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Surface integral differential

  1. Apr 11, 2009 #1
    1. The problem statement, all variables and given/known data

    For the parametrically defined surface S given by r(u,v) = <cos(u+v), sin(u+v), uv>, find the following differential:

    In double integral over S of f(x, y, z)dS, dS =



    2. Relevant equations
    Above



    3. The attempt at a solution
    I thought I needed to put x, y, and z all in terms of two variables, (all three in terms of x and y, or y and z, or x and z), so that I can find dz/dx and dz/dy, but I don't know how to do this. :(
     
  2. jcsd
  3. Apr 12, 2009 #2

    HallsofIvy

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    You are given x, y and z "in terms of two variables": x= cos(u+v), y= sin(u+v), z= uv. Do everything in terms of u and v, not x, y, and z.


    The simplest way to find dS is this:
    You are given [itex]\vec{r}= cos(u+v)\vec{i}+ sin(u+v)\vec{j}+ uv\vec{k}[/itex].
    Find the derivative of that vector with respect to each of the parameters:
    [tex]\vec{r}_u= -sin(u+v)\vec{i}+ cos(u+v)\vec{j}+ v\vec{k}[/tex]
    [tex]\vec{r}_v= -sin(u+v)\vec{i}+ cos(u+v)\vec{j}+ u\vec{k}[/tex]

    The "fundamental vector product" is the cross product of those two derivative vectors. It is perpendicular to the surface at each point and dS is its length times dudv.
     
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