16.26 min read

Symmetry of Mixed Partials

In the calculation $f(x, y) = xy^3 + x^2 - y$ , you can compute $f_{xy} = 3y^2$ and $f_{yx} = 3y^2$ . They are equal! Is this a coincidence?

No — for well-behaved functions, mixed partials are always equal. The order you differentiate in does not matter: differentiating first by $x$ then $y$ gives the same result as first by $y$ then $x$ .

This is the Mixed Partials Theorem (sometimes called Clairaut's Theorem or Schwarz's Theorem). The proof is non-trivial and requires the $D^2$ assumption, but the conclusion is beautifully clean.

The consequence: the Hessian matrix (coming next) is symmetric. All the theory of symmetric matrices — eigenvalues, spectral decomposition, definiteness — applies to it.

Formal View

Theorem 16.1 — Mixed Partials (Clairaut's Theorem)

Let

f(\mathbf{x}) \in D^2

. Then for all

i, j

\frac{\partial^2 f}{\partial x_i \partial x_j}(\mathbf{x}) = \frac{\partial^2 f}{\partial x_j \partial x_i}(\mathbf{x})

The order of differentiation does not matter for twice-differentiable functions.

Why This Matters

Symmetry of the Hessian is the bridge that lets us apply all our tools from symmetric matrix theory to second-order optimization.

Hessian symmetry means only $n(n+1)/2$ unique entries to compute, not $n^2$
Symmetric Hessian guarantees real eigenvalues — critical for classifying critical points
Integrability conditions in physics use mixed-partial symmetry (Maxwell's relations)
Conservative vector fields satisfy $\frac{\partial F_y}{\partial x} = \frac{\partial F_x}{\partial y}$ — same idea

Learning Resources

Symmetry of second partial derivatives

Khan Academy

Demonstrates mixed partial equality with examples and discusses when it can fail.

7 min

Second derivatives and the Hessian matrix

Steve Brunton

Shows how symmetry of mixed partials leads to a symmetric Hessian.

14 min

Quiz

Question 1

For $f \in D^2$ , we always have $f_{xy} = f_{yx}$ .

Question 2

Why does symmetry of mixed partials matter for optimization?

Question 3

For $f(x,y) = e^{x+y}$ , are $f_{xy}$ and $f_{yx}$ equal?

Question 4

For $f$ on $\mathbb{R}^n$ with $f \in D^2$ , how many distinct second partial derivative values are there?

Common Mistakes

Assuming $f_{xy} = f_{yx}$ without checking that $f \in D^2$ — counterexamples exist for pathological functions.
Confusing the theorem direction: symmetry of mixed partials follows from twice-differentiability, not the other way around.