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Math Help - Finding normal vector for line/surface integrals

  1. #1
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    Finding normal vector for line/surface integrals

    For instance, if curve C lying on the intersection of the cylinder x^2 + y^2 = 2y and the plane y = z, how would I find the normal vector n for this surface?


    You often have to find the normal vector when evaluating a line or surface integral. I have trouble with this. Does anyone have any tips? Or a guide to go about finding it?
    Last edited by messianic; December 5th 2009 at 11:52 PM.
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  2. #2
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    The normal is just the gradient at some point on the surface of the equation F(x,y,z)=0 so that's easy right:

    n=\nabla F]_{p_0}

    But we want the point p_0 to be at the intersection of the equations F(x,y,z)=x^2+y^2-2y=0 and g(x,y)=z=y. You can complete the square of the first one and see in the x-y plane that it's a circle with radius one at center (0,1). So parametrically, that's x=\cos(t), y=1+\sin(t) and since we want y=z, then z=1+\sin(t). Then the yellow contour is:

    \{\cos(t),1+\sin(t),1+sin(t)\} .

    So I can let any point p_0 be on that contour and then calculate the normal there:

    \nabla F]_{p_0}

    So I let t=0, calculate that normal, then translate it to p_0. That's the red one. Here's the Mathematica code I used in case you want to work with it:

    Code:
    p1 = ParametricPlot3D[{Cos[t], 1 + Sin[t], 1 + Sin[t]}, {t, 0, 
        2 \[Pi]}, PlotStyle -> {Thickness[0.008], Yellow}];
    xpt = 1;
    ypt = 1;
    zpt = 1;
    myNormal = {2 x, 2 y - 2, 0} /. {x -> xpt, y -> ypt, z -> zpt};
    myLine = Graphics3D[{Thickness[0.01], Red, 
        Line[{{xpt, ypt, zpt}, {xpt + myNormal[[1]], ypt + myNormal[[2]], 
           zpt + myNormal[[3]]}}]}];
    cp1 = ContourPlot3D[{x^2 + y^2 == 2 y}, {x, -1, 2}, {y, -1, 
        3}, {z, -2, 3}];
    cp2 = ContourPlot3D[{y == z}, {x, -1, 2}, {y, -1, 3}, {z, -2, 2}, 
       ContourStyle -> Opacity[0.2]];
    Show[{cp1, cp2, p1, myLine}, BoxRatios -> {1, 1, 1}, 
     PlotRange -> {{-3, 3}, {-3, 3}, {-3, 3}}]
    Attached Thumbnails Attached Thumbnails Finding normal vector for line/surface integrals-mytube.jpg  
    Last edited by shawsend; December 6th 2009 at 07:24 AM.
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