Recipe

Test a 3D field for conservativeness and find its potential function

Use when: you have a 3D field $\mathbf{F}=(P,Q,R)$ on a simply connected domain and need to know whether it is conservative, and if so, produce an actual potential function $f$

Goal: either a confirmed potential function $f$ with $\nabla f=\mathbf{F}$ , or a definitive proof that no such $f$ exists

Taught in The 3D conservative-field test · extends Test a 2D field for conservativeness and find its potential function

Quick recipe

Confirm the domain is simply connected
Compute $\operatorname{curl}\mathbf{F}$
Integrate $P$ with respect to $x$
Differentiate with respect to $y$ and match to $Q$
Differentiate with respect to $z$ and match to $R$
Check $f$ against the original field

1
Check that every closed loop in $\mathbf{F}$ 's domain can shrink to a point without leaving the domain.

Why: only on a simply connected domain does $\operatorname{curl}\mathbf{F}=\mathbf{0}$ everywhere guarantee that a potential function exists, not merely that the field looks locally irrotational.
2
Compute all three components of the curl.

Example

Let $\mathbf{F}=(y+z,\ x+z,\ x+y)$ . Then $\operatorname{curl}\mathbf{F} = (R_y-Q_z,\ P_z-R_x,\ Q_x-P_y) = (1-1,\ 1-1,\ 1-1) = (0,0,0)$ .
- $\operatorname{curl}\mathbf{F}=\mathbf{0}$ everywhere — $\mathbf{F}$ is conservative; continue to Step 3 to build $f$ .
- $\operatorname{curl}\mathbf{F}\ne\mathbf{0}$ somewhere — $\mathbf{F}$ is NOT conservative; stop here, since no potential function exists.
Why: this packages the three cross-partial comparisons the 2D test only needed one of into a single vector check.
3
Treat $y$ and $z$ as constants and integrate $P$ in $x$ . The result is a first draft of $f$ , plus an unknown function $g(y,z)$ .

Example

$f(x,y,z) = \displaystyle\int(y+z)\,dx = xy+xz+g(y,z)$ .

Why: this recovers all of $f$ 's $x$ -dependence directly from $P=f_x$ , exactly as in the 2D construction, one variable held fixed instead of one.
4
Differentiate the Step 3 draft with respect to $y$ , set it equal to $Q$ , and solve for $g_y$ . Integrate that to update $g$ , leaving a still-unknown function of $z$ alone.

Example

$f_y = x+g_y$ . Matching $Q=x+z$ gives $g_y=z$ , so $g(y,z) = yz+h(z)$ and $f(x,y,z)=xy+xz+yz+h(z)$ .

Why: this is the 2D construction's second step, run one variable at a time instead of finishing the whole $y$ -dependence in one integration.
5
Differentiate the updated draft with respect to $z$ , set it equal to $R$ , solve for $h'(z)$ , then integrate to finish $f$ . Add an arbitrary constant $C$ .

Example

$f_z = x+y+h'(z)$ . Matching $R=x+y$ gives $h'(z)=0$ , so $h(z)=C$ and $f(x,y,z) = xy+xz+yz+C$ .

Why: only the third pass pins down the last unknown piece, since neither earlier integration could see $z$ -only information.
6
Differentiate the finished $f$ in $x$ , $y$ , and $z$ , and confirm all three match $P$ , $Q$ , and $R$ .

Example

$f_x=y+z=P$ , $f_y=x+z=Q$ , and $f_z=x+y=R$ . All three match.

Why: this catches an algebra slip in the three-pass construction before $f$ is used downstream.

Common mistakes

Running the curl test on a domain with a hole around a line (like the outside of an infinite cylinder), where the converse direction of the theorem does not apply.
Stopping after matching only two of the three components against $P$ , $Q$ , and $R$ .
Forgetting that the unknown after the first integration is a function of TWO variables, $g(y,z)$ , not just $g(y)$ .
Treating a zero curl as sufficient without first checking that the domain is simply connected.