Second-Order Partial Derivatives

Concept Explanation

Geometric Meaning of Pure vs. Mixed Partials

• Pure Partials ($f_{xx}, f_{yy}$): Measure the concavity (rate of change of slope) of the surface $z = f(x, y)$ along vertical slices parallel to the coordinate axes:
  • If $f_{xx} > 0$, the surface is concave upward along the $x$-direction (curving like a bowl or trough).
  • If $f_{xx} < 0$, the surface is concave downward along the $x$-direction (curving like a dome or ridge).
• Mixed Partials ($f_{xy}, f_{yx}$): Measure the twisting or cross-coupling shear of the surface:
  • $f_{xy}$ describes how the slope in the $x$-direction tilts as you step in the $y$-direction.
  • Non-zero mixed partials indicate that changes in one variable directly influence the system's sensitivity to another variable.

Notation Order Comparison

Notice the difference between subscript (Lagrange) notation and differential operator (Leibniz) notation:

• Subscript Notation reads Left-to-Right:
  $$f_{xy} = (f_x)_y \implies \text{differentiate with respect to } x \text{ first, then } y$$
• Leibniz Notation reads Right-to-Left (from inside out):
  $$\frac{\partial^2 f}{\partial y \partial x} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right) \implies \text{differentiate with respect to } x \text{ first, then } y$$

   Subscript Order:    f_{ x  y }   ===>  (1) x first,   (2) y second
                          |  |
                          1  2

   Leibniz Order:      rac{\partial^2 f}{\partial y \partial x}  ===>  (1) x first,   (2) y second
                                                     |        |
                                                     2        1

The Hessian Matrix and Curvature

The Hessian matrix $\mathbf{H}_f$ is the multivariable generalization of the single-variable second derivative $f''(x)$:

The Hessian determinant (or discriminant) is: In multivariable optimization (Second Derivative Test):

• If $D > 0$ and $f_{xx} > 0$, the surface has positive Gaussian curvature in all directions, corresponding to a local minimum.
• If $D > 0$ and $f_{xx} < 0$, the surface has negative curvature in all directions, corresponding to a local maximum.
• If $D < 0$, the surface bends upward in one direction and downward in another, forming a saddle point.
• If $D = 0$, the test is inconclusive.

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Engineering Applications

• Structural Engineering: Strain energy functions $U(x, y)$ in steel columns evaluate stability under axial loading $x$ and thermal expansion $y$ using Hessian determinants to verify structural resistance against buckling (see Structural Column Strain Energy Stability (Civil & Mechanical Engineering)).
• Electrical & Computer Engineering: Closed-loop automatic gain control circuits in audio amplifiers use Hessian matrices of the amplification function $G(v, r)$ to locate optimal bias voltages and feedback resistances (see Automated Audio Amplifier Loop Gain (Electrical & Computer Engineering)).
• Robotics & Kinematics: Joint torque equations utilize second partials to compute centrifugal and Coriolis coupling terms in dynamic multi-link robot arms.
• Geotechnical Engineering: Subsurface settlement basins model soil compaction where pure partials quantify settlement curvature under foundation footings.

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Example Problems

Example 1: Full Second-Order Partial Evaluation and Clairaut Verification

Compute all four second-order partial derivatives for $f(x, y) = x^3 y^4 - 5x^2 y + 2y^2$ and verify Clairaut's Theorem.

Solution

1. Compute First-Order Partials:

   $$f_x = \frac{\partial}{\partial x}[x^3 y^4 - 5x^2 y + 2y^2] = 3x^2 y^4 - 10xy$$
   $$f_y = \frac{\partial}{\partial y}[x^3 y^4 - 5x^2 y + 2y^2] = 4x^3 y^3 - 5x^2 + 4y$$

2. Compute Pure Second Partials:

   $$f_{xx} = \frac{\partial}{\partial x}[3x^2 y^4 - 10xy] = 6x y^4 - 10y$$
   $$f_{yy} = \frac{\partial}{\partial y}[4x^3 y^3 - 5x^2 + 4y] = 12x^3 y^2 + 4$$

3. Compute Mixed Partials:

   $$f_{xy} = \frac{\partial}{\partial y}[3x^2 y^4 - 10xy] = 12x^2 y^3 - 10x$$
   $$f_{yx} = \frac{\partial}{\partial x}[4x^3 y^3 - 5x^2 + 4y] = 12x^2 y^3 - 10x$$

4. Clairaut Verification: Since $f_{xy} = 12x^2 y^3 - 10x = f_{yx}$, Clairaut's Theorem is verified.

Example 2: Hessian Matrix Assembly and Determinant Evaluation

For the potential energy function $V(x, y) = x^3 + y^3 - 3xy$, construct the Hessian matrix $\mathbf{H}_V(x, y)$ and evaluate its determinant at the critical points $(0, 0)$ and $(1, 1)$.

Solution

1. First-Order Partials:
   $$V_x = 3x^2 - 3y, \qquad V_y = 3y^2 - 3x$$
2. Second-Order Partials:
   $$V_{xx} = 6x, \qquad V_{yy} = 6y, \qquad V_{xy} = -3$$
3. Hessian Matrix:
   $$\mathbf{H}_V(x, y) = \begin{bmatrix} 6x & -3 \\ -3 & 6y \end{bmatrix}$$
   $$D(x, y) = \det(\mathbf{H}_V) = (6x)(6y) - (-3)^2 = 36xy - 9$$
4. Evaluate at Points:
   • At $(0, 0)$: $D(0, 0) = 36(0)(0) - 9 = -9 < 0 \implies$ Saddle Point.
   • At $(1, 1)$: $D(1, 1) = 36(1)(1) - 9 = 27 > 0$. Since $V_{xx}(1, 1) = 6 > 0 \implies$ Local Minimum.

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Connections & References

• Parent Concepts: Partial Derivatives, Single-Variable Higher-Order Derivatives
• Theorems: Clairaut's Theorem
• Future Applications: Multivariable Optimization and Extrema
• Course Sessions: Calc-3 Session 11