Recent content by HMPARTICLE

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    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

    Thanks, Ray. Maybe i should have made what i HAD done already a little more clear. I have already done what you said, regarding a 4x4 matrix and eliminating the first row and column, what results is another lower triangular matrix. the question asks me to show that that if i > j then ##B_{ij}##...
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    Triangular Matrices

    Have i not followed the forums rules? if not could someone please point out the error of my ways
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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

    Many thanks again, you also! :)
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    Cardinal arithmetic

    Thank you ever so much. The first line If m = n+1 then g(x) = m+1. I find this a little confusing. if m is some element of ##S_{n+1}## and m = n+1, i get this part.... but then to say that g(x) = m+1. This may sound really stupid, but to me, i'm not quite seeing it. Claim: ##g## is a 1-1...
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    Cardinal arithmetic

    Because y is an element of the set X U {x}
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    Cardinal arithmetic

    for any ##y \in X \cup \{x\}## we define; ##g(y):= f(y)## for all ## y \in X##, ##g(x):=n+1 ##
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    Cardinal arithmetic

    Actually i don't have a teacher/ lecturer, i made the choice to drop out of university to look after a dying grandparent, this resulted in me taking a degree with the open university which does not cover analysis of this kind. so i am self-studying Analysis 1 by terrence tao. When i do come...
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    Cardinal arithmetic

    I am setting out to prove that g is onto; let ## m \in S_{n+1} ##, we must show that there exist ## y \in X \cup \{x\}## such that ## g(y) = m ##. since ##f## is a bijection between X and ##S_n## there is some ##y \in X ## such that ##f(y) = m## and if ##m= n+1 ## then ##g(x) = m ## by the...
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    Cardinal arithmetic

    I see the difference. to state that for all ## y \in X, f(y) = m ## is a false statement as it implies that the image of every y in X is m. I understand the difference. I do apologise i wrongfully assumed that you were trying to point out some apparent difference between 'there exists' and...
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    Cardinal arithmetic

    Well. first, what is the difference? there is some, there exists, i cant quite make the difference, could you give me an example of where one is true and the other is not? moving on, there is some ##y \in X##such that## f(y) = m## and if ## m = n +1 ## then g(x) = m by the definition of ##g##...
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