Evaluate the surface integral of vector field f (x, y,z) = x i + y j+(x+y) k over the portion s of the paraboloid z = x^2 + y^2 lying above the disk x^2 + y^2 ? 1. use outward pointing normals.

Parameterize the surface by



where we use the polar coordinate transformation and , from which it follows that . For this region, we have and .

The surface integral is then



where and are the respective partial derivatives of . We have







Now, presumably you know what a paraboloid looks like. An outward-pointing normal vector to the surface would probably refer to any normal that points downward, so check what happens to the cross product above at, let's say, the point (something close to the origin; the origin itself won't work because ):



Now in Cartesian space, the point would lie above the paraboloid, since



which means this normal vector has an inward orientation. To rectify this, swap the sign and replace with (which differs by a sign because the cross product is anticommutative).



So the value of the surface integral is

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