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  1. H

    Substituting spherical coordinates to evaluate an integral

    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...
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    Using the path equation to determine the path of a satellite

    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...
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    Central Force problems using radial motion equation

    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...
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    The x,y,z coordinates of CM of a solid cylinder

    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.
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    Triangular Matrices

    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...
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    Cardinal arithmetic

    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...
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    Simple question on 2d projectiles

    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...
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    Power set?

    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
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    Complex numbers: Find the Geometric image

    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...
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    Linear transformation

    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...
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    Complex numbers and completing the square

    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...
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    Proof: Multiplication is commutative

    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...
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    Proof troubles

    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...
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    Addition is associative proof

    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++) +...
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    Group Theory Question

    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...
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    Proof Trouble.

    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...
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    LaTeX Latex problem with alignment character and &

    \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.
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    Logic Question

    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...
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    Subsets question

    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 +...
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    Complex number equations.

    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)...
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    Annoying inequalities question

    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...
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    Quark decay

    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...
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    Math degree or mechanical eng?

    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.
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    Linear Equations and Matrices

    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...
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    Mechanics problem

    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...
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    Complex number homework

    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+...
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    Mechanics Question

    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...
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    Am I going blind?

    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?
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    How to Mathematically describe a plane?

    The linear combinations of V = (1,1,0) and W = (0,1,1) fill a plane. My question is how do i describe that plane? (not geometrically).
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