22. Parametric Surfaces and Surface Integrals

Homework

1. Consider the piece of the paraboloid \(z=x^2+y^2\) below \(z=4\). Assume the paraboloid is oriented down and out and has density \(\delta=x^2+y^2\). Also consider the vector fields \(\vec{F}=\langle x,y,xz+yz\rangle\) and \(\vec{G}=\langle y,-x,xz+yz\rangle\).

   

   1. Parameterize the paraboloid starting from cylindrical coordinates.

   2. Find the tangent vectors to the paraboloid and find a parametric equation for the tangent plane to the paraboloid at the point \((r,\theta)=(1,\pi)\).

   3. Find the normal vector to the paraboloid and check its orientation. Then find the normal equation for the tangent plane to the paraboloid at the point \((r,\theta)=(1,\pi)\).

   4. Find the length of the normal and the surface area of the paraboloid.

   5. Find the centroid of the paraboloid.

   6. Find the mass of the paraboloid.

   7. Find the center of mass of the paraboloid.

   8. Find the average density of the paraboloid.

   9. Evaluate the vector fields \(\vec F\) and \(\vec\nabla\times\vec G\) on the paraboloid.

   10. Compute \(\displaystyle \iint_P\vec{F}\cdot\,d\vec{S}\) over the piece of the paraboloid with \(r \le 2\) .

   11. Compute \(\displaystyle \iint_P \vec\nabla \times \vec G\cdot d\vec S\) over the piece of the paraboloid with \(r \le 2\) .

2. Consider the ellipse which is the piece of the plane \(z-x-5y=10\) that lies inside the cylinder \(x^2+y^2=81\).

   1. Starting from cylindrical coordinates, find a parametrization for the ellipse \[ \vec R(r,\theta)=\langle x(r,\theta), y(r,\theta), z(r,\theta)\rangle \] and give the range for \(r\) and \(\theta\).

   2. Find the tangent vectors, \(\vec e_r\) and \(\vec e_\theta\), the normal vector \(\vec N\) and its length \(|\vec N|\).

   3. Find the area of the ellipse.

3. The parametric surface \[ \vec R(r,\theta)=\langle r\cos\theta, r\sin\theta, \theta\rangle \] for \(0 \le \theta \le 6\pi\) and \(0 \le r \le 4\), is a helicoid (spiral ramp), as shown.
   Find its surface area.
   HINT: The integral requires a trig substitution.

   

4. Compute \(\displaystyle \iint_H \vec\nabla \times\vec F\cdot d\vec S\) for the vector field \(\vec F=\langle -x^2y,xy^2,xy\rangle\) over the hemisphere \(x^2+y^2+z^2=9\) for \(z \ge 0\) oriented up and out.

5. Consider the solid cylinder \(x^2+y^2 \le 16\) for \(4 \le z \le 8\). Find the flux of the vector field \(\vec F=\langle 2x, 2y, 4z \rangle\) over the complete surface of the cylinder (including the top and bottom) oriented outward.