Due date: Monday 04. March 2024, 10 AM.
Exercise 1
Let \(U\subset \mathbb{R}^n\) be open, \(X : U \to T\mathbb{R}^n\) be a smooth vector field, \(\gamma : [a,b] \to U\subset \mathbb{R}^n\) a smooth curve and \(f : U \to \mathbb{R}\) a smooth function.
Show that for all \(t \in [a,b]\) we have \[(f\circ \gamma)^{\prime}(t)=\mathrm{d}f(\dot{\gamma}(t)).\]
The path integral of \(X\) along \(\gamma\) is defined as \[\int_\gamma X:=\int_a^b \langle X(\gamma(t)),\dot{\gamma}(t)\rangle\mathrm{d}t.\] Show that if \(X=\operatorname{grad}f,\) then \[\int_{\gamma} \operatorname{grad}f=f(\gamma(b))-f(\gamma(a)).\]
Exercise 2
Let \(U\subset \mathbb{R}^2\) be open and \(f : U \to \mathbb{R}^2\) a smooth map. We say \(f\) is conformal or angle preserving if \(f_* : T_p\mathbb{R}^2 \to T_{f(p)}\mathbb{R}^2\) has positive determinant for all \(p \in U\) and if for all \(\vec{v}_p,\vec{w}_p\) in \(T_p\mathbb{R}^2\setminus\{\vec{0}_p\}\) the angle between \(\vec{v}_p\) and \(\vec{w}_p\) is the same as the angle between \(f_*(\vec{v}_p)\) and \(f_*(\vec{w}_p).\)
Show that the smooth map \[f : \mathbb{R}^2\setminus \{(0,0)\} \to \mathbb{R}^2, \qquad p=(x,y) \mapsto (x^2-y^2,2xy)\] is conformal.
Write \(f=(u,v)\) for smooth functions \(u,v : U \to \mathbb{R}^2.\) Derive equations that the partial derivatives of \(u\) and \(v\) need to satisfy so that \(f : U \to \mathbb{R}^2\) is conformal.
Exercise 3
On \(\mathbb{R}^n\) consider the smooth vector field – called the Euler vector field – given by \[X(p)=x_1\frac{\partial}{\partial x_1}(p)+\cdots+x_n\frac{\partial}{\partial x_n}(p),\] where we write \(p=(x_1,\ldots,x_n).\)
A smooth function \(f : \mathbb{R}^{n}\setminus\{0_{\mathbb{R}^n}\} \to \mathbb{R}\) is called positively homogeneous of degree \(c \in \mathbb{R}\) if \[f(s p)=s^{c}f(p)\] for all \(s \in \mathbb{R}^+\) and all \(p \in \mathbb{R}^{n}\setminus\{0_{\mathbb{R}^n}\}.\) Show that if \(f\) is positively homogeneous of degree \(c,\) then \[X(f)=c f.\]