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A $C^{1}$ field on an open subset of $R^{3}$ is closed exactly when its curl vanishes

Statement

Let $U \subseteq R^{3}$ be open and let $F : U \to R^{3}$ be $C^{1}$ . Then a $C^{1}$ field on an open subset of $R^{3}$ is closed if and only if its curl vanishes identically: $F$ is closed in the sense of Exact and closed C1 vector fields exactly when $curl F (p) = 0$ for every $p \in U$ .

Facts & Assumptions

Given: The open set $U \subseteq R^{3}$ and the $C^{1}$ field $F : U \to R^{3}$ of the Statement, with the three coordinates named $x, y, z$ as on this page.

[F1]

The curl of a $C^{1}$ field $F$ on an open $U \subseteq R^{3}$ is $curl F = (\partial_{y} F_{z} - \partial_{z} F_{y}, \partial_{z} F_{x} - \partial_{x} F_{z}, \partial_{x} F_{y} - \partial_{y} F_{x})$ (Divergence and curl of a $C^{1}$ vector field).

[F2]

A $C^{1}$ field $F = (F_{0}, \dots, F_{n - 1})$ on an open $U \subseteq R^{n}$ , with coordinates and partial derivatives indexed from $0$ , is closed when $\partial_{j} F_{i} = \partial_{i} F_{j}$ for all $i, j < n$ (Exact and closed C1 vector fields).

[F3]

For $x \in R^{n}$ one writes $x_{k} : = x (k)$ for $k < n$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

Proof

technique · direct

1.1

By [F3] the coordinates of a point of $R^{3}$ are indexed $0, 1, 2$ , and the names $x, y, z$ used on this page are those three indices in that order; so the closedness condition of [F2] at $n = 3$ is the system of equations $\partial_{j} F_{i} = \partial_{i} F_{j}$ ranging over all pairs $i, j$ drawn from ${x, y, z}$ .

givenF2F3

1.2

In that system the equations with $i = j$ read $\partial_{i} F_{i} = \partial_{i} F_{i}$ and hold for every field, and the equation indexed $(i, j)$ is the same equation as the one indexed $(j, i)$ . Hence the system is equivalent to its three equations indexed by the unordered pairs ${y, z}$ , ${z, x}$ and ${x, y}$ .

F2algebra

2.1

Written out, those three equations are $\partial_{y} F_{z} = \partial_{z} F_{y}$ , $\partial_{z} F_{x} = \partial_{x} F_{z}$ and $\partial_{x} F_{y} = \partial_{y} F_{x}$ . Their left-minus-right differences $\partial_{y} F_{z} - \partial_{z} F_{y}$ , $\partial_{z} F_{x} - \partial_{x} F_{z}$ and $\partial_{x} F_{y} - \partial_{y} F_{x}$ are, by [F1], exactly the first, second and third coordinates of $curl F$ .

step 1.1step 1.2F1algebra

3.1

For the forward direction, suppose $F$ is closed. By steps 1.1 and 1.2 the three equations of step 2.1 hold at every $p \in U$ , so by [F1] each of the three coordinates of $curl F (p)$ is zero; hence $curl F$ vanishes identically on $U$ .

step 2.1F2F1

3.2

For the converse direction, suppose $curl F (p) = 0$ for every $p \in U$ . By [F1] each of the three differences of step 2.1 is zero at every $p$ , so the three equations of step 2.1 hold on $U$ ; by steps 1.1 and 1.2 these are equivalent to the full system of [F2], so $F$ is closed.

step 2.1F1F2

4.1

Steps 3.1 and 3.2 are the two implications, so $F$ is closed if and only if its curl vanishes identically.

step 3.1step 3.2∎

Remarks

Why the count of equations matters. Closedness in $R^{n}$ is a condition on all ordered pairs of indices, and the curl in $R^{3}$ has three coordinates. Step 1.2 is what shows that these are the same amount of information: the diagonal equations are automatic and each off-diagonal equation is listed twice. In $R^{n}$ with $n \neq 3$ the number of independent equations is $n (n - 1) / 2$ , so there is no vector of that many coordinates in the same space to collect them into, and closedness is then stated only as the system itself.

Depends on

Used by

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Sources

J. Feldman, A. Rechnitzer and E. Yeager, CLP-4 Vector Calculus (University of British Columbia), section 4.1 (standard reference, not scraped)

A C1C1 field on an open subset of R3R3 is closed exactly when its curl vanishes