Problem Set 11

  1. Easy
    Determine whether the equation \(\sin(xyz) = z\) may be solved for \(z\) as a function of \(x\) and \(y\) in a neighbourhood of the point \((x,y,z) = (\pi/2, 1,1). \)

  2. Easy
    Find conditions on \(x\) and \(y\) which guarantee that one can locally solve the following for \(u(x,y)\) and \(v(x,y)\): \[xu^2+yv^2=9\] \[xv^2-yu^2=7\]

  3. Easy
    Consider the function \(f: \mathbb R^3 \to \mathbb R^3\) given by \[ f(x,y,z) = \left(ye^x + \sin(\pi y)\cos(z) ,\ \cos(yz),\ z^2\right). \] Determine whether \(f\) is invertible in a neighbourhood of \((0,1,\pi/2).\)

  4. Medium
    Show that the following system always has a solution for sufficiently small \(a\), \[ x+y + \sin(xy) = a \] \[ \sin(x^2+y) = 2a \]

  5. Medium
    Let \(f: \mathbb R \to \mathbb R\) be a non-constant \(C^1\) function such that \(f'(0) \neq 0\) and \(f(x+y) = f(x)f(y)\). Define \(F:\mathbb R^2 \to \mathbb R\) by \(F(x,y) = f(x) f(y)\). Determine what conditions (if any) must be imposed upon \(y\) to ensure that \(y\) can be solved as a function of \(x\) on the set \(\{(x,y): F(x,y) = 1\}\). Bonus: Write down an explicit formula for \(y\) as a function of \(x\).

  6. Medium
    Define the set \[ M_2(\mathbb{R}) = \left\{\begin{pmatrix} a & b \\ c & d \end{pmatrix}: a,b,c,d \in\mathbb{R} \right\} \] to be the set of \(2\times2\) matrices. Define a map \(g: M_2(\mathbb{R}) \to M_2(\mathbb R)\) by \(g(A) = A^2\). Determine whether \(g\) is invertible in a neighbourhood of \(I = \begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}\).

  7. Hard
    In this problem, we will show that for a special class of polynomials, slightly perturbing the coefficients will preserve the number of roots of the equation.
    1. Let \(f: \mathbb R \to \mathbb R\) be a degree \(n\) polynomial. Show that if all the roots of \(f\) are distinct, then for any root \(r\) we necessarily have \(f'(r) \neq 0\). [Hint: We say that a root \(r\) has multiplicity \(k\) if \(f(x) = (x-r)^k q(x)\) and \(q(r)\neq0\). All the roots of a polynomial are distinct if every root has multiplicity \(1\)].
    2. For fixed \((c_0,\ldots,c_{n-1}\) let \(f(x) = x^n + c_{n-1}x^{n-1} + \cdots + c_1 x + c_0\) be a function with distinct roots. Identify the \((c_0,\ldots,c_{n-1})\) with a point in \(\mathbb R^n\). Show that for each root \(r\), there is a neighbourhood of \(U_r\) of \((c_0,c_1,\ldots,c_{n-1}) \in \mathbb R^n\) and a neighborhood \(V_r\) of \(r\) such that if \((d_0,\ldots,d_{n-1}) \in U_r\) then \(x^n + d_{n-1} x^{n-1} + \cdots + d_1 x + d_0\) has a root in \(V_r\).
    3. Use part \(2\) to conclude that a degree \(n\) polynomial with exactly \(m < n\) roots, all of which are distinct, has the same number of roots under small perturbation of its coefficients.
    4. Does this result still hold if the roots are no longer distinct? Prove the result or give a counter-example.

  8. Hard
    A map \(\mathbf f: \mathbb R^n \to \mathbb R^n\) is said to be open if whenever \(U\) is open then \(\mathbf f(U)\) is open. Show that if \(\mathbf f\) is \(C^1\) and \(D\mathbf f(\mathbf x_0)\) is invertible for all \(\mathbf x_0 \in \mathbb R^n\) then \(\mathbf f\) is an open map.

  9. Bonus
    Let \(M_2(\mathbb R)\) be defined as in Problem (6) and define the subset \[ O_2(\mathbb R) = \left\{ X \in M_2(\mathbb R): XX^T = \text{Id}\right\}. \] Show that \(O_2(\mathbb R)\) defines a \(C^1\)-manifold of \(M_2(\mathbb R)\) and determine its dimension.

  10. Bonus
    We say that a function \(f: \mathbb R^n \to \mathbb R\) is strictly convex if for every \(t \in [0,1]\) and \(\mathbf x, \mathbf y \in \mathbb R^n\) \[ f(t\mathbf x + (1-t) \mathbf y) < t f(\mathbf x) + (1-t) f(\mathbf y). \] Let \(F: \mathbb R^2 \to \mathbb R\) be a \(C^2\) function which is strictly convex, non-negative, and satisfies \(F(0,0) = 0\). Show that there is an everywhere concave-down function \(f: \mathbb R \to \mathbb R\), such that in a neighbourhood of \((0,1)\), \(F(x, f(x)) = F(0,1)\).

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