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📝 Equality of mixed partial derivatives theorem (13 MCQs)

📖 From Calculus • 14. Partial Derivatives Calculus • 13 questions available

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What is Equality of mixed partial derivatives theorem?

Definition:
Clairaut's Theorem states $f_{xy} = f_{yx}$ at points where both mixed partials are continuous, ensuring order independence.

Example:
For smooth $f(x,y) = x^2y^3 + \sin(xy)$ , mixed partials equal everywhere; pathological counterexamples exist only where continuity fails.

Reason:
This symmetry simplifies computation and theoretical proofs, reducing independent second derivatives from four to three for two-variable functions.

Easy

Medium

Hard

📝 All Equality of mixed partial derivatives theorem MCQs

Q1. A student claims that for any function $f(x,y)$ , the mixed partial derivatives $f_{xy}$ and $f_{yx}$ are always equal at every point in the domain. Which scenario best refutes this universal claim by highlighting the necessary conditions?

A.The function is defined on a non-rectangular domain.

B.The second-order partial derivatives exist but are not continuous at the point in question. ✅

C.The first-order partial derivatives are zero at the origin.

D.The function involves trigonometric terms rather than polynomials.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: This question targets error analysis regarding Clairaut's Theorem. Students often memorize that mixed partials are equal without recalling the crucial hypothesis of continuity. Equality fails specifically when second derivatives exist but lack continuity, making option B the precise mathematical refutation of the universal claim.

Q2. Consider a physical potential field modeled by $V(x,y)$ . If experimental data suggests $V_{xy}(a,b) \neq V_{yx}(a,b)$ at a specific coordinate, what is the most rigorous mathematical interpretation of this discrepancy within the context of smooth modeling?

A.The measurement instruments have calibration errors causing asymmetric readings.

B.The potential field is not twice continuously differentiable at

(a,b)

, indicating a singularity or phase transition. ✅

C.Mixed partial derivatives are never equal in physical systems involving vector fields.

D.The order of differentiation was incorrectly applied during data processing.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: This scenario-based question links abstract calculus to physical modeling. In smooth physical theories, equality holds; thus, observed inequality implies a breakdown in smoothness (C2 condition). It tests conceptual understanding that mathematical theorems describe idealized smoothness, and violations signal physical singularities rather than computational mistakes.

Q3. Given $f(x,y) = x^3 y^2 + \sin(xy)$ , a student computes $f_{xy}$ and obtains $6x^2 y + \cos(xy) - xy\sin(xy)$ . Without recomputing $f_{yx}$ , how can one verify this result using higher-order reasoning?

A.Differentiate with respect to

y

then

x

to confirm symmetry.

B.Check if the expression is symmetric under variable exchange since the original function is symmetric. ✅

C.Assume equality holds and check dimensional consistency of each term.

D.Evaluate the limit of the difference quotient along the line

y=x

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: This application question avoids brute-force computation. Since $f(x,y)$ is symmetric ( $f(x,y)=f(y,x)$ ), its mixed partials must yield identical functional forms. Recognizing structural symmetry saves time and demonstrates deep conceptual understanding of how function properties constrain derivative behavior beyond mere calculation algorithms.

Q4. Analyze the following incorrect reasoning: 'Since $f_x(0,0) = 0$ and $f_y(0,0) = 0$ , the mixed partials $f_{xy}(0,0)$ and $f_{yx}(0,0)$ must both be zero.' What is the fundamental flaw in this deduction?

A.It confuses first-order critical points with second-order derivative values. ✅

B.It assumes continuity of first derivatives implies equality of second derivatives.

C.It incorrectly applies the product rule to constant values.

D.It fails to account for the domain being open versus closed.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: This error analysis question addresses a common misconception. Vanishing first derivatives at a point provide no information about the rate of change of those derivatives (second partials). Students must distinguish between the value of a function and the value of its derivative, recognizing that critical points do not dictate curvature or mixed rates.

Q5. A contour plot of $z=f(x,y)$ shows perfect reflectional symmetry across the line $y=x$ . At a point $P$ on this line of symmetry, what can be definitively concluded about $f_{xy}(P)$ and $f_{yx}(P)$ based solely on graphical evidence?

A.They must be equal due to geometric symmetry implying functional symmetry. ✅

B.They must be opposite in sign due to orthogonal axes.

C.Nothing can be concluded without an explicit algebraic formula.

D.They are equal only if the contour lines are circles.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: This graph-based question requires interpreting visual symmetry as mathematical structure. Reflectional symmetry across $y=x$ means $f(x,y)=f(y,x)$ . For such functions, mixed partials are necessarily equal at symmetric points. This connects visual intuition to analytical properties without requiring symbolic manipulation, testing spatial-mathematical translation skills.

Q6. When verifying equality of mixed partials for $g(r,\theta)$ in polar coordinates, a student transforms to Cartesian, checks equality, and concludes it holds in polar. Why might this approach be conceptually insufficient for points at the origin?

A.Polar coordinates are undefined at the origin, so transformation validity breaks down there. ✅

B.The Jacobian determinant is always zero in polar coordinates.

C.Cartesian equality never implies polar equality anywhere.

D.Mixed partials are coordinate-independent scalars by definition.

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: This challenging question integrates coordinate transformations with theorem prerequisites. While equality is intrinsic, the transformation method assumes valid differentiability in both systems. At the origin, polar coordinates have a singularity; thus, Cartesian verification doesn't guarantee polar mixed partials are well-defined or equal there, testing nuanced understanding of coordinate singularities.

Q7. For the piecewise function $h(x,y)$ defined as $xy(x^2-y^2)/(x^2+y^2)$ away from origin and 0 at origin, computing $h_{xy}(0,0)$ via limit definition yields 1 while $h_{yx}(0,0)$ yields -1. What does this demonstrate about the relationship between existence and continuity?

