Chapter 4

Let \(e_{1}, \ldots, e_{n}\) be the usual basis of \(\mathbf{R}^{n}\) and let \(\varphi_{1}, \ldots, \varphi_{n}\) be the dual basis. (a) Show that \(\varphi_{i_{1}} \wedge \cdots \wedge \varphi_{i_{k}}\left(e_{i_{1}}, \ldots, e_{i_{k}}\right)=1\). What would the right síde be if the factor \((k+l) ! / k ! l !\) did not appear in the definition of \(\wedge\) ? (b) Show that \(\varphi_{i_{1}} \wedge \cdots \wedge_{\varphi_{i_{k}}}\left(v_{1}, \ldots, v_{k}\right)\) is the determinant of the \(k \times k\) minor of \(\left(\begin{array}{c}v_{1} \\ \cdot \\ \cdot \\\ v_{k}\end{array}\right)\) obtained by selecting columns \(i_{1}, \ldots, i_{k}\)

If \(f: V \rightarrow V\) is a linear transformation and \(\operatorname{dim} V=n\), then \(f^{*}: \Lambda^{n}(V) \rightarrow \Lambda^{n}(V)\) must be multiplication by some constant \(c\). Show that \(c=\operatorname{det} f\)

4-3. If \(\omega \in \Lambda^{n}(V)\) is the volume element determined by \(T\) and \(\mu\), and \(w_{1}, \ldots, w_{n} \in V\), show that $$ \left|\omega\left(w_{1}, \ldots, w_{n}\right)\right|=\sqrt{\operatorname{det}\left(g_{i j}\right)} $$ where \(g_{i j}=T\left(w_{i}, w_{j}\right) .\) Hint: If \(v_{1}, \ldots, v_{n}\) is an orthonormal basis and \(w_{i}=\sum_{j=1}^{n} a_{i j} v_{j}\), show that \(g_{i j}=\sum_{k=1}^{n} a_{i k} a_{k j}\)

\({ }^{*}\) Deduce the following properties of the cross product in \(\mathbf{R}^{3}\) : $$ \begin{array}{llll} \text { (a) } e_{1} \times e_{1}=0 & e_{2} \times e_{1}=-e_{3} & & e_{3} \times e_{1}=e_{2} \\ e_{1} \times e_{2} & =e_{3} & e_{2} \times e_{2}=0 & & e_{3} \times e_{2}=-e_{1} \\ e_{1} \times e_{3} & =-e_{2} & e_{2} \times e_{3}=e_{1} & & e_{3} \times e_{3}=0 \end{array} $$ (b) \(v \times w=\left(v^{2} w^{3}-v^{3} w^{2}\right) e_{1}\) $$ \begin{aligned} +\left(v^{3} w^{1}\right.&\left.-v^{1} w^{3}\right) e_{2} \\ &+\left(v^{1} w^{2}-v^{2} w^{1}\right) e_{3} \end{aligned} $$ (c) \(|v \times w|=|v| \cdot|w| \cdot|\sin \theta|\), where \(\theta=\angle(v, w)\). \(\langle v \times w, v\rangle=\langle v \times w, w\rangle=0\) (d) \(\langle v, w \times z\rangle=\langle w, z \times v\rangle=\langle z, v \times w\rangle\) \(v \times(w \times z)=\langle v, z\rangle w-\langle v, w\rangle z\) \((v \times w) \times z=\langle v, z\rangle w-\langle w, z\rangle v\) (e) \(|v \times w|=\sqrt{\langle v, v\rangle \cdot\langle w, w\rangle-\langle v, w\rangle^{2}}\)

If \(T\) is an inner product on \(V\), a linear transformation \(f: V \rightarrow V\) is called self-adjoint (with respect to \(T\) ) if \(T(x, f(y))=T(f(x), y)\) for \(x, y \in V .\) If \(v_{1}, \ldots, v_{n}\) is an orthonormal basis and \(A=\left(a_{i j}\right)\) is the matrix of \(f\) with respect to this basis, show that \(a_{i j}=a_{j i}\)

(a) If \(f: \mathbf{R}^{n} \rightarrow \mathbf{R}^{m}\) and \(g: \mathbf{R}^{m} \rightarrow \mathbf{R}^{p}\), show that \((g \circ f)_{*}=g_{*} \circ f_{*}\) and \((g \circ f)^{*}=f^{*} \circ g^{*}\) (b) If \(f, g: \mathbf{R}^{n} \rightarrow \mathbf{R}\), show that \(d(f \cdot g)=f \cdot d g+g \cdot d f\).

