Multivariable and vector calculus

A hiker's altitude is $h(x, y) = 3x + 2y^2$ (in suitable units). What is the greatest rate of increase of $h$ at the point $(0, 1)$, over all unit directions?

Let $f(x, y) = \dfrac{xy}{x^2 + y^2}$ for $(x, y) \ne (0, 0)$ and $f(0, 0) = 0$. Which statement is true?

Using polar coordinates, $\displaystyle\iint_{x^2 + y^2 \le 4} 3\sqrt{x^2 + y^2}\;dA = k\pi$. Find $k$.

Find the maximum value of $f(x, y) = x + 2y$ on the circle $x^2 + y^2 = 5$.

The vortex field $F = \big(-y, x\big)/(x^2 + y^2)$ has zero curl on $\R^2 \setminus \{0\}$, yet its circulation around the unit circle is $2\pi$. Which hypothesis of the theorem “a $C^1$ field with zero curl on a domain $U$ is conservative on $U$” fails?

For $f(x, y) = x^3 - 3x + y^2$, which statements are true? (Select all that apply.)

Let $F(x, y, z) = (x + y^2,\; y + z^2,\; z + x^2)$. The outward flux of $F$ through the sphere of radius $2$ centred at the origin is $k\pi$. Find $k$.

In Lean, xyQuot p = p.1 * p.2 / (p.1 ^ 2 + p.2 ^ 2). Which statement formalises “both partial derivatives exist at the origin, but the function is not differentiable there”?

Write circulation(P, Q, r, n) that approximates $\oint (P\,dx + Q\,dy)$ counterclockwise around the circle of radius r centred at the origin, using the parametrisation $(r\cos t, r\sin t)$ and the rectangle (left-point) rule with n equal steps in $t \in [0, 2\pi]$. Test it on the rotation field $(-y, x)$ (circulation $2\pi r^2$), a gradient field (circulation $0$) and the vortex (circulation $2\pi$).