# Let F(x, y, z) = (y, -3x, y z) and G = F. Calculate the surface integral I = G dS, where the surface S is the triangular portion of the plane defined by the three vertices (x, y, z) = (2, 0,...

Let F(x, y, z) = (y, -3x, y z) and G = F. Calculate the surface integral I = G dS, where the surface S is the triangular portion of the plane defined by the three vertices (x, y, z) = (2, 0, 0), (x, y, z) = (0, 1, 0), (x, y, z) = (0, 0, 4).

a) Sketch the surface S and write the cartesian equation of the plane.

b) Write a parametrization of the surface S.

c) Calculate I.

I suppose we start with the equation for I and use the equation for G?

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Denote the given points:

First, find the unit vector orthogonal to (for a plane it is a constant vector). An orthogonal vector is

Its unit vector is

**a**)

Therefore the equation of the plane is

or

**b**) The parametrization of is

Therefore

**c**) To find the integral, find first:

Then compute the dot product

To finally find the surface integral, note that on the -plane under the variable is from to

and the variable is from to

Also note that and the surface integral becomes

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