one natural number b such that b++ =a.
3. My attempt;
Axiom 2.4
Different natural numbers must have different successors; if n, m are natural numbers and n is not equal to m, then n++ is not equal to m++. equivalently if...
Homework Statement
For any natural numbers a,b,c, we have (a+b) + c = a + (b + c)
Homework Equations
Definition 2.2.1
Let m be a natural number. To add zero to m, we define 0 + m := m. Now suppose inductively that we have defined how to add n to m. Them we can add n++ to m by defining (n++) +...
Homework Statement
Determine all the subgroups of (A,x_85) justify.
where A = {1, 2, 4, 8, 16, 32, 43, 64}.
The Attempt at a Solution
To determine all of the subgroups of A, we find the distinct subgroups of A.
<1> = {1}
<2> = {1,2,4..} and so on?
<4> = ...
...
is this true? are there any...
Homework Statement
Suppose that , for any $$ \epsilon > 0, a < b + \epsilon $$ . Then $$ a\le b $$
The Attempt at a Solution
I have the proof, its not a question that was assigned to me, it was an example used.
According to the proof i can choose ANY epsilon greater than 0, so lets choose...
\begin{align*}
AB &= \sqrt{(2-(-1))^2+(3-4)^2}\\
&= \sqrt{3^2+(-1)^2}\\
&= \sqrt{10}.
\end{align*}
is what i am entering.
This is the error i am getting;
misplaced alignment character &.
I am baffled.
Which one of these statements is true?
$$ \exists y >0 : \forall x > 0, y < x $$
or
$$ \forall x > 0 \exists y > 0 : y < x $$
I think the second statement is correct, since for all x greater than 0, there exists at least one value of y > 0 such that y <x.
The first statement doesn't...
1. The problem statement.
consider the following sets;
C = {(x, y) ∈ R^2 : y ≥ (x + 2)^2},
D = {(x, y) ∈ R^2 : y ≥ 4x + 4}.
show that C is a subset of D.
3. Attempt at solution.
Let (x,y) be an arbitrary element of C, then
y ≥ x^2 + 4x + 4.
Rearranging the inequality gives
y - 4 ≥ x^2 +...
Homework Statement
a) Find the modulus and argument of 6^(1/2) + 2^(1/2)i
b) Solve the equation z^(3/4) = 6^(1/2) + 2^(1/2)i
Homework Equations
The Attempt at a Solution
For part a) i used Pythagoras to find the modulus.
( (6^(1/2))^2 + (2^(1/2))^2 )^(1/2) = (6 + 2)^(1/2)...
1.On the same axis sketch the graphs of y = (x-a)^-1 and y = 4|x-a|
This part i have completed, the first equation has a horizontal asymptote at x = a, it being rectangular hyperbola. the second equation drawn too.
2. Solve (x-a)^-1 < 4|x-a|, giving your answers in a.
Now the second part...
Homework Statement
One of the following equations represent a possible decay of the K + kaon.
K+ → π+ + π0
K+ → μ+ + νμ
State, with a reason, which one of these decays is not possible.
The Attempt at a Solution
At first i thought the first decay was not possible, however, i recently...
I enjoy both mechanics and pure mathematics equally!
I'd just like to hear everyone's thoughts on a 4yr Meng degree programme and a 4 yr MMath degree programme. One being a masters in mechanical engineering and another in well... Mathematics.
Right, i don't believe this is a homework question. The only reason I am stating this is because PF are stringent with their rules.
I'm quite confused and I'm not sure how to explicitly state my problem.
The vertices of a triangle are (a,b) (c,d) and (e,f).
This can be arranged into a...
1. The problem
A 3 kg object is moving in a plane with its x and y coordinates given by x = 5t^2 -1 and y = 3t^3 +2 where x and y are in meters and t is in seconds. Find the magnitude of the new force acting on this object at t= 2s
2. My attempt.
So my first attempt is to find a vector...
1. The problem.
Given that z= 3-4i
Show that z^2 = 3-4i
Hence or otherwise find the roots of the equation (z+i)^2=3-4i
2. My attempt.
The first part of the problem is strait forward z^2= (2-i)(2-i) then expand to get the desired result.
Now the second part
(z+i)^2=3-4i. Becomes
z^2+...
1. The problem statement.
Setting new world records in a 100-m race, Maggie and Judy cross the finish line in a dead heat, both taking 10.2s. Accelerating uniformly, Maggie took 2.0s and Judy 3s to attain maximum speed, which they maintained for the rest of the race.
What was the acceleration...
My eye prescription is gradually increasing, two years are I was 3.5 in both eyes now I am a 4? Will this eventually converge so that I remain at a constant prescription?