Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 2

The multivariable chain rule gives $u_s=2s{u}_x+2t{u}_y$ and $u_t=-2t{u}_x+2s{u}_y$. Thereforeso $\boxed{m=2}$. The $u_y$ terms cancel identically, independently of the particular function $u$.

The partial derivatives areThus the Jacobian determinant isThis vanishing reflects the functional dependence found next.

The tangent addition formula givesHence $\boxed{f(v)=\tan v}$. The restriction $xy\ne 1$ excludes the poles of tangent that can arise in the possible range $-\pi <v<\pi$. One should not replace this identity by $v=\arctan u$ globally: the principal inverse tangent can differ from $v$ by an integer multiple of $\pi$.

A scalar potential is $U(x,y)=x^{3}y+y^4$, since $\nabla U=(3x^{2}y,x^3+4y^3)$. The fundamental theorem for line integrals therefore reduces the line integral to its endpoints. The curve starts at $(0,1)$ and ends at $(2,1)$, so

The region is the part of the disk $x^2+y^2\le a^2$ in the first quadrant above $y=x$. In polar coordinates, it is $0\le r\le a$ and $\pi /4\le \theta \le \pi /2$. The integrand becomes $r$, and the area Jacobian determinant contributes another $r$. Thus