A.Existence of mixed partials at a point does not guarantee their continuity or equality. ✅

B.Mixed partials can never exist for rational functions at the origin.

C.The limit definition is invalid for piecewise functions.

D.Continuity of first derivatives is sufficient but not necessary for equality.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: This Olympiad-style counterexample probes the boundary of Clairaut's Theorem. It demonstrates that mixed partials can exist yet differ, precisely because they lack continuity at the point. Students must understand that existence alone is insufficient; continuity is the bridging condition, reinforcing the distinction between pointwise existence and local regularity.

Q8. In thermodynamics, state functions like entropy $S(T,P)$ satisfy $S_{TP} = S_{PT}$ . If a proposed equation of state yields unequal mixed partials, what is the most appropriate scientific conclusion?

A.The equation of state is mathematically inconsistent with the definition of a state function. ✅

B.Experimental measurements of temperature and pressure are inaccurate.

C.Entropy is not a true state function in this regime.

D.Partial derivatives in thermodynamics follow different rules than in pure mathematics.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: This mixed-concepts question bridges calculus and physics. State functions are exact differentials by definition, requiring equality of mixed partials (Maxwell relations). Inequality implies the proposed model violates fundamental thermodynamic consistency, not experimental error. It tests application of mathematical criteria to validate physical theories, emphasizing interdisciplinary reasoning.

Q9. A student argues: 'I computed $f_{xy}$ and got a complex expression. Computing $f_{yx}$ gave the same expression after simplification, so Clairaut's Theorem is verified.' Why is this reasoning circular or incomplete?

A.Verification requires checking continuity of second derivatives, not just algebraic coincidence. ✅

B.Simplification errors could mask actual inequality.

C.Clairaut's Theorem only applies to polynomial functions.

D.The student should have used numerical approximation instead.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: This error analysis question exposes flawed verification logic. Algebraic agreement suggests equality but doesn't prove the theorem's hypotheses hold; it merely confirms the conclusion for that specific function. True verification requires establishing continuity independently. Students must distinguish between observing a consequence and validating the underlying sufficient conditions, preventing tautological reasoning.

Q10. Suppose $f(x,y)$ has continuous second partials everywhere except possibly at $(0,0)$ . You know $f_{xy} = f_{yx}$ for all $(x,y) \neq (0,0)$ . Can you conclude equality holds at $(0,0)$ without direct computation?

A.Yes, if

f_{xy}

has a removable discontinuity at the origin that can be continuously extended. ✅

B.No, pointwise behavior cannot be inferred from neighborhood behavior without additional constraints.

C.Yes, equality on a dense set implies equality everywhere for differentiable functions.

D.No, unless the first derivatives are also continuous at the origin.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: This conceptual question tests understanding of limits versus point values. Equality almost everywhere plus continuity allows extension to the exceptional point via limits. Without continuity, neighborhood equality doesn't force point equality. It requires multi-step reasoning about topological density, continuity extensions, and the precise role of hypotheses in extending local properties to singular points.

Q11. When optimizing $f(x,y)$ , the Hessian matrix requires $f_{xy} = f_{yx}$ . If numerical gradient estimation yields asymmetric off-diagonal entries, which diagnostic step best distinguishes discretization error from genuine non-equality?

A.Refine the grid spacing and check if asymmetry diminishes toward zero. ✅

B.Compute third-order partials to verify higher-order smoothness.

C.Switch to symbolic differentiation immediately.

D.Assume asymmetry is always numerical noise and symmetrize the matrix.

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: This application question addresses practical computational challenges. Genuine non-equality persists under refinement, while discretization error typically scales with step size. Testing convergence behavior distinguishes algorithmic artifacts from mathematical reality. It combines numerical analysis with theoretical knowledge, requiring students to design validation protocols rather than blindly trusting or dismissing computational outputs.

Q12. Consider $f(x,y) = x^2 \arctan(y/x)$ for $x \neq 0$ and 0 otherwise. Before computing mixed partials at the origin, what preliminary analysis determines whether Clairaut's Theorem is applicable?

A.Investigate whether

f_{xy}

and

f_{yx}

are continuous at the origin via limit analysis. ✅

B.Verify that

f

is differentiable at the origin using the linear approximation definition.

C.Check if

f

is bounded in a neighborhood of the origin.

D.Determine if the arctangent function is analytic at zero.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: This multi-step reasoning question emphasizes prerequisite checking before computation. Applicability hinges on continuity of second derivatives, not mere differentiability or boundedness. Students must recognize that verifying theorem hypotheses requires analyzing the behavior of the very quantities the theorem concerns, fostering disciplined mathematical practice over procedural automation.

Q13. Two surfaces $z=f(x,y)$ and $z=g(x,y)$ intersect tangentially at $P$ with identical tangent planes. Does this tangency imply $f_{xy}(P) = g_{xy}(P)$ ?

A.No, tangency constrains only first derivatives; second derivatives describe curvature which can differ independently. ✅

B.Yes, identical tangent planes force all higher-order derivatives to match.

C.Only if both surfaces are quadrics.

D.Yes, because mixed partials determine the tangent plane orientation.

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: This conceptual question separates first and second-order geometric information. Tangency ensures gradient equality but says nothing about Hessian components. Surfaces can share tangent planes yet have different curvatures and mixed partials. It tests understanding that derivative orders encode distinct geometric data, preventing overgeneralization from lower-order contact conditions.

🔗 Related Topics (MCQs)

📝 Functions of Two or More Variables MCQs 📝 Multivariable function notation and terminology MCQs 📝 Functions of two variables from tables MCQs 📝 Level curves and contour plots MCQs 📝 Graphing functions of two variables MCQs

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