This week marks about a third of the way through the course. Next week offers a significant change as we wrap up differentiation and we will begin integration. Carefully think back over the past month. What is going well in Math~4182H? How can I improve the course? What strategies will you pursue to improve your learning?
For a function \(f : A \to B\) and \(b \in B\), what is meant by \(f^{-1}(b)\)?
For a function \(f : A \to B\) with \(C \subseteq A\), carefully explain what is meant by \(f |_C\).
What is a regular value of a function \(f : \R^n \to \R^m\)? (This is not exactly S04P04.)
Suppose \(0\) is a regular value of \(f : \R^n \to \R\). For a point \(p \in f^{-1}(0)\), define the tangent space \(T_p f^{-1}(0)\). (Compare your answer to S02P04.)
Define \(f : \R^3 \to \R\) by \(f(x,y,z)=x^2+y^2-z\). Check that \(0\) is a regular value of \(f\), and write down the tangent plane to \(f^{-1}(0)\) at some point \((x_0,y_0,z_0) \in f^{-1}(0)\).
It is possible to parameterize a curve with “corners” using \(C^1\) functions.
Specifically, define \(f : \R \to \R\) via \[ f(x) = \begin{cases} \phantom{-}x^2 & \text{if } x \geq 0, \\ -x^2 & \text{if }x < 0. \end{cases} \] Check that \(f\) is \(C^1\).
What shape does \(S = \{ \left(f(\cos \theta), f(\sin \theta)\right) \in \R^2 \mid \theta \in \R \}\) make?
The \(C^1\) functions \(u,v:\R^2\to\R\) satisfy the Cauchy–Riemann equations (S01P14). Suppose \(a \in \R^2\) and that \(u(a)\) is a regular value of \(u\) and \(v(a)\) is a regular value of \(v\). Show that the level curve \(u^{-1}(u(a))\) meets the level curve \(v^{-1}(v(a))\) orthogonally at the point \(a\).
Suppose \(f,g:\R^n\to\R\) are \(C^{1}\). Let \(x\in\R^n\) satisfy \(g(x)=0\), and assume that \(x\) is a local maximizer of \(f\) subject to the constraint \(g=0\); that is, there exists a neighborhood \(U\) of \(x\) such that \[ f(x)\ge f(y)\quad\text{for all }y\in U\cap g^{-1}(0). \] Assume also that \(\nabla g(x)\neq 0\).
Show that there exists \(\lambda\in\R\) such that \[ \nabla f(x)=\lambda\,\nabla g(x). \] This is the method of Lagrange multipliers.
Consider a smooth function \(f : \R^2 \to \R\) and the circle \[ S^1 = \{ (x,y) \in \R^2 : x^2 + y^2 = 1 \}. \] I can think of two ways to find the extreme values of \(f |_{S^1}\). First, I could use the method of Lagrange multipliers as in S05P09. Second, I could parameterize \(S^1\) and consider a function of a single variable \(g : \R \to \R\) given by \[ g(\theta) = f(\cos \theta, \sin\theta). \] Carefully relate finding the extrema of \(g\) to applying the method of Lagrange multipliers to the original function \(f\).
Let \(n\ge 2\) and let \(x_1,\dots,x_n>0\). Define \[ A=\frac{x_1+\cdots+x_n}{n} \quad\text{and}\quad G=(x_1x_2\cdots x_n)^{1/n}. \] Show that \(G \le A\). This is the inequality of arithmetic and geometric means, otherwise known as the AM–GM inequality.
Hint: You can use S05P09 to do this.
Let \(A = (a_{ij})\) be an \(n\times n\) symmetric matrix with real entries, meaning \(a_{ij} = a_{ji}\). Define \(S^{n-1} = \{ (x_1,\ldots,x_n) \in \R^n \mid \sum_{i=1}^n {x_i}^2 = 1 \}\) and define \(f : \R^n \to \R\) by \[ f(x) = x^{\mathsf T}Ax. \] The function \(f\) attains a maximum value somewhere (why?), say at \(v \in \R^n\). Using S05P09, show that \(Av = \lambda v\) for some \(\lambda \in \R\).
This provides a proof that every symmetric matrix has an eigenvalue, a consequence of the spectral theorem.
Suppose \(f : \R \to \R\) is differentiable at \(0\) with \(f'(0) \neq 0\). Then there exists \(\epsilon > 0\) so that \(f |_{(-\epsilon,\epsilon)}\) is injective.
Suppose \(\gamma : \R \to \R^n\) is a curve such that, for all \(t \in \R\), we have \(\gamma'(t) \neq 0\). Then for all \(t \in \R\), it is the case that \(\gamma'(t)\) is orthogonal to \(\gamma''(t)\).
Suppose \(f : \R^2 \to \R\) is smooth and \(0\) is a regular value of \(f\). Then there exist functions \(c_i : (0,1) \to \R^2\) so that \(f^{-1}(0)\) is the union of the images of the \(c_i\)'s.