Hey there! Sign in to join this conversationNew here? Join for free
    Offline

    11
    ReputationRep:
    (Original post by cooldudeman)
    Thanks so much for this. Feel free to ignore this if you're not bothered to explain anymore but I still don't understand how you got the limits of t.

    How did you deduce that the limits for x is (0,1)?
    Also is the z^2=1-x a plane or just a curve?
    It sounds as if you haven't sketched out the two surfaces. If you're having trouble doing this, you can get graphing software to do it for you. I did in Microsoft Mathematics, which can do basic 3D parametric plots and so on.

    In brief, you have a circle of diameter 1, centre (1/2,0,0) (x^2+y^2=x) which in 3-space defines a cylinder parallel to the z-axis. (The z variable doesn't appear, so we get the same set of (x,y) points for any z value, just as when we write y=2 to define a line in 2-space, we get a line with y height 2 for any x value).

    This cylinder intersects the unit 3-sphere x^2+y^2+z^2=1. Note that the circle lies totally in the +ve x half-plane, so for any point on the cylinder x \ge 0.

    If you draw this out, it should be clear where the cylinder and sphere intersect. Since the cylinder equation defines a surface (not a volume) then the intersection is a curve in 3-space.

    By algebra you find that the curve is defined by the intersection of the surface z^2=1-x with x^2+y^2+z^2=1. Since we can only plot real values of z on a graph, we have 0 \le z^2 = 1-x \Rightarrow x \le 1. Since we already know that 0 \le x for any point on the circle, we have 0 \le x \le 1. This gives us the limits for x.

    All of this is pretty clear once you can see a nice picture of the situation.
    • Thread Starter
    Offline

    13
    ReputationRep:
    (Original post by atsruser)
    It sounds as if you haven't sketched out the two surfaces. If you're having trouble doing this, you can get graphing software to do it for you. I did in Microsoft Mathematics, which can do basic 3D parametric plots and so on.

    In brief, you have a circle of diameter 1, centre (1/2,0,0) (x^2+y^2=x) which in 3-space defines a cylinder parallel to the z-axis. (The z variable doesn't appear, so we get the same set of (x,y) points for any z value, just as when we write y=2 to define a line in 2-space, we get a line with y height 2 for any x value).

    This cylinder intersects the unit 3-sphere x^2+y^2+z^2=1. Note that the circle lies totally in the +ve x half-plane, so for any point on the cylinder x \ge 0.

    If you draw this out, it should be clear where the cylinder and sphere intersect. Since the cylinder equation defines a surface (not a volume) then the intersection is a curve in 3-space.

    By algebra you find that the curve is defined by the intersection of the surface z^2=1-x with x^2+y^2+z^2=1. Since we can only plot real values of z on a graph, we have 0 \le z^2 = 1-x \Rightarrow x \le 1. Since we already know that 0 \le x for any point on the circle, we have 0 \le x \le 1. This gives us the limits for x.

    All of this is pretty clear once you can see a nice picture of the situation.
    on this Microsoft maths programme, how do you plot the equation of the sphere and cylinder? I went on graphing and then equations and functions but they have either z or r as the subject. I cant rearrange the cylinder to z cuz there is no z so I cant plot it. Cant I just plot them in the form as they are given? I have been trying to find a programme/website that can do this for ages.
    Offline

    11
    ReputationRep:
    (Original post by cooldudeman)
    on this Microsoft maths programme, how do you plot the equation of the sphere and cylinder?.
    1. Go to Graphing > Equations and Functions
    2. Change to settings to 3D and Cartesian
    3. In the first box, enter z^2=1-x
    4. Add a second box. In this box, enter x^2+y^2+z^2=1
    5. Click on the Graph button at the bottom of the Equations and Functions section
    6. Click on "Proportional Display" on the ribbon thing.
    7. Click on "Hide Outer Frame" if you prefer to declutter it a bit.

    You will now be able to see the 3-sphere and parabolic surface, and their curve of intersection.
 
 
 
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • Poll
    Has a teacher ever helped you cheat?
    Useful resources

    Make your revision easier

    Maths

    Maths Forum posting guidelines

    Not sure where to post? Read the updated guidelines here

    Equations

    How to use LaTex

    Writing equations the easy way

    Student revising

    Study habits of A* students

    Top tips from students who have already aced their exams

    Study Planner

    Create your own Study Planner

    Never miss a deadline again

    Polling station sign

    Thinking about a maths degree?

    Chat with other maths applicants

    Can you help? Study help unanswered threads

    Groups associated with this forum:

    View associated groups
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

    Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

    Write a reply...
    Reply
    Hide
    Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.