📝 Differentials of multivariable functions (14 MCQs)

📖 Explanation: When applying the total differential $dV = \frac{\partial V}{\partial r}dr + \frac{\partial V}{\partial h}dh$ and substituting $h=4r$, the relative error becomes $\frac{dV}{V} = 3\frac{dr}{r} + \frac{dh}{h}$. Even with equal percentage measurement errors, the radius term carries a factor of 3 due to the geometric relationship, making radius precision critically more important for volume accuracy in this constrained design scenario.

📖 Explanation: The displacement from $(3,4)$ to $(3.1, 3.9)$ gives $dx = +0.1$ but $dy = -0.1$. The partial derivatives are $f_x = 3/5$ and $f_y = 4/5$, so $df = (3/5)(0.1) + (4/5)(-0.1) = -0.02$. While mathematically correct for the linear approximation, the actual function value increases because the path moves closer to the origin's radial direction; the error arises from expecting the linear model to capture curvature over this step size.

📖 Explanation: Stability analysis examines system response to infinitesimal perturbations near equilibrium. The total differential linearizes the equation of state locally, transforming the nonlinear constraint into a linear relation among differentials. This linearization allows construction of Jacobian matrices and application of Lyapunov or eigenvalue methods, which fundamentally require the differential structure rather than discrete finite changes that would obscure the local geometric behavior of the state surface.

📖 Explanation: Contour line density directly reflects gradient magnitude: $|\nabla f| \approx \Delta z / \Delta s$ between adjacent contours. Closely spaced contours at $A$ indicate steep slope, meaning $|\nabla f(A)|$ is large. Since $dz = \nabla f \cdot d\mathbf{s}$ and walking in steepest ascent aligns $d\mathbf{s}$ with $\nabla f$, we get $dz = |\nabla f| ds$. Thus, for identical $ds$, the point with denser contours yields a larger vertical differential, visually encoding rate of change without explicit formulas.

📖 Explanation: This problem mixes constrained differentials with variable elimination. Computing $du = 2xy\,dx + x^2\,dy$ and $dv = y^2\,dx + 2xy\,dy$ first preserves structural clarity. Then imposing $dx = -dy$ from $x+y=5$ yields two equations in $dx, dy$. Solving this system ensures consistency because premature substitution can obscure dependencies or introduce domain restrictions. The key insight is maintaining the differential framework while respecting constraints through simultaneous relations rather than sequential reduction.

📖 Explanation: The equation $df = 0$ yields $f_x dx + f_y dy = 0$, which describes the tangent direction to level curves at each point, not the global curve. For $f = x^3 - 3xy^2$, solving $df = 0$ gives a differential equation whose solutions are indeed the level curves, but this works because the form is exact. The student’s error is assuming $df = 0$ directly equals the level set rather than its infinitesimal generator. Distinguishing between the differential condition and its integral curves requires understanding that $df = 0$ is a necessary local condition, not a sufficient global definition.

📖 Explanation: The function $f = \sqrt{x^2+y^2}$ has significant curvature, but $g = x^2 + y^2$ is quadratic with constant second derivatives, making its linear approximation via differentials exact up to second order. Computing $dg = 2x\,dx + 2y\,dy$ at $(3,4)$ gives $\Delta g \approx 2(3)(0.02) + 2(4)(-0.02) = -0.04$, so $g \approx 25 - 0.04 = 24.96$, and $\sqrt{24.96} \approx 4.996$. This leverages algebraic simplification to reduce approximation error inherent in directly linearizing a composite radical function.

📖 Explanation: The differential formula assumes errors combine linearly and infinitesimally, effectively treating $\Delta Q/Q$ as a first-order Taylor expansion. If all errors align in the worst-case direction (e.g., $a$ and $b$ overestimated while $c$ underestimated), and the relative errors are large enough that higher-order terms matter, the linear sum underestimates the true extremal deviation. This highlights that differential-based error propagation is valid for small, random errors but fails for systematic biases or large uncertainties where nonlinear effects dominate the error budget.

📖 Explanation: Positional uncertainty is quantified by the covariance ellipsoid defined by $\Sigma$. The differential displacement variance in direction $\mathbf{v}$ is $\mathbf{v}^T \Sigma \mathbf{v}$. Minimizing this quadratic form subject to $\|\mathbf{v}\|=1$ yields the smallest eigenvalue direction. Aligning motion with this eigenvector ensures that accumulated differential errors grow slowest, as the projection of noise onto the trajectory is minimized. This connects differential geometry of uncertainty with optimal control, showing how spectral properties of $\Sigma$ dictate robust path planning through local error propagation.

📖 Explanation: At $(1,1,1)$, all partials $F_x = F_y = F_z = 0$, indicating a singular point where the gradient vanishes. The implicit function theorem requires $\nabla F \neq \mathbf{0}$ to guarantee a smooth local graph $z = f(x,y)$. Here, the surface has a conical singularity (it factors as $(x+y+z)(x^2+y^2+z^2-xy-yz-zx)=0$), so no well-defined tangent plane or differential exists. Recognizing singularity versus mere zero denominator prevents misapplying differential calculus to non-manifold points.

📖 Explanation: If $df = dg$, then $d(f-g) = 0$, meaning $f-g$ is locally constant. On a simply connected domain, local constancy implies global constancy because there are no holes to support different constants in separate regions. Without simple connectivity, $f-g$ could take different constant values in disconnected components or around non-contractible loops. Thus, the correct inference is $f = g + C$ with a single global constant, and topological assumptions are crucial for upgrading local differential equality to global functional equality.

📖 Explanation: The total differential $d\mathcal{L} = (f_x - \lambda g_x)dx + (f_y - \lambda g_y)dy - (g-c)d\lambda$. Setting $d\mathcal{L} = 0$ for arbitrary independent variations $dx, dy, d\lambda$ forces each coefficient to vanish, yielding $f_x = \lambda g_x$, $f_y = \lambda g_y$, and $g = c$ simultaneously. This elegantly packages the constrained optimality conditions as a single stationarity requirement in an augmented space, demonstrating how differentials unify constraints and objectives through dimensional extension rather than sequential enforcement.

📖 Explanation: The fundamental relation $dU = TdS - PdV$ applies strictly to closed simple compressible systems. If $(\partial U/\partial V)_T \neq -P$, additional work modes or mass transfer may be present. For open systems, $dU = TdS - PdV + \mu dN$, and holding $T$ constant while varying $V$ at fixed $N$ still gives $-P$, but if $N$ varies implicitly with $V$ in experiments, the measured derivative includes $\mu (\partial N/\partial V)_T$. Thus, apparent violation signals unaccounted extensive variables, emphasizing that differential forms encode system boundaries and must match experimental constraints.

📖 Explanation: At $(0,0)$, $f_x = y e^{xy} = 0$ and $f_y = x e^{xy} = 0$, so $df = 0$. However, $f(x,y) = 1 + xy + \frac{(xy)^2}{2} + \cdots$, so $\Delta f \approx xy$ for small displacements, which is second-order. Unlike polynomials where vanishing gradient implies flatness captured by higher-degree monomials, exponentials have infinite series with coupled variables; the leading non-zero term $xy$ is invisible to first-order differentials. This illustrates that $df = 0$ does not imply local constancy for non-polynomial functions, demanding awareness of Taylor series structure beyond linearization.