# Search results

1. ### 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...
2. ### 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...
3. ### 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...
4. ### 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. ### 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}##...
6. ### Triangular Matrices

Have i not followed the forums rules? if not could someone please point out the error of my ways
7. ### 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. ### Cardinal arithmetic

Many thanks again, you also! :)
9. ### 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. ### Cardinal arithmetic

Because y is an element of the set X U {x}
11. ### 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. ### 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. ### 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. ### 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...
15. ### 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##...

19. ### 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. ### 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. ### 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. ### 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. ### Cardinal arithmetic

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

26. ### 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. ### 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...
28. ### Simple question on 2d projectiles

DONE! Thank you so much guys!
29. ### 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. ### 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...