The Lumenary · Early undergraduate

Multivariable and vector calculus

Name: ______________________

Date: ______________________

You will learn to

Compute partial and directional derivatives, gradients and Jacobian matrices, and explain why the derivative of a map from $\R^n$ to $\R^m$ is a linear map
Prove the chain rule and that continuous partial derivatives imply differentiability, and give a function whose partial derivatives exist but which is not differentiable
Solve constrained optimisation problems with Lagrange multipliers and justify the method geometrically
Evaluate multiple integrals with Fubini's theorem and the change of variables formula, including the Jacobian factor, and compute the Gaussian integral
Evaluate line and surface integrals, use the gradient theorem, and apply Green's theorem, Stokes's theorem and the divergence theorem

1.
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?
2.
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?
- A.Both partial derivatives exist at $(0, 0)$ , but $f$ is not differentiable there.
- B. $f$ is differentiable at $(0, 0)$ with $Df(0, 0) = 0$ , since both partials are $0$ .
- C.The partial derivatives do not exist at $(0, 0)$ .
- D. $f$ is continuous at $(0, 0)$ but not differentiable.
3.
Using polar coordinates, $\displaystyle\iint_{x^2 + y^2 \le 4} 3\sqrt{x^2 + y^2}\;dA = k\pi$ . Find $k$ .
4.
Find the maximum value of $f(x, y) = x + 2y$ on the circle $x^2 + y^2 = 5$ .
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?
- A. $U$ must be simply connected (e.g. a disc or a star-shaped set); the punctured plane is not.
- B. $F$ is not $C^1$ on $\R^2 \setminus \{0\}$ .
- C.The curl is not really zero; the computation is wrong.
- D.None: the theorem is false, as this field shows.
6.
For $f(x, y) = x^3 - 3x + y^2$ , which statements are true? (Select all that apply.)
- A. $(1, 0)$ is a local minimum.
- B. $(-1, 0)$ is a saddle point.
- C. $(1, 0)$ is the global minimum of $f$ on $\R^2$ .
- D. $(0, 0)$ is a critical point.
7.
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 $k$ .
8.
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”?
- A.HasDerivAt (fun t => xyQuot (t, 0)) 0 0 ∧ HasDerivAt (fun t => xyQuot (0, t)) 0 0 ∧ ¬ DifferentiableAt ℝ xyQuot 0
- B.HasDerivAt (fun t => xyQuot (t, t)) 0 0 ∧ ¬ DifferentiableAt ℝ xyQuot 0
- C.DifferentiableAt ℝ (fun t => xyQuot (t, 0)) 0 → DifferentiableAt ℝ xyQuot 0
- D.¬ HasDerivAt (fun t => xyQuot (t, 0)) 0 0
9.
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 $2\pi$ ).
Computational — Practice

Answer key — Multivariable and vector calculus

1.
$\nabla h = (3, 4y) = (3, 4)$ at $(0, 1)$ . For a unit vector $u$ , $D_u h = \nabla h \cdot u \le |\nabla h| = 5$ by Cauchy–Schwarz, with equality for $u = (3, 4)/5$ . So the greatest rate is $5$ .
2.
Along the axes $f \equiv 0$ , so both partials exist and are $0$ . Along the diagonal $f(t, t) = t^2/(2t^2) = 1/2$ , so $f(t, t) \not\to f(0, 0)$ : $f$ is discontinuous at $0$ , and a differentiable function is continuous. So (a).
3.
$\int_0^{2\pi}\int_0^2 3r \cdot r\,dr\,d\theta = 2\pi \cdot [r^3]_0^2 = 2\pi \cdot 8 = 16\pi$ , so $k = 16$ . Forgetting the Jacobian gives $2\pi \cdot 6 = 12\pi$ .
4.
From $1 = 2\lambda x$ and $2 = 2\lambda y$ we get $y = 2x$ , and then $5x^2 = 5$ , so $(x, y) = (1, 2)$ or $(-1, -2)$ , with values $5$ and $-5$ . The circle is compact, so the maximum exists and is $5$ . (Check with Cauchy–Schwarz: $x + 2y \le \sqrt5\,\sqrt{x^2 + y^2} = 5$ .)
5.
Curl-free implies conservative on simply connected domains (Poincaré's lemma), not on the punctured plane, which has a hole. The circulation $2\pi$ around the origin is exactly the obstruction; this is the start of de Rham cohomology.
6.
The critical points are $(\pm1, 0)$ . At $(1, 0)$ the Hessian $\mathrm{diag}(6, 2)$ is positive definite: a strict local minimum. At $(-1, 0)$ it is $\mathrm{diag}(-6, 2)$ : a saddle. There is no global minimum since $f(x, 0) \to -\infty$ . So (a) and (b).
7.
$\operatorname{div} F = 1 + 1 + 1 = 3$ , constant. By the divergence theorem the flux is $3 \cdot \frac43\pi \cdot 2^3 = 32\pi$ , so $k = 32$ . A direct surface integral would be much harder: the terms $y^2, z^2, x^2$ contribute zero net flux, by symmetry or by the divergence theorem.
8.
Option (a), which is the conjunction of xyQuot_partials and xyQuot_not_differentiableAt in the lesson. The partial derivatives are one-variable derivatives of t ↦ xyQuot (t, 0) and t ↦ xyQuot (0, t); differentiability is DifferentiableAt ℝ xyQuot 0.
9.
```
import math

def circulation(P, Q, r, n=1000):
    total = 0.0
    for i in range(n):
        t = 2 * math.pi * i / n
        x, y = r * math.cos(t), r * math.sin(t)
        dx, dy = -r * math.sin(t), r * math.cos(t)
        total += P(x, y) * dx + Q(x, y) * dy
    return total * 2 * math.pi / n
```
For smooth periodic integrands this simple rule is extremely accurate. The rotation field has curl $2$ , so by Green's theorem its circulation is $2 \cdot \pi r^2$ ; the gradient field gives $0$ by the gradient theorem; the vortex gives $2\pi$ for every radius — the signature of the hole at the origin.