Parametrically-Defined Surfaces

• Stewart

  Find a parametric representation of the sphere

  \(\displaystyle x^2+y^2+z^2 = a^2.\)

• Stewart

  Find a parametric representation of the cylinder

  \(\displaystyle x^2+y^2 = 4 \qquad 0 \leq z \leq 1.\)

• Stewart

  Write down formulas for a vector-valued function that represents the elliptic paraboloid \(z=x^2+2y^2.\)

• Stewart

  Find a parametric representation of the surface \(z = 2\sqrt{x^2+y^2},\) that is, the top half of the cone \(z^2 = 4x^2+4y^2.\)

• Stewart

  Find parametric equations for the surface generated by rotating the curve \(y=\sin(x)\) for \(0\leq \theta \leq 2\pi\) about the \(x\)-axis. Use these equations to sketch the surface of revolution.

• Stewart

  What is a formula for plane tangent to this surface at the point \((1,1,3)?\)

  \(\displaystyle x=u^2 \quad y=v^2 \quad z=u+2v \)

• Stewart

  Calculate the surface area of a sphere of radius \(a.\)

Review Before Next Class

• Just stare at the formulas for line integrals over functions and line integrals over vector fields for about three minutes each.