On an open set \(U \subset \R^n\), what does a \(1\)-form refer to?
What is a 2-form? How does the wedge product of 1-forms yield a 2-form?
Define the exterior derivative taking \(k\)-forms to \((k+1)\)-forms.
What does it mean for an open set \(U\subseteq\R^2\) to be simply connected?
The astroid is the curve parametrized by \(\gamma(t) = (\cos^3 t, \sin^3 t)\) for \(t\in[0,2\pi]\).
By Green's theorem, the area enclosed by a positively oriented simple closed curve \(C\) is \[ \text{area} = \int_C x\,dy = -\int_C y\,dx = \frac{1}{2}\int_C\left(x\,dy - y\,dx\right) \] Use this to compute the area enclosed by the astroid.
Show that \(d(df) = 0\) for every \(C^2\) function \(f:U\to\R\).
What well-known theorem does this encode? How does it relate to S08P12?
Let \(P_1=(x_1,y_1),\dots,P_n=(x_n,y_n)\) be the vertices of a polygon listed counterclockwise, with the convention \(P_{n+1}=P_1\). Use S09P05 to derive the shoelace formula, namely that \[ \operatorname{area} = \frac{1}{2}\sum_{i=1}^{n}\det\begin{pmatrix}x_i & x_{i+1}\\ y_i & y_{i+1}\end{pmatrix}. \]
Prove Green's theorem for the case where \(S\) is a type~I region, i.e., \[ S = \{(x,y) : a \le x \le b, \; \phi(x) \le y \le \psi(x)\} \] for continuous functions \(\phi \le \psi\) on \([a,b]\). Specifically, show that for any \(C^1\) function \(P\), \[ \int_{\partial S} P\,dx = -\iint_S \frac{\partial P}{\partial y}\,dA. \]
Recall the Cauchy–Riemann equations from S01P14. Suppose \(f(z) = u(x,y)+iv(x,y)\) is complex differentiable on a simply connected open set \(U\subseteq\R^2\), and let \(C\) be a piecewise smooth simple closed curve in \(U\) bounding a region \(S\subset U\). Define the complex line integral \[ \oint_C f(z)\,dz = \oint_C(u\,dx-v\,dy)+i\oint_C(v\,dx+u\,dy). \] Apply Green's theorem to the real and imaginary parts and use the Cauchy–Riemann equations to prove Cauchy's integral theorem: \(\displaystyle\oint_C f(z)\,dz=0\).
For a \(C^1\) vector field \(F = (P,Q)\) on a region \(S\subset\R^2\) with piecewise smooth boundary, let \(\hat n\) denote the outward-pointing unit normal to \(\partial S\). Show that if \(\partial S\) is parameterized counterclockwise by \((x(t),y(t))\), then \(\hat n\,ds = (dy,-dx)\). Then derive from Green's theorem the two-dimensional divergence theorem, namely \[ \iint_S \operatorname{div} F\,dA = \oint_{\partial S} F\cdot\hat n\,ds. \]
Let \(S\subset\R^2\) be a bounded region with piecewise smooth boundary, and let \(u\) and \(v\) be \(C^2\) on a neighborhood of \(\overline{S}\). Write \(\frac{\partial v}{\partial n} = \nabla v \cdot \hat n\) for the outward normal derivative.
Apply S09P10 with \(F = u\,\nabla v\) to deduce Green's first identity, namely \[ \iint_S \left(u\,\Delta v + \nabla u\cdot\nabla v\right)dA = \oint_{\partial S} u\,\frac{\partial v}{\partial n}\,ds. \]
Now prove Green's second identity by subtracting to obtain \[ \iint_S \left(u\,\Delta v - v\,\Delta u\right)dA = \oint_{\partial S}\left(u\,\frac{\partial v}{\partial n} - v\,\frac{\partial u}{\partial n}\right)ds. \]
Now suppose \(u\) is harmonic, i.e., \(\Delta u = 0\). Set \(v = u\) in the first identity and show that if \(u\) vanishes on \(\partial S\), then \(u \equiv 0\) on \(S\). This proves a uniqueness theorem that a harmonic function on \(S\) is determined by its boundary values.
If \(\omega\) is a closed \(C^1\) 1-form on an open set \(U\subseteq\R^2\), then \(\omega\) is exact on \(U\).
Suppose \(\gamma_1:[0,1]\to\R^2\) and \(\gamma_2:[0,1]\to\R^2\) are \(C^1\) simple closed curves whose images coincide as subsets of~\(\R^2\). Then for every \(C^1\) 1-form \(\omega\), \( \displaystyle\oint_{\gamma_1} \omega = \oint_{\gamma_2} \omega\).