Test for exactness

Statement

Let $R = (\alpha, \beta) \times (\gamma, \delta)$ be an open rectangle in $\mathbb{R}^2$ . Let $M, N : R \to \mathbb{R}$ be continuous functions such that the partial derivatives $\partial M / \partial y$ and $\partial N / \partial x$ exist and are continuous on $R$ . Then the equation $M\, dx + N\, dy = 0$ is exact on $R$ if and only if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \quad \text{on } R.$

Remarks

The condition is necessary on each open set: if $\psi_x = M$ and $\psi_y = N$ , the mixed partial derivatives of $\psi$ are equal, so $M_y = N_x$ . For the converse, fix $(x_0, y_0) \in R$ and set $\psi(x, y) = \int_{x_0}^{x} M(s, y_0)\, ds + \int_{y_0}^{y} N(x, t)\, dt$ . The converse holds on each simply connected open set. It can fail on other sets: $(-y\, dx + x\, dy)/(x^2 + y^2) = 0$ satisfies the condition on $\mathbb{R}^2 \setminus \{0\}$ , but it is not exact there.

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Differential equation concept
Function concept
Interval concept
Ordinary differential equation definition
Solution of an ordinary differential equation definition
Exact equation definition
Test for exactness statement

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Integrating factor for a non-exact equation

Tag	`0MSCQNFTM8`
URI	`m2://0MSCQNFTM8`
Kind	statement, theorem
Writing	W0 stub
MSC	34A05
Status	This copy of the site has no build data. Run `m2 status 0MSCQNFTM8`.
Uses	Exact equation

Test for exactness

Statement

Remarks

Reading path

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