Problem Set 20

  1. Easy
    Let \(S\) be the capless cylinder \(S = \left\{(x,y): x^2+y^2=9, 0 \leq z\leq 5\right\}\), and \(\mathbf F(x,y,z) = (2x,2y,2z)\). Determine the flux of \(\mathbf F\) through \(S\).

  2. Easy
    Let \(S\) be the disk of radius \(3\), sitting in the plane \(z=3\). Determine the flux of \(\mathbf F(x,y,z) = (0,0,x^2+y^2)\) through \(S\).

  3. Easy
    Evaluate the flux of \(\mathbf F(x,y,z) = (3x^2,2y,8)\) over the plane \(-2x+y+z=0\) for \((x,y) \in [0,2]\times [0,2]\), oriented pointing in the \(-z\)-direction.

  4. Easy
    Let \(S\) be the triangle with vertices \((1,0,0), (0,2,0), (0,1,1)\), and \(\mathbf F(x,y,z) = (xyz,xyz,0)\). Find the flux of \(\mathbf F\) through \(S\).

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