Level 1 — RecognitionMultivariable Calculus

Multivariable Calculus

⏱ 20 minutes30 marksprintable — key stays hidden on paper

Difficulty: Level 1 (Recognition) Time limit: 20 minutes Total marks: 30

Section A — Multiple Choice (1 mark each)

Choose the single best answer.

Q1. The level curves of $f(x,y) = x^2 + y^2$ are:

(a) parabolas
(b) concentric circles
(c) straight lines
(d) hyperbolas

Q2. For $f(x,y) = x^2 y$ , the partial derivative $f_x$ is:

(a) $2xy$
(b) $x^2$
(c) $2x$
(d) $x^2 y$

Q3. Clairaut's theorem guarantees $f_{xy} = f_{yx}$ provided:

(a) $f$ is bounded
(b) both mixed partials exist and are continuous near the point
(c) $f$ is a polynomial only
(d) $f_x = f_y$

Q4. The gradient $\nabla f$ at a point is:

(a) tangent to the level curve
(b) perpendicular to the level curve, pointing toward steepest ascent
(c) always the zero vector at a maximum's boundary
(d) parallel to the $x$ -axis

Q5. The directional derivative of $f$ in the direction of unit vector $\mathbf{u}$ is:

(a) $\nabla f \times \mathbf{u}$
(b) $\nabla f \cdot \mathbf{u}$
(c) $|\nabla f|\,|\mathbf{u}|$
(d) $\nabla f + \mathbf{u}$

Q6. In the second derivative test, if $D = f_{xx}f_{yy} - f_{xy}^2 < 0$ at a critical point, the point is:

(a) a local minimum
(b) a local maximum
(c) a saddle point
(d) inconclusive

Q7. The Jacobian factor when converting a double integral to polar coordinates is:

(a) $1$
(b) $r$
(c) $r^2$
(d) $\rho^2\sin\phi$

Q8. The volume element in spherical coordinates $dV$ equals:

(a) $r\,dr\,d\theta\,dz$
(b) $\rho^2\sin\phi\,d\rho\,d\phi\,d\theta$
(c) $dx\,dy\,dz$
(d) $\rho\,d\rho\,d\phi\,d\theta$

Q9. A vector field $\mathbf{F}$ is conservative if and only if (on a simply connected domain):

(a) $\operatorname{div}\mathbf{F} = 0$
(b) $\operatorname{curl}\mathbf{F} = \mathbf{0}$
(c) $|\mathbf{F}| = 1$
(d) $\mathbf{F} = \nabla \times \mathbf{G}$

Q10. The divergence of $\mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}$ is:

(a) $P_x + Q_y + R_z$
(b) $P_y - Q_x$
(c) $(R_y - Q_z,\, P_z - R_x,\, Q_x - P_y)$
(d) $P + Q + R$

Q11. The Fundamental Theorem for line integrals states $\int_C \nabla f\cdot d\mathbf{r}$ equals:

(a) $0$ always
(b) $f(\text{end}) - f(\text{start})$
(c) the area enclosed by $C$
(d) $\operatorname{curl}f$

Q12. Green's theorem relates a line integral around a closed curve $C$ to:

(a) a surface integral over a sphere
(b) a double integral over the enclosed plane region
(c) a triple integral
(d) a gradient

Section B — Matching (1 mark each, Q13 = 4 marks total)

Q13. Match each theorem (left) with what it generalizes/connects (right).

Theorem	Connection
(i) Divergence Theorem	(A) circulation around boundary curve = flux of curl through surface
(ii) Stokes' Theorem	(B) flux out of closed surface = triple integral of divergence
(iii) Green's Theorem	(C) $\int_C \nabla f\cdot d\mathbf{r} = f(B)-f(A)$
(iv) FTC for line integrals	(D) 2D version of Stokes/Divergence in the plane

Section C — True/False WITH Justification (2 marks each: 1 for verdict, 1 for reason)

Q14. If $\lim_{(x,y)\to(0,0)} f(x,y)$ gives different values along $y=0$ and $y=x$ , then the limit does not exist. (T/F + justify)

Q15. A function that has both partial derivatives $f_x$ and $f_y$ at a point must be differentiable there. (T/F + justify)

Q16. Reversing the order of integration in a double integral over a rectangle always changes the value of the integral. (T/F + justify)

Q17. The gradient vector is the direction in which a function increases most rapidly. (T/F + justify)

Q18. If $\operatorname{curl}\mathbf{F} = \mathbf{0}$ everywhere on $\mathbb{R}^3$ , then $\mathbf{F}$ is conservative. (T/F + justify)

Q19. At an interior local maximum of a differentiable function, $\nabla f = \mathbf{0}$ . (T/F + justify)

Q20. For $f(x,y,z)=x^2+y^2+z^2$ , the level surfaces are spheres centered at the origin. (T/F + justify)

Answer keyMark scheme & solutions

Section A (1 mark each)

Q1 — (b). $x^2+y^2=c$ describes circles of radius $\sqrt c$ ; concentric.

