📝 Partial derivatives of three variable functions (14 MCQs)

📖 Explanation: The chemical potential is rigorously defined as the partial derivative of internal energy with respect to particle number while keeping entropy and volume constant. This constraint ensures that the measured energy change arises solely from adding particles, not from heat transfer or mechanical work, reflecting true thermodynamic conjugacy.

📖 Explanation: When computing partial derivatives in applied models, one must distinguish between explicit and implicit dependencies. If $T = g(K)$, then the total derivative $dQ/dK$ includes both $\partial Q/\partial K$ and $(\partial Q/\partial T)(dT/dK)$. Ignoring the latter yields an incomplete marginal productivity measure, leading to flawed investment decisions in economic modeling.

📖 Explanation: The student’s conclusion that $f_{xyz} = 0$ is numerically correct, but the justification is misleading. Mixed partials depend on sequential differentiation, not just monomial structure. Here, $f_y = x^2 + zw$, then $f_{yx} = 2x$, and $f_{yxz} = 0$. By Clairaut’s theorem, all orders yield zero, but the reasoning must invoke differentiation rules, not syntactic presence of variables.

📖 Explanation: Contour line spacing inversely correlates with gradient magnitude in the projected plane. Dense contours along $a$ imply rapid change in $h$ with respect to $a$, hence large $|\partial h/\partial a|$. Sparse contours along $b$ indicate gentle variation, so smaller $|\partial h/\partial b|$. This interpretation holds even in higher dimensions when other variables are fixed, making 2D slices valid for local partial analysis.

📖 Explanation: The partial derivative $\partial T/\partial t$ measures temperature change at a fixed location, ignoring movement of air masses. Actual observed change follows the material derivative $DT/Dt = \partial T/\partial t + \vec{v} \cdot \nabla T$. Thus, $\partial T/\partial t$ alone predicts real change only when advection vanishes ($\vec{v} = 0$). In most atmospheric contexts, neglecting advection leads to significant forecast errors, highlighting the distinction between Eulerian and Lagrangian perspectives.

📖 Explanation: Symmetry of the function does not imply equality of arbitrary mixed partials. While $F_{uv} = F_{vu}$ by Clairaut’s theorem, $F_{uv} = wx$ and $F_{wx} = uv$ involve different variable pairs and are generally unequal. Symmetry means $F(u,v,w,x) = F(\sigma(u,v,w,x))$ for permutations $\sigma$, but mixed partials transform accordingly. Equality occurs only under specific value constraints, not as a general property, revealing a common misconception about symmetry and derivatives.

📖 Explanation: In observational or imperfectly controlled experiments, estimated coefficients approximate partial derivatives only if other variables are truly held constant. Drift in $T$ and $H$ introduces omitted variable bias, making regression estimates reflect combined effects rather than pure $\partial \sigma / \partial \epsilon$. Valid partial derivative estimation requires either experimental control or statistical adjustment (e.g., multiple regression with all relevant predictors), emphasizing the gap between mathematical definition and empirical practice in multivariable systems.

📖 Explanation: For a monomial $p^a q^b r^c s^d$, the mixed partial $\partial^{a+b+c+d} G / \partial p \partial q \partial r \partial s$ equals $a! b! c! d!$ only if differentiating each variable exactly its exponent times. Here, differentiating once per variable gives $(1)(2q)(3r^2)(4s^3)$ evaluated at 1, yielding $1 \cdot 2 \cdot 3 \cdot 4 = 24$. The key insight is recognizing that single differentiation per variable preserves non-zero result, and factorial logic applies only for repeated differentiation of same variable.

📖 Explanation: When taking $\partial^2 f / \partial x \partial y$, treat $z$ as constant during both differentiations since it is an independent variable. First, $\partial f / \partial y = x z \sin(z)$. Then $\partial / \partial x$ of that is $z \sin(z)$. The product rule involving $\sin(z)$ and $z$ is irrelevant here because neither depends on $x$ or $y$. This tests understanding of variable independence in partial differentiation versus ordinary calculus contexts.

📖 Explanation: Constrained optimization restricts movement to a manifold defined by constraints. The standard Hessian measures curvature in full space, including infeasible directions. The bordered Hessian augments the matrix with constraint gradients to evaluate curvature only along tangent directions of the feasible set. Its signature determines constrained extremum type via modified Sylvester’s criterion. Applying unconstrained tests ignores geometric restrictions, potentially misclassifying saddle points as minima or vice versa, underscoring the need for specialized tools in multivariable constrained analysis.

📖 Explanation: Correct application of chain rule gives $\partial A / \partial w_1 = \text{sech}^2(w_1 w_2 + w_3 w_4) \cdot w_2$. Treating $w_1 w_2$ as atomic ignores the inner derivative $\partial (w_1 w_2)/\partial w_1 = w_2$, yielding only $\text{sech}^2(\cdot)$. This omission scales the true gradient by $1/w_2$ (when $w_2 \neq 0$), distorting update steps. In deep learning, such errors propagate through layers, causing training instability or stagnation, illustrating precise partial derivative computation’s role in algorithmic correctness.

📖 Explanation: The heat equation links temporal change to the full Laplacian $\nabla^2 P = P_{xx} + P_{yy} + P_{zz}$. Knowing only $P_{xx} > 0$ is insufficient; $P_{yy}$ and $P_{zz}$ could dominate negatively. Even if net Laplacian were positive, $k$’s sign determines increase/decrease. Vanishing first derivatives indicate a critical point, but curvature in all directions governs diffusion. This highlights that partial information about spatial derivatives cannot predict dynamics without complete second-order context in PDEs.

📖 Explanation: Logarithmic differentiation transforms products into sums: $\ln f = xyz$ for single exponential, offering no gain. But if $f = \prod_{i=1}^n e^{g_i(x,y,z)}$, then $\ln f = \sum g_i$, and $\partial f / \partial x = f \cdot \sum \partial g_i / \partial x$. This avoids applying product rule across $n$ terms, reducing algebraic complexity. The advantage emerges in multiplicative models common in probability or economics, where log-space computation streamlines partial derivatives despite initial transformation cost.

📖 Explanation: Clairaut’s theorem guarantees equality of mixed partials only when those partials are continuous. In theory, physical fields are smooth, but numerical methods use discrete grids where continuity isn’t assured. Finite-difference schemes may produce asymmetric results if step sizes differ or if underlying data has noise. Thus, while mathematically true under ideal conditions, practical computation demands verification of smoothness or use of symmetric stencils. This bridges abstract analysis and applied numerics, showing why theoretical assumptions matter in implementation.