Which sets are simply connected?

\(\displaystyle \R^2\setminus(\Z\times\Z) \qquad \R^3\setminus(\Z\times\Z\times\Z) \qquad \R^3\setminus\{(\cos t,\sin t,0): t\in\R\} \qquad \R^4\setminus\{(\cos t,\sin t,0,0): t\in\R\} \)

Difficulty: 2.

Let \(\displaystyle G \subset \R^p\) be open and connected. Show that the scalar potentials of a vector field \(\displaystyle G\to\R^p\) can differ only in constants.

Difficulty: 3.

The electric field of a homogeneously charged line is orthogonal to the line and its strength at distance \(\displaystyle d\) from the line is \(\displaystyle 2k\rho/d\). Determine the electric potential difference (voltage) between two points.

Difficulty: 3.

Which of the following vector fields are gradient fields? For those that are not give a closed curve on which the line integral of the field does not vanish.

\(\displaystyle (\ch y; x\sh y) \qquad (\ch x; y\sh x) \qquad \left(\frac{x}{x^2+y^2};\frac{y}{x^2+y^2};\right) \)

Difficulty: 4.

Which of the following is simply connected?

\(\displaystyle \R^2\setminus\{(0,0)\} \qquad \R^3\setminus\{(0,0,0)\} \qquad \R^3\setminus\{t,0,0): t\in\R\} \qquad \R^4\setminus\{t,0,0,0): t\in\R\} \)

Difficulty: 5.

Let \(\displaystyle H=\R^3\setminus\{(x,y,0): \; x^2+y^2=1\}\). Give a differentiable irrotational vector field \(\displaystyle H\to\R^3\) which is not a gradient field.

Difficulty: 5.

Which of the following vector fields are gradient fields? For those that are not give a closed curve on which the line integral of the field does not vanish.

\(\displaystyle (x,y) \qquad (y,x) \qquad \left(\frac{-y}{x^2+y^2}, \frac{x}{x^2+y^2}\right) \qquad \left(\frac{x}{\sqrt{x^2+y^2}}, \frac{y}{\sqrt{x^2+y^2}}\right) \)

Difficulty: 5.

Let \(\displaystyle G=\R^3\setminus\{(x,x,x): x\in\R\}\). Find a differentiable vector field \(\displaystyle X:G\to\R^3\) that is irrotational (\(\displaystyle \curl X={\bf{0}}\)) but is not a gradient field.

Difficulty: 5.

Redo the proof of Goursat's lemma for rectangles.

Difficulty: 6.

Let \(\displaystyle G\subset\R^2\) be open and let \(\displaystyle \phi_u(t)\) \(\displaystyle [0,1]^2\to G\) be continuously differentiable family of curves. Show that for a continuously differentiable \(\displaystyle f:G\to\R^2\) irrotational vector field the \(\displaystyle I(u)=\int_{\phi_u} \langle f,\mathrm{d}x\rangle \) parametric line integral satisfies \(\displaystyle I'(u)=0\).

Difficulty: 8.

Is \(\displaystyle H=\R^3\setminus\{(\cos t, \sin t, e^t):t\in\R\}\) simply connected?

Difficulty: 9. Answer (final result) is provided for this problem.

Let \(\displaystyle G\subset\R^p\) be open, let \(\displaystyle f:G\to\R^p\) be differentiable and irrotational and let \(\displaystyle g,h:[0,1]\to G\) be continuously differentiable curves with the same initial and endpoints. (I.e. \(\displaystyle g(0)=h(0)\) and \(\displaystyle g(1)=h(1)\).) Assume that \(\displaystyle g\) and \(\displaystyle h\) are homotopic, \(\displaystyle \exists \phi:[0,1]^2\to\R^p\) continuous such that \(\displaystyle \phi(t,0)=g(t)\), \(\displaystyle \phi(t,1)=h(t)\), and \(\displaystyle \phi(0,u)=g(0)=h(0)\), \(\displaystyle \phi(1,u)=g(1)=h(1)\) for all \(\displaystyle u\in[0,1]\).

(a) Show from Goursat's lemma that \(\displaystyle \int_g\langle f,\mathrm{d}x\rangle =\int_h\langle f,\mathrm{d}x\rangle \).

(b) Assume in addition that \(\displaystyle \phi\) is continuously differentiable \(\displaystyle I(u) = \int_{\phi(\cdot,u)} \langle f,\mathrm{d}x\rangle \). Prove directly that \(\displaystyle I'=0\).

Difficulty: 10.

Let \(\displaystyle G=\R^2\setminus\{(-1,0), (1,0)\}\), and \(\displaystyle g\) be the curve shown on the figure.

(a) Show that the line integral of any differentiable irrotational vector field \(\displaystyle f:G\to\R^2\) along \(\displaystyle g\) is zero.

(b) Is \(\displaystyle g\) homotopic to a point in \(\displaystyle G\)?

(c) Is \(\displaystyle g\) homologous to \(\displaystyle 0\) in \(\displaystyle G\)?

Difficulty: 10.