Multivariable Calculus 13 | Schwarz's Theorem [dark version]

Q: What exact conditions on $f$ make $\partial_{x_i x_j} f$ equal $\partial_{x_j x_i} f$?

The function $f:U\to\mathbb{R}$ must be defined on an open set $U\subset\mathbb{R}^n$. All second-order partial derivatives must exist at every point in $U$, and the collection of these second partial derivatives must be continuous functions on $U$. Continuity is crucial for passing limits through the derivative expressions when difference quotients shrink to zero.

Q: How does applying the mean value theorem in different orders produce two different mixed derivatives?

In the $n=2$ setup, one computation applies the mean value theorem first in the $h_1$ direction (leading to intermediate points $C_1, C_2$) and then in the $h_2$ direction, yielding $\partial_{x_1 x_2} f$. The swapped computation applies it first in the $h_2$ direction (with intermediate points $\eta_2, \eta_1$) and then in the $h_1$ direction, yielding $\partial_{x_2 x_1} f$. Both start from the same difference-quotient identity, so the only difference is the derivative order.

Q: Why is continuity needed right at the end of the proof?

After canceling factors and taking $h_1,h_2\to 0$, the intermediate points converge to the target point. Continuity of the second partial derivatives allows the limit to be taken inside the derivative expressions, so both routes evaluate to the same second mixed derivative at the point (e.g., $(0,0)$). Without continuity, the limiting step could fail.

TL;DR

Schwartz’s theorem asserts symmetry of mixed second partial derivatives: $\partial_{x_{i} x_{j}} f = \partial_{x_{j} x_{i}} f$ under the right regularity conditions.

Briefing Cornell Notes

Briefing

Schwartz’s theorem guarantees that, under standard regularity conditions, mixed second partial derivatives of a multivariable function are symmetric: $\frac{\partial ^{2} f}{\partial x _{i} \partial x _{j}} = \frac{\partial ^{2} f}{\partial x _{j} \partial x _{i}}$ for all points in an open domain. The practical payoff is immediate—when computing higher-order derivatives, there’s no need to treat $x_{i}$ then $x_{j}$ as a fundamentally different operation than $x_{j}$ then $x_{i}$ . That symmetry can cut down the number of distinct derivatives that must be calculated and tracked.

The theorem is framed for a function $f : U \to R$ defined on an open set $U \subset R^{n}$ . The key assumptions are that every second-order partial derivative exists throughout $U$ , and that these second partial derivatives form continuous functions on $U$ . Continuity is the ingredient that makes the symmetry robust: it allows limits to pass through the derivative expressions when the proof drives certain difference quotients to zero.

A proof strategy is built around reducing the multivariable problem to one-dimensional mean value theorem steps. To keep the argument concrete, the discussion first restricts to the case $n = 2$ , focusing on indices $i = 1$ and $j = 2$ . By translating coordinates, the point of interest $\tilde{x}$ is taken as the origin, simplifying notation without changing the substance of the argument.

The core move replaces partial derivatives with difference quotients. For instance, the proof considers expressions like $\frac{f ( h _{1} , h _{2} ) - f ( h _{1} , 0 )}{h _{1}}$ and then sends $h_{1} \to 0$ (and similarly for $h_{2} \to 0$ ). To apply the one-dimensional mean value theorem, the proof introduces auxiliary two-variable functions designed so that each difference quotient becomes the slope of a one-variable function. One such function, $U (h_{1}, h_{2}) = f (h_{1}, h_{2}) - f (h_{1}, 0)$ , is used to apply the mean value theorem in the $h_{1}$ direction first, producing intermediate points (labeled $C_{1}$ and $C_{2}$ ) lying between the relevant endpoints.

Then the same original difference-quotient expression is handled in the opposite order. A second auxiliary function, $V$ , is defined so that the mean value theorem can be applied in the $h_{2}$ direction first, yielding intermediate points (labeled $η_{2}$ and $η_{1}$ ). In both routes, the mean value theorem produces factors of $h_{1}$ or $h_{2}$ multiplying appropriate first derivatives of the auxiliary functions, which correspond to second mixed partial derivatives of $f$ in the two different orders.

Because the two computations start from the same difference-quotient identity, the $h_{1}$ and $h_{2}$ factors cancel. Finally, letting $h_{1} \to 0$ and $h_{2} \to 0$ forces the intermediate points to converge to the origin. Continuity of the second partial derivatives is what permits taking limits inside the derivative expressions, concluding that the two mixed second derivatives agree at $(0, 0)$ . The same reasoning extends to any point in $U$ , establishing Schwartz’s symmetry result throughout the domain.

With the theorem in hand, mixed second derivatives—and by extension higher-order mixed derivatives—can be reordered freely wherever the continuity and existence conditions hold, streamlining later computations across multivariable calculus problems.