Q2 — (a). Treat $y$ as constant: $\partial_x(x^2 y)=2xy$ .

Q3 — (b). Clairaut requires continuity of the mixed second partials near the point.

Q4 — (b). $\nabla f\perp$ level curve and points in the steepest-ascent direction.

Q5 — (b). $D_{\mathbf u}f=\nabla f\cdot\mathbf u$ for unit $\mathbf u$ .

Q6 — (c). $D<0 \Rightarrow$ saddle point.

Q7 — (b). $dx\,dy = r\,dr\,d\theta$ .

Q8 — (b). Spherical volume element $\rho^2\sin\phi\,d\rho\,d\phi\,d\theta$ .

Q9 — (b). On a simply connected domain, $\operatorname{curl}\mathbf F=\mathbf 0\iff$ conservative.

Q10 — (a). $\operatorname{div}\mathbf F=P_x+Q_y+R_z$ .

Q11 — (b). Path-independence: value = potential difference of endpoints.

Q12 — (b). Green: line integral around $C$ = double integral over enclosed region.

Section B

Q13 (4 marks, 1 each): (i)→(B), (ii)→(A), (iii)→(D), (iv)→(C).

Section C (2 marks each: verdict 1, reason 1)

Q14 — TRUE. If two paths give different limiting values, the two-variable limit cannot exist (uniqueness of limits fails). (1 T + 1 reason)

Q15 — FALSE. Existence of partials does not imply differentiability; a standard counterexample is $f(x,y)=\frac{xy}{x^2+y^2}$ (with $f(0,0)=0$ ), which has both partials at $0$ but is not even continuous there. Differentiability requires the linear approximation error to vanish faster than $\|(h,k)\|$ . (1 F + 1 reason)

Q16 — FALSE. By Fubini's theorem, for an integrable function over a rectangle both iterated integrals give the same value; only the computation path changes. (1 F + 1 reason)

Q17 — TRUE. The maximum of $D_{\mathbf u}f=\nabla f\cdot\mathbf u=|\nabla f|\cos\theta$ occurs at $\theta=0$ , i.e. along $\nabla f$ . (1 T + 1 reason)

Q18 — TRUE (on $\mathbb R^3$ ). $\mathbb R^3$ is simply connected, so curl-free implies the field is a gradient. (1 T + 1 reason) (Note: false on non-simply-connected domains.)

Q19 — TRUE. At an interior extremum of a differentiable function, all partials vanish (Fermat's condition), so $\nabla f=\mathbf 0$ . (1 T + 1 reason)

Q20 — TRUE. $x^2+y^2+z^2=c$ ( $c>0$ ) are spheres of radius $\sqrt c$ centered at origin. (1 T + 1 reason)

[
  {"claim":"f_x of x^2*y is 2xy","code":"x,y=symbols('x y'); result = diff(x**2*y, x) == 2*x*y"},
  {"claim":"Mixed partials of x^2*y equal (Clairaut)","code":"x,y=symbols('x y'); f=x**2*y; result = diff(f,x,y) == diff(f,y,x)"},
  {"claim":"Polar Jacobian is r","code":"r,t=symbols('r t'); x=r*cos(t); y=r*sin(t); J=Matrix([[diff(x,r),diff(x,t)],[diff(y,r),diff(y,t)]]); result = simplify(J.det()) == r"},
  {"claim":"Spherical volume element is rho^2 sin(phi)","code":"rho,ph,th=symbols('rho phi theta'); x=rho*sin(ph)*cos(th); y=rho*sin(ph)*sin(th); z=rho*cos(ph); J=Matrix([[diff(x,rho),diff(x,ph),diff(x,th)],[diff(y,rho),diff(y,ph),diff(y,th)],[diff(z,rho),diff(z,ph),diff(z,th)]]); result = simplify(J.det()) == rho**2*sin(ph)"}
]