State the Inverse Function Theorem.
State the Implicit Function Theorem.
What is a critical point of a function \(f : \R^n \to \R\)?
What is a regular value of a function \(f : \R^n \to \R\)?
Recall S01P06.
Denote the set of \(2\)-by-\(2\) matrices with real entries by \(\Mtwo(\R)\).
Determine the regular values of \(\det : \Mtwo(\R) \to \R\).
Recall S01P07.
Let \(\GL_2(\R)\subset \Mtwo(\R)\) be the set of invertible \(2\)-by-\(2\) matrices. Consider the map \(\mathrm{inv}:\GL_2(\R)\to \GL_2(\R)\) defined by \(\mathrm{inv}(A)=A^{-1}\).
Fix \(A_0\in \GL_2(\R)\). Define \[ F:\Mtwo(\R)\times \Mtwo(\R)\to \Mtwo(\R),\qquad F(A,B)=AB-I. \] Note that \(F(A,B)=0\) holds exactly when \(B=A^{-1}\).
Use “implicit differentiation” in order to compute the derivative \(D(\mathrm{inv})_{A_0}(H)\), i.e., the derivative of the inverse map at \(A_0 \in \GL_2(\R)\) and in the direction \(H\in \Mtwo(\R)\).
Define \(\Phi:(0,\infty)\times\R\to\R^2\) by \[ \Phi(r,\theta)=(r\cos\theta,r\sin\theta). \] Compute \(\det D\Phi(r,\theta)\).
Your friend says “so every point of \(\R^2\setminus\{0\}\) has a neighborhood on which \((x,y)\) can be written as \((r,\theta)\) uniquely.”
Is this precisely right?
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Consider the elliptic curve \[ E=\{(x,y)\in\mathbb R^2 : y^2 = x^3 - x\}. \] Let \(F(x,y)=y^2-(x^3-x)\), so that \(E=F^{-1}(0)\).
Pick \(x_0\) so that \(p=(x_0,0)\in E\), i.e., pick \(x_0\in\{-1,0,1\}\).
Explain why you cannot solve locally for \(y\) as a function of \(x\) around such a point \(p\), but you can solve locally for \(x\) as a function of \(y\).
More precisely, show there is an \(\epsilon > 0\) and a \(C^1\) function \(f:(-\epsilon,\epsilon)\to\mathbb R\) such that \(f(0)=x_0\) and, when \(-\epsilon < y < \epsilon\), we have \((f(y),y) \in E\). In this case, compute \(f'(0)\).
The folium of Descartes is the curve in \(\R^2\) carved out by \[ x^3+y^3=3xy. \] Assume \(y\neq 0\) and set \(t=x/y\).
Express the points on the curve with \(y\neq 0\) in parametric form \((x(t),y(t))\) where \(t = x/y\). Compute \(x'(t)\). For which values of \(t\) is \(x'(t)\neq 0\)?
Use the Inverse Function Theorem to identify points of the folium where the curve is the graph of a differentiable function \(y = y(x)\). Locate the two places where this fails, and explain what happens there.
Fix an integer \(k\ge 2\), and consider the map \(P: \Mtwo(\R)\to \Mtwo(\R)\) given by \(P(X)=X^k\). This is the \(k\)-th power map.
Show that \(P\) is differentiable and compute the derivative of \(P\) at \(I\). Show that the derivative at \(I\) is an isomorphism. Conclude that for every sufficiently small matrix \(H\), the equation \(X^k=I+H\) has a solution.
Is it unique? Does every matrix have a \(k\)-th root?
Interpret \(\mathbb{R}^{n+1}\) as the coefficients of a degree \(n\) polynomial. Pick \(P \in \mathbb{R}^{n+1}\) so that there is a root \(r\) of the polynomial \(P\), i.e., \[ P_0 + P_1 \, r + P_2 \, r^2 + \cdots + P_n r^n = 0. \] Does there exist a function \(f : U \to \R\) defined on a neighborhood \(U \ni P\) so that \(f(Q)\) is a root of the polynomial \(Q\)? In this case, compute \(f'(P)\).
Let \(F:\R\to\R\) be \(C^1\). If \(F'(x) \neq 0\) for all \(x\in\R\), then \(F\) is surjective.
Let \(F:\R^n\to\R^n\) be \(C^1\). If \(\det DF(x)\neq 0\) for all \(x\in\R^n\), then \(F\) is injective.
Recall S01P14.
Suppose \(F:\R^2 \to\R^2\) is \(C^1\) and \(F(x,y) = (u(x,y),v(x,y))\) for functions \(u, v : \R^2 \to \R\). Suppose \(F\) satisfies the Cauchy–Riemann equations, i.e., \(u_x=v_y\) and \(u_y=-v_x\). Assume \(DF(x_0,y_0)\) is invertible at some \((x_0,y_0)\), and let \(G=(p,q)\) be the \(C^1\) local inverse of \(F\) near \((x_0,y_0)\). Then \(G\) also satisfies the Cauchy–Riemann equations.