Cornell Notes

Schwartz’s theorem says mixed second partial derivatives are equal when they exist everywhere on an open set and are continuous there: $\partial_{x_{i} x_{j}} f = \partial_{x_{j} x_{i}} f$ . The proof reduces the symmetry question to one-dimensional mean value theorem applications by rewriting difference quotients and introducing auxiliary functions (like $U (h_{1}, h_{2}) = f (h_{1}, h_{2}) - f (h_{1}, 0)$ ). One route applies the mean value theorem in the $h_{1}$ direction first, then in the $h_{2}$ direction; the other route swaps the order. After canceling the difference-quotient factors and sending $h_{1}, h_{2} \to 0$ , continuity lets the limits pass through, forcing both mixed derivatives to match at the point. The argument extends from the origin to every point in $U$ .

What exact conditions on $f$ make $\partial_{x_{i} x_{j}} f$ equal $\partial_{x_{j} x_{i}} f$ ?

The function

f : U \to R

must be defined on an open set

U \subset R^{n}

. All second-order partial derivatives must exist at every point in

U

, and the collection of these second partial derivatives must be continuous functions on

U

. Continuity is crucial for passing limits through the derivative expressions when difference quotients shrink to zero.

Why does the proof introduce auxiliary functions like $U (h_{1}, h_{2}) = f (h_{1}, h_{2}) - f (h_{1}, 0)$ ?

Auxiliary functions turn a multivariable difference quotient into the slope of a one-variable function, enabling direct use of the one-dimensional mean value theorem. For

U

, holding

h_{2}

fixed makes

U

a function of

h_{1}

alone, so the mean value theorem produces an intermediate point where a tangent slope matches the secant slope.

How does applying the mean value theorem in different orders produce two different mixed derivatives?

In the

n = 2

setup, one computation applies the mean value theorem first in the

h_{1}

direction (leading to intermediate points

C_{1}, C_{2}

) and then in the

h_{2}

direction, yielding

\partial_{x_{1} x_{2}} f

. The swapped computation applies it first in the

h_{2}

direction (with intermediate points

η_{2}, η_{1}

) and then in the

h_{1}

direction, yielding

\partial_{x_{2} x_{1}} f

. Both start from the same difference-quotient identity, so the only difference is the derivative order.

What role do the intermediate points ( $C_{1}, C_{2}, η_{1}, η_{2}$ ) play?

They are the “mean value theorem points” that lie between the endpoints of the intervals used in the difference quotients. Their exact values aren’t the goal; what matters is that they stay within bounds shrinking to zero as

h_{1}, h_{2} \to 0

. That boundedness ensures the intermediate points converge to the point where symmetry is being proved.

Why is continuity needed right at the end of the proof?

After canceling factors and taking

h_{1}, h_{2} \to 0

, the intermediate points converge to the target point. Continuity of the second partial derivatives allows the limit to be taken inside the derivative expressions, so both routes evaluate to the same second mixed derivative at the point (e.g.,

(0, 0)

). Without continuity, the limiting step could fail.

Review Questions

State Schwartz’s theorem precisely, including the assumptions on existence and continuity.
In the $n = 2$ proof, how does the mean value theorem get applied to turn a difference quotient into a derivative at an intermediate point?
Where in the argument does continuity enter, and what would break if the second partial derivatives were not continuous?

Key Points

1
Schwartz’s theorem asserts symmetry of mixed second partial derivatives: $\partial_{x_{i} x_{j}} f = \partial_{x_{j} x_{i}} f$ under the right regularity conditions.
2
The domain $U$ must be open in $R^{n}$ , ensuring neighborhoods around each point for the derivative definitions to behave properly.
3
All second-order partial derivatives must exist throughout $U$ , not just at a single point.
4
Continuity of the second partial derivatives on $U$ is the key technical requirement that justifies taking limits in the proof.
5
The proof uses one-dimensional mean value theorem steps by rewriting partial derivatives as difference quotients.
6
Auxiliary functions (like $U (h_{1}, h_{2}) = f (h_{1}, h_{2}) - f (h_{1}, 0)$ ) convert multivariable expressions into one-variable functions with fixed parameters.
7
The argument is first demonstrated at the origin for $n = 2$ and then extended to all points in $U$ .

Highlights

Mixed second derivatives become interchangeable when they exist everywhere and are continuous on an open set.

The proof’s engine is the one-dimensional mean value theorem applied twice—once in each coordinate direction.

Auxiliary functions turn difference quotients into slopes, producing intermediate points that converge as the step sizes shrink.

Continuity is used exactly when limits are pushed through the derivative expressions, locking in equality of the two derivative orders.

Topics

Schwartz’s Theorem
Mixed Partial Derivatives
Mean Value Theorem
Continuity Assumptions
Multivariable Calculus