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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.
  5. H

    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
  7. H

    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...
  8. H

    Cardinal arithmetic

    Many thanks again, you also! :)
  9. H

    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...
  10. H

    Cardinal arithmetic

    Because y is an element of the set X U {x}
  11. H

    Cardinal arithmetic

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

    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...
  13. H

    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...
  14. H

    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##...
  16. H

    Cardinal arithmetic

    g(y) = f(y) for all y in X, g(y) = n + 1 when x = y. ok, now to prove the 'there exists' part; let $$ m \in S_{n+1} $$ we must show that there exists $$ y \in X \cup \{x\} such that g(y) = m $$, now $$ g(y) = f(y) = m $$ for all $$ y \in X. $$and $$ g(y) = n+ 1 = m for y = x $$ so $$ g(y) =...
  17. H

    Cardinal arithmetic

    I'm sorry, a little more detail please? so is it the wording thats the problem in the first issue? so if i was to say 'we have' instead of 'there exists', secondly that was a typo, i meant to type $$ y_1 = y_2 $$ instead of $$ i_1 = i_2 $$. surely the last one proves it that it is injective...
  18. H

    Cardinal arithmetic

    right for the onto proof, i have said that; given any $$ i \in \{i \in N : 1 \le i \le n + 1 \} $$ there exists $$ y \in X \cup \{x\} $$ such that $$ g(y) = i $$ it follows that g is onto, this quite simply has to be true! for the one to one proof; suppose that $$ g(y_{1}) = g(y_{2})$$ then $$...
  19. H

    Cardinal arithmetic

    g(y) = n+1 ? because thats the element its mapped to when y = x? i know this is true, surely? ONE- ONE; so each element in the domain of g is mapped to a unique element in the co-domain. so y = x iff f(y) = f(x). ONTO; the image set of g is equal to the co-domain. could i use the fact that f...
  20. H

    Cardinal arithmetic

    g(y): = g(x) when y = x? To prove that g is a bijection , i just have to show that X U {x} has equal cardinality with the set $$ \{i \in N : 1 \le i \le n +1 \} $$ right?
  21. H

    Cardinal arithmetic

    right, here is some work on a function g; I will now define a function g, such that $$g: X\cup\{x\} \rightarrow \{ i \in N:1 \le i \le n+1\} $$ by the following rule for any $$ y \in X \cup \{x\}, $$ we define $$ g(y):=f(y) for y \in X $$ and $$ g(y):=f(y) + 1 when y = x $$ since f is a...
  22. H

    Cardinal arithmetic

    I cant think of anything more basic other than actually letting $$ X = \{x_1, x_2....x_n\}$$ then showing that $$ X \cup {x} = \{x_1, x_2....x_{n+1}\} $$. since x is not an element of X. Other than that LCKurtz...
  23. H

    Cardinal arithmetic

    i'm really sorry, i'm gonna need a little help in that case, i'm stuck
  24. H

    Cardinal arithmetic

    Let X be a finite set with cardinality n, denoted by #X, then there exists a bijection; $$f:X \rightarrow \{ i \in N : 1 \le i \le n\} $$. Since x is not an element of X then $$X \cup \{x\} $$ is the set of elements that contains X or {x}, then there must exist a bijection $$ g:X \cup \{x\}...
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    Cardinal arithmetic

    Oh there is another one; Let n be a natural number. A set X is said to have cardinality n, iff it has equal cardinality with $$ \{i \in N : 1 \le i \le n\} $$. We also say that X has n elements iff it has cardinality n.
  26. H

    Cardinal arithmetic

    A set is finite iff it has cardinality n for some natural number n; otherwise, the set if called infintie. If X is a finite set, we use #(X) to denote the cardinality of X. That is what your asking for, i think?
  27. H

    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

    DONE! Thank you so much guys!
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    Simple question on 2d projectiles

    if i was to use the suggested co-ordinate system wouldnt the cannon just fire at a 90 degree angle?
  30. H

    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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