Let \(c\) be a differentiable curve in \(R^{n}\), that is, a differentiable function \(c:[0,1] \rightarrow \mathbf{R}^{n} .\) Define the tangent vector \(v\) of \(c\) at \(t\) as \(c_{*}\left(\left(e_{1}\right)_{t}\right)=\left(\left(c^{1}\right)^{\prime}(t), \ldots,\left(c^{n}\right)^{\prime}(t)\right)_{c(t)} . \quad\) If \(f: \mathbf{R}^{n} \rightarrow \mathbf{R}^{m}\), show that the tangent vector to \(f \circ c\) at \(t\) is \(f_{*}(v)\)

. If \(f: \mathbf{R}^{n} \rightarrow \mathbf{R}^{n}\), define a vector field \(\mathbf{f}\) by \(\mathbf{f}(p)=f(p)_{p} \in \mathbf{R}_{p}^{n}\). (a) Show that every vector field \(F\) on \(\mathbf{R}^{n}\) is of the form \(\mathbf{f}\) some \(f\). (b) Show that div \(\mathbf{f}=\) trace \(f^{\prime}\).

If \(F\) is a vector field on \(\mathbf{R}^{3}\), define the forms $$ \begin{aligned} \omega_{P}^{1} &=F^{1} d x+F^{2} d y+F^{3} d z \\ \omega_{F}^{2} &=F^{1} d y \wedge d z+F^{2} d z \wedge d x+F^{3} d x \wedge d y (a) Prove that $$ \begin{aligned} d f &=\omega_{\mathrm{grad} f}^{1} \\ d\left(\omega_{F}^{1}\right) &=\omega_{\text {curl } F}^{2} \\ d\left(\omega_{F}^{2}\right) &=(\operatorname{div} F) d x \wedge d y \wedge d z \end{aligned} $$ (b) Use (a) to prove that $$ \begin{gathered} \text { curl grad } f=0 \\ \text { div curl } F=0 \end{gathered} $$ (c) If \(F\) is a vector field on a star-shaped open set \(A\) and \(\operatorname{curl} F=0\), show that \(F=\operatorname{grad} f\) for some function f: \(A \rightarrow \mathbf{R} .\) Similarly, if \(\operatorname{div} F=0\), show that \(F=\) curl \(G\) for some vector field \(G\) on \(A\).

Let \(f: U \rightarrow \mathbf{R}^{n}\) be a differentiable function with a differentiable inverse \(f^{-1}: f(U) \rightarrow \mathbf{R}^{n}\). If every closed form on \(U\) is exact, show that the same is true for \(f(U)\). Hint: If \(d \omega=0\) and \(f^{*} \omega=d \eta\) consider \(\left(f^{-1}\right) * \eta\) \end{aligned} $$

Let \(S\) be the set of all singular \(n\)-cubes, and \(\mathbf{Z}\) the integers. An \(n\)-chain is a function \(f: S \rightarrow \mathbf{Z}\) such that \(f(c)=0\) for all but finitely many c. Define \(f+g\) and \(n f\) by \((f+g)(c)=\) \(f(c)+g(c)\) and \(n f(c)=n \cdot f(c) .\) Show that \(f+g\) and \(n f\) are \(n\)-chains if \(f\) and \(g\) are. If \(c \in S\), let \(c\) also denote the function \(f\) such that \(f(c)=1\) and \(f\left(c^{\prime}\right)=0\) for \(c^{\prime} \neq c .\) Show that every \(n\)-chain \(f\) can be written \(a_{1} c_{1}+\ldots+a_{k} c_{k}\) for some integers \(a_{1}, \ldots, a_{k}\) and singular \(n\)-cubes \(c_{1}, \ldots, c_{k}\)

