Number of Partial Derivatives of Order n/Examples/Order 2 of 2 Variables

Examples of Use of Number of Partial Derivatives of Order $n$

Let $u = \map f {x, y}$ be a real function of $2$ variables.

There are $4$ partial derivatives of $u$ of order $2$.

These are:

\(\ds \dfrac {\partial^2 u} {\partial x^2}\)

\(=\)

\(\ds \map {f_{1 1} } {x, y}\)

\(\ds \dfrac {\partial^2 u} {\partial y \partial x}\)

\(=\)

\(\ds \map {f_{1 2} } {x, y}\)

\(\ds \dfrac {\partial^2 u} {\partial x \partial y}\)

\(=\)

\(\ds \map {f_{2 1} } {x, y}\)

\(\ds \dfrac {\partial^2 u} {\partial y^2}\)

\(=\)

\(\ds \map {f_{2 2} } {x, y}\)

Proof

From Number of Partial Derivatives of Order $n$, there are $m^n$ partial derivatives of order $n$ of a function of $m$ independent variables.

In this case, $m = 2$ and $n = 2$.

Thus there are $2^2 = 4$ partial derivatives of $u$ of order $2$.

$\blacksquare$

Sources

• 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 1$. Introduction: $1.3$ Higher Order Derivatives