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Math Help - vector calculus..cylindrical coordinates

  1. #1
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    Exclamation vector calculus..cylindrical coordinates

    how to you transform cartesian equations into cylindrical coordinates?
    for example, write the following cartesian equation in cylindrical coordinates and graph:
    x^2 + y^2 + (z-1)^2= 1

    this seems like it is really simple, so i'm sorry for posting it, but there aren't any examples in my book so i'm not sure what to do. thanks so much
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  2. #2
    MHF Contributor Calculus26's Avatar
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    let t= theta

    x= rcos(t) y = rsin(t) z = z


    x^2 + y^2 + (z-1)^2= 1

    becomes r^2 +(z-1)^2 = 1
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  3. #3
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    but what about z?
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  4. #4
    MHF Contributor Calculus26's Avatar
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    z is the same in cylindrical coord as in rect coord

    i.e. the distance above or below the x-y plane or if you will the r-theta plane
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  5. #5
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    oh okay that makes sense..but how am i supposed to graph that?
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  6. #6
    MHF Contributor Calculus26's Avatar
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    It is still a sphere centered at (0,0,1)

    the idea of the exercise is to recognize the from a sphere takes in cylindrical coordinates

    For eg a sphere centered at the origin is r^2 + z^2 = 1

    as another eg the saddle z = 1- x^2 + y^2

    takes the form z= 1 - r^2cos^2(t) + r^2sin^(t)

    z = 1 -r^2cos(2t)


    and so on.
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  7. #7
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    okay that sorta makes sense to me haha so is the radius 1?

    my next problem is z=2(x^2+y^2)..so that would become z=2r^2? is that a sphere too?

    sorry for so many questions
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  8. #8
    MHF Contributor Calculus26's Avatar
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    You've got it--don't worry about the number of questions

    it is always hard when you have to start thinking in terms of a new coordinate system--wait till you get to spherical coords.
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  9. #9
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    lol thats the other half of this homework..spherical coords...i almost like them better!

    so z=2r^2 has a center of (2,2,0) and i'm not sure about the radius?
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  10. #10
    MHF Contributor Calculus26's Avatar
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    No


    z=2(x^2+y^2).. is a parabaloid with vertex (0,0,0)

    z =2r^2 -- its still a parabaloid opening up with vertex at (0,0,0)

    just written in terms of cylindrical coords

    A saddle is still a saddle etc
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  11. #11
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    thanks so much..its just not clicking for me somehow..i hate 3d!
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  12. #12
    MHF Contributor Calculus26's Avatar
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    the thing to remember is that all the surfaces you learned in rectangular coordinates don't change when you go to cylindrical and spherical coordinates.

    The form of the equations change in the new coordinates but you still are describing the same exact surfaces.

    You may hate it now but believe me when you get to triple integrals you'll thank your lucky stars for the new coordinate systems

    Hang in there
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