If \(c\) is a singular 1-cube in \(\mathbf{R}^{2}-0\) with \(c(0)=c(1)\), show that there is an integer \(n\) such that \(c-c_{1, n}=\partial c^{2}\) for some 2 -chain \(c^{2}\). Hint: First partition \([0,1]\) so that each \(c\left(\left[t_{i-1}, t_{i}\right]\right)\) is contained on one side of some line through \(0 .\)

. If \(\omega\) is a 1-form \(f d x\) on \([0,1]\) with \(f(0)=f(1)\), show that there is a unique number \(\lambda\) such that \(\omega-\lambda d x=d g\) for some function \(g\) with \(g(0)=g(1)\). Hint: Integrate \(\omega-\lambda d x=d g\) on \([0,1]\) to find \(\lambda\).

If \(\omega\) is a 1 -form on \(\mathbf{R}^{2}-0\) such that \(d \omega=0\), prove that $$ \omega=\lambda d \theta+d g $$ for some \(\lambda \in \mathbf{R}\) and \(g: \mathbf{R}^{2}-0 \rightarrow \mathbf{R}\). Hint: If $$ c_{R .1} *(\omega)=\lambda_{R} d x+d\left(g_{R}\right) $$ show that all numbers \(\lambda_{R}\) have the same value \(\lambda\).

If \(\omega \neq 0\), show that there is a chain \(c\) such that \(\int_{c} \omega \neq 0\). Use this fact, Stokes' theorem and \(\partial^{2}=0\) to prove \(d^{2}=0\).

(a) Let \(c_{1}, c_{2}\) be singular 1-cubes in \(\mathbf{R}^{2}\) with \(c_{1}(0)=c_{2}(0)\) and \(c_{1}(1)\) \(=c_{2}(1)\). Show that there is a singular 2 -cube \(c\) such that \(\partial c=\) \(c_{1}-c_{2}+c_{3}-c_{4}\), where \(c_{3}\) and \(c_{4}\) are degenerate, that is, \(c_{3}([0,1])\) and \(c_{4}([0,1])\) are points. Conclude that \(\int_{c_{1} \omega}=\int_{c_{2} \omega}\) if \(\omega\) is exact. Give a counterexample on \(\mathbf{R}^{2}-0\) if \(\omega\) is merely closed. (b) If \(\omega\) is a 1-form on a subset of \(\mathbf{R}^{2}\) and \(\int_{c_{1} \omega}=\int_{c_{2} \omega \text { for all } c_{1}}\) \(c_{2}\) with \(c_{1}(0)=c_{2}(0)\) and \(c_{1}(1)=c_{2}(1)\), show that \(\omega\) is exact.

\- (A first course in complex variables.) If \(f: \mathbf{C} \rightarrow \mathbf{C}\), define \(f\) to be differentiable at \(z_{0} \in \mathbf{C}\) if the limit $$ f^{\prime}\left(z_{0}\right)=\lim _{z \rightarrow z_{0}} \frac{f(z)-f\left(z_{0}\right)}{z-z_{0}} $$ exists. (This quotient involves two complex numbers and this definition is completely different from the one in Chapter 2.) If \(f\) is differentiable at every point \(z\) in an open set \(A\) and \(f^{\prime}\) is continuous on \(A\), then \(f\) is called analy tic on \(A\). (a) Show that \(f(z)=z\) is analytic and \(f(z)=\bar{z}\) is not (where \(\overline{x+i y}=x-i y) .\) Show that the sum, product, and quotient of analytic functions are analytic. (b) If \(f=u+i v\) is analytic on \(A\), show that \(u\) and \(v\) satisfy the Cauchy- Riemann equations: $$ \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y} \quad \text { and } \quad \frac{\partial u}{\partial y}=\frac{-\partial v}{\partial x} $$