Problem Set 12

1. Easy
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.
1. The ellipse $$ax^2+by^2 = c^2$$,
2. The set $$\left\{ (t^2+t,2t-1): t \in \mathbb R\right\}$$,
3. The plane $$3x-4y+3z=10$$,
4. The sphere of radius $$r$$, $$S_r = \left\{ \mathbf x \in \mathbb R^3: \vert \mathbf x\vert=r \right\}$$,
5. The cylinder $$\left\{(x,y,z): x^2+y^2=4\right\}$$,
6. The intersection of the plane $$x+z=1$$ with the sphere $$x^2+y^2+z^2=1$$,
7. If $$f: [a,b] \to \mathbb R$$ let $$S$$ be the space defined by revolving the graph of $$f$$ about the $$x$$-axis.

2. Easy
Determine whether the following spaces are smooth:
1. The set $$C = \left\{(\cos(t), \sin\left(2t\right): t \in [0,2\pi) \right\}.$$
2. Let $$S$$ be the surface defined by the image of $$g: \mathbb R^2 \to \mathbb R^3$$, $$g(s,t) = (3s, s^2-2t, s^3+t^2)$$.
3. Let $$S = F^{-1}(0)$$ where $$F(x,y,z) = 3xy + x^2+z$$.
4. Let $$S = F^{-1}(0)$$ where $$F(x,y,z) = \cos(xy) + e^z$$

3. Medium
Let $$U=\{(x,y,z): x>0, y>0, z>0\}$$ be the first octant, and let $$g: U \to \mathbb R^4$$ be given by $g(t,u,v) = (tu, tv, uv, tuv).$ Determine whether the image of $$g$$ defines a smooth surface.

4. Medium
For what values of $$c, c_1,c_2$$ do the following equations define a $$C^1$$ surface?
1. $$x^2+y^2+z^2 = c_1$$, $$x^2+y^2-z^2=c_2$$,
2. $$xyz=c$$

5. Medium
Let $$F_1,F_2: \mathbb R^n \to \mathbb R$$ be $$C^1$$ functions and let $$F_3(\mathbf x) = F_1(\mathbf x) F_2(\mathbf x)$$. If $$S_i = \left\{\mathbf x \in \mathbb R^n: F_i(\mathbf x) = 0\right\}$$ show that $$S_3 = S_1 \cup S_2.$$ [In particular, this shows that it when analyzing the zero-locus of a function which is the product of two functions, it is sufficient to look at each constituent function separately.]

6. Medium
Consider the curve $$\gamma: \mathbb R \to \mathbb R^2$$ give by $$\gamma(t) = (2e^{-t/2} \cos(t), 2e^{-t/2} \sin(t))$$.
1. Show that $$\gamma(t)$$ defines a $$C^1$$ curve.
2. Calculate the speed of this curve as a function of $$t$$. [Hint: The velocity is $$\gamma'(t)$$ so the speed is $$\|\gamma'(t)\|$$]
3. We define the unit tangent vector to the curve to be $$T(t) = \gamma'(t)/\|\gamma'(t)\|$$. Compute the unit tangent vector.
4. We will see later in the course that the arc-length of a curve on the interval $$[0,t]$$ is given by $s(t) = \int_0^t \|\gamma'(u)\| du.$ Compute the arc-length function $$s(t)$$ for the curve $$\gamma$$.
5. Inverting the arc-length formula gives a function $$t(s)$$ (time as a function of arc-length). The reparameterization of the curve $$\gamma(t)$$ using $$t=t(s)$$ is known as the arclength parameterization. Compute the arc-length parameterization of $$\gamma(t)$$; that is, compute $$\gamma(t(s))$$.

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