0 (Warmup). Let \(\gamma = \dfrac {-x_2\,dx_1 + x_1\,dx_2}{x_1^2 + x_2^2}\in \Omega ^1(\R ^2\setminus \{0\})\). Show that \(d\gamma =0\).

[Solution: By the product rule

\[ d\gamma = \frac {d(-x_2\,dx_1+x_1\,dx_2)}{x_1^2 + x_2^2} - \frac {2x_1\,dx_1 + 2x_2\, dx_2}{(x_1^2 +x_2^2)^2} \wedge (-x_2\,dx_1+x_1\,dx_2) = 0, \]

since \(d(-x_2\,dx_1+x_1\,dx_2)=2dx_1\wedge dx_2\) and \((x_1\,dx_1 + x_2\, dx_2) \wedge (-x_2\,dx_1+x_1\,dx_2)=(x_1^2+x_2^2) dx_1\wedge dx_2\).]

1. Let \(U = \{ p \in \R ^4 : x_2(p) \neq 0\}\), and \(\varphi = \left (x_1, \, x_2^2 x_3, \, x_4/x_2 \right ): U \to \R ^3\), where \(x_1,x_2,x_3,x_4\) are the coordinate functions on \(U\sub \R ^4\). Let \(\alpha = y_1 dy_2 {\wedge } dy_3 \in \Omega ^2(\R ^3)\) where \(y_1, y_2, y_3 : \R ^3 \to \R \) denote the coordinate functions.

• (i) Express \(\varphi ^*\alpha \) and \(\varphi ^*d\alpha \) in standard form, i.e., as a sum of terms \(f dx_I\).

  [Hint: First expand \(\varphi ^*\alpha \) as \((\varphi ^*y_1) d(\varphi ^*y_2) \wedge d(\varphi ^*y_3) \) and similarly \(\varphi ^*d\alpha \).]

• (ii) Compute directly \(d(\varphi ^*\alpha )\). What do you observe?

  [Hint: To save work, just compute the terms that don’t appear in \(\varphi ^*d\alpha \).]

2. For \(U\) open in \(\R ^n\), \(\alpha \in \Omega ^k(U)\), \(p\in U\), and \(v_1,\ldots ,v_k\in \R ^n\), show that

\[ d\alpha _p(v_0,\ldots ,v_k)=\sum _{i=0}^k (-1)^i D\alpha _p(v_i)(v_0,\ldots , \widehat v_i,\ldots , v_k) \]

where \(v_0,\ldots ,\widehat v_i,\ldots , v_k\) denotes the list obtained from \(v_0,\ldots , v_k\) by omitting \(v_i\). Equivalently

\[ d\alpha _p=\sum _{i=0}^k\sgn (\sigma _i) \, \sigma _i\cdot D\alpha _p^\vee , \]

where \(\sigma _i = (0\;1\;\cdots \; i)^{-1}\in G:=\sym (\{0,1,\ldots k\})\cong S_{k+1}\), and for \(\sigma \in G\) and \(\beta \in \Alt ^{k+1}(\R ^n)\), \((\sigma \cdot \beta )(v_0,\ldots , v_k) =\beta (v_{\sigma (0)},\ldots , v_{\sigma (k)})\) (which is \(\beta (v_i,v_0,\ldots ,\widehat v_i,\ldots , v_k)\) when \(\sigma =\sigma _i\)).

[Hint: One approach, using the second formula, is to let \(H\cong S_k\) be the subgroup of \(G\) fixing \(0\), and observe that \(\sigma _iH:i=0,\ldots k\) is a left coset partition of \(G\). Now split the sum into sums over each coset: what is \(\tau \cdot D\alpha _p^\vee \) for \(\tau \in H\) ?]

3 (Less essential). Let \(U \subseteq \R ^3\) an open subset. Given a vector-valued function \(v = (v_1, v_2, v_3) : U \to \R ^3\), define \(v^\flat \in \Omega ^1(U)\) and \(v \inner \Det \in \Omega ^2(U)\) by applying the corresponding operations on vectors pointwise.

• (i) Let \(\div (v) : U \to \R \) be defined by

  \[ \div (v) = \pd {v_1}{x_1} + \pd {v_2}{x_2} + \pd {v_3}{x_3} . \]

  Show that

  \[ d(v \inner \Det ) = \div (v) \Det \in \Omega ^3(U) . \]

• (ii) Let \(\curl (v) : U \to \R ^3\) be defined by

  \[ \curl (v) = \left ( \pd {v_3}{x_2} {-} \pd {v_2}{x_3},\; \pd {v_1}{x_3} {-} \pd {v_3}{x_1},\; \pd {v_2}{x_1} {-} \pd {v_1}{x_2}\right ) . \]

  Show that

  \[ d(v^\flat ) = \curl (v) \inner \Det \in \Omega ^2(U) . \]

[Hint: Write all the forms in standard form, e.g., \(v \inner \mathrm {Det} \) can be expressed as \(v_1 dx_2 {\wedge } dx_3 - v_2 dx_1 {\wedge } dx_3 + v_3 dx_1 {\wedge } dx_2\).]

4. Let \(x_1,x_2\colon \R ^2\setminus \{0\}\to \R \) be the coordinate functions. Which of the following elements of \(\Omega ^1(\R ^2 \setminus \{0\})\) are closed? Which are exact?

• (i) \(\alpha = -2x_1x_2\,dx_1 + x_1^2dx_2\)

• (ii) \(\beta = x_2dx_1 + x_1dx_2\)

• (iii) \(\gamma = \dfrac {-x_2dx_1 + x_1dx_2}{x_1^2 + x_2^2}\)

  [Hint: For (iii), consider \(\varphi : \mathbb {R} \to \R ^2\setminus \{0\}\) defined by \(t \mapsto (\mathrm {cos} \, t, \mathrm {sin} \, t) \). Suppose that \(\gamma = df \) for some \(f : \mathbb {R}^2 \setminus \{0\} \to \mathbb {R}\). What can you say about \(\varphi ^*f\) and \(\varphi ^*\gamma \)? Or about \(\int _0^{2\pi } \frac {d(f\circ \varphi )}{dt} dt\) ?]