Past exam of the mathematical methods II course of the University of Oxford / 2021 resit / 9

Regard the planar vector field as $(A(x,y),B(x,y),0)$, with no dependence on $z$. Its curl isThus, for continuously differentiable $A$ and $B$, the necessary and sufficient pointwise condition isTo deduce global path independence from this local condition also requires a suitable domain, for example a simply connected domain.

Set $P=10x^4-2x{y}^3$ and $Q=-3x^2{y}^2$. Their partial derivatives satisfy $P_y=-6x{y}^2=Q_x$ throughout ${\mathbb{R}}^2$. The vector field is smooth on the entire simply connected plane, so zero curl proves path independence.

Choose the broken straight path $(0,0)\to (2,0)\to (2,1)$. Along its horizontal segment $y=0$ and $dy=0$, givingAlong its vertical segment $x=2$ and $dx=0$, givingTherefore the line integral is

An exact differential is an expression $Pdx+Qdy$ that equals $df=f_{x}dx+f_{y}dy$ for a single-valued differentiable function $f$ on the domain. Its line integral is an endpoint difference by the fundamental theorem for line integrals.

Integrating $f_x=P$ with respect to $x$ gives $f=2x^5-x^2{y}^3+g(y)$. Matching $f_y=-3x^2{y}^2+g^{\prime }(y)$ to $Q$ requires $g^{\prime }(y)=0$. HenceThe arbitrary additive constant cancels. This constructs the vector-field potential globally and provides a second proof of path independence.

Assume a simple, piecewise smooth closed curve oriented counterclockwise around its enclosed region $D$. Apply Green's theorem with $P=-y$ and $Q=x$:The orientation assumption is essential. A clockwise unit circle, $x=\cos \theta$, $y=-\sin \theta$ for $0\le \theta \le 2\pi$, gives $-2\pi$ although its enclosed area is $\pi$. For a self-intersecting curve the same line integral measures twice the signed area, with multiplicity, rather than necessarily twice an ordinary enclosed area.

For positive semiaxes $a,b$, the ellipse has the counterclockwise parametrization $x=a\cos \theta$, $y=b\sin \theta$. ThenThereforeIf nonzero signed constants are allowed in place of positive semiaxes, the geometric area is $\pi |ab|$.