For each of the following manifolds, determine which can be written as the graph of a function, the zero-locus of a function, and parametrically. Give the appropriate functions in each case.
- The ellipse \(ax^2+by^2 = c^2\),
- The set \(\left\{ (t^2+t,2t-1): t \in \mathbb R\right\}\),
- The plane \(3x-4y+3z=10\),
- The sphere of radius \(r\), \( S_r = \left\{ \mathbf x \in \mathbb R^3: \vert \mathbf x\vert=r \right\}\),
- The
cylinder \(\left\{(x,y,z): x^2+y^2=4\right\}\),
- The intersection of the
plane \(x+z=1\) with the sphere \(x^2+y^2+z^2=1\),
- If \(f: [a,b]
\to \mathbb R\) let \(S\) be the space defined by revolving the graph
of \(f\) about the \(x\)-axis.