Let \(C\) be a smooth curve in \(\R^n\) parametrized by \(g:[a,b]\to\R^n\) with \(g'(t)\neq 0\). Define the arc length of \(C\).
Let \(F\) be a continuous vector field defined in a neighborhood of a smooth oriented curve \(C\) in \(\R^n\).
Define the line integral \(\displaystyle\int_C F\cdot dx\).
What does it mean for a vector field \(F:U\to\R^n\) (with \(U\subseteq\R^n\) open) to be conservative?
What is a potential function for \(F\)?
What is a piecewise smooth curve? A closed curve? A simple curve?
One arch of the cycloid is the curve \(g(t)=(t-\sin t,\; 1-\cos t)\) with \(t\in[0,2\pi]\).
Find the arc length of one arch of the cycloid.
The logarithmic spiral is the curve \(g(t) = (e^{-t}\cos t,\; e^{-t}\sin t)\) for \(t \in [0, \infty)\). Show that this curve has finite total arc length and compute it.
Define \(F : \R^2 \to \R^2\) via \(F(x,y)=(x^2 y,\;-xy^2)\). Is \(F\) conservative?
The angle form on \(\R^2\setminus\{0\}\) is \( \omega = \displaystyle\frac{-y\,dx+x\,dy}{x^2+y^2} \).
Integrate \(\omega\) over various curves, like
Show that on the half-plane \(U=\{(x,y):x>0\}\), the form \(\omega\) equals \(d\!\left(\arctan(y/x)\right)\). Conclude that \(\omega\) is conservative on \(U\), and find a potential.
Suppose \(U\subseteq\R^n\) is open and \(F:U\to\R^n\) is \(C^1\) and conservative.
Give names to the components of \(F\) via \(F(u) = (F_1(u),F_2(u),\ldots,F_n(u))\).
Show that \( \displaystyle\frac{\partial F_i}{\partial x_j}=\frac{\partial F_j}{\partial x_i}\) on \(U\) for all \(i,j\).
A \(C^1\) vector field \(F = (F_1, \ldots, F_n)\) on an open set \(U \subseteq \R^n\) is called closed if \(\dfrac{\partial F_i}{\partial x_j} = \dfrac{\partial F_j}{\partial x_i}\) for all \(i, j\). It is called exact if it is conservative (cf.~S08P03). Explain, using S08P11, why every exact field is closed.
Suppose \(f:\R^n\to\R\) is \(C^1\), and let \(C\) be a piecewise smooth curve from \(a\) to \(b\) in \(\R^n\). Prove the fundamental theorem for line integrals, namely that \[ \int_C \nabla f\cdot dx = f(b)-f(a). \]
An open set \(U\subseteq\R^n\) is star-shaped with respect to the origin if for every \(x\in U\) and every \(t\in[0,1]\), we have \(tx\in U\). Let \(F:U\to\R^n\) be \(C^1\) and closed (cf.~S08P12). Define \(f:U\to\R\) by \[ f(x)=\int_0^1 F(tx)\cdot x\,dt. \] Prove that \(\nabla f = F\). This is the Poincar\'e lemma: on star-shaped domains, closed implies exact.
Parametrize the unit circle counterclockwise. Compute \(\displaystyle\oint x\,dy\) and \(\displaystyle \oint y\,dx\).
Then parametrize the boundary of the rectangle \([0,a]\times[0,b]\) counterclockwise. Compute the same two integrals.
Do the same for the triangle with vertices \((0,0)\), \((1,0)\), \((0,1)\).
What do you observe? What are these integrals calculating?
Now consider the “figure-eight” curve \(\gamma(t)=(\sin t,\;\sin 2t)\) for \(t\in[0,2\pi]\).
Compute \(\displaystyle\oint x\,dy\). What does this mean for your observations in S08P15?
Prove that arc length is independent of parametrization. More precisely, suppose \(g:[a,b]\to\R^n\) is \(C^1\) with \(g'(t)\neq 0\), and let \(\varphi:[c,d]\to[a,b]\) be \(C^1\) with \(\varphi'>0\) (an orientation-preserving reparametrization). Show that \(\tilde g=g\circ\varphi\) has the same arc length as \(g\). What happens if \(\varphi'<0\)?
Let \(C\) be a piecewise smooth curve in \(\R^n\) and \(F\) a continuous vector field on \(C\). Then \[ \left|\int_C F\cdot dx\right| < \int_C |F|\,ds. \]
For any \(C^1\) curve \(\gamma:[a,b]\to\R^n\), the arc length satisfies \(L(\gamma)\ge|\gamma(b)-\gamma(a)|\), with equality if and only if \(\gamma\) parametrizes a line segment.