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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