Conservative Vector Fields — Theory
1. What is a conservative vector field?
A vector field is conservative if it is the gradient of a scalar function.
That scalar function is called a potential function.
\[
\mathbf F=\nabla \phi.
\]
In two dimensions, if \(\phi=\phi(x,y)\), then
\[
\nabla\phi
=
\left\langle
\frac{\partial \phi}{\partial x},
\frac{\partial \phi}{\partial y}
\right\rangle.
\]
In three dimensions,
\[
\nabla\phi
=
\left\langle
\phi_x,\phi_y,\phi_z
\right\rangle.
\]
2. Conservative fields in two dimensions
Suppose
\[
\mathbf F(x,y)=\langle P(x,y),Q(x,y)\rangle.
\]
If \(\mathbf F\) is conservative, then there is a potential function \(\phi(x,y)\) such that
\[
\phi_x=P,
\qquad
\phi_y=Q.
\]
A common test is:
\[
P_y=Q_x.
\]
Equivalently, the scalar curl must be zero:
\[
Q_x-P_y=0.
\]
3. Conservative fields in three dimensions
Suppose
\[
\mathbf F(x,y,z)=\langle P(x,y,z),Q(x,y,z),R(x,y,z)\rangle.
\]
If \(\mathbf F\) is conservative, then
\[
\mathbf F=\nabla\phi.
\]
A common test is:
\[
\nabla\times\mathbf F=\mathbf 0.
\]
Written in components,
\[
\nabla\times\mathbf F
=
\left\langle
R_y-Q_z,\,
P_z-R_x,\,
Q_x-P_y
\right\rangle.
\]
Therefore, for a conservative field,
\[
R_y=Q_z,
\qquad
P_z=R_x,
\qquad
Q_x=P_y.
\]
4. Path independence
The most important property of a conservative vector field is path independence.
If \(\mathbf F=\nabla\phi\), then the line integral from \(A\) to \(B\) depends only on
the endpoints:
\[
\int_C \mathbf F\cdot d\mathbf r
=
\phi(B)-\phi(A).
\]
This means that any two paths from the same start point \(A\) to the same end point \(B\)
give the same line integral.
5. Example: \(\mathbf F=\langle 2x,2y\rangle\)
Here
\[
P=2x,
\qquad
Q=2y.
\]
Check the conservative condition:
\[
P_y=0,
\qquad
Q_x=0.
\]
Since \(P_y=Q_x\), the field passes the 2D curl test.
Now find the potential function. Since \(\phi_x=2x\),
\[
\phi=x^2+h(y).
\]
Differentiate with respect to \(y\):
\[
\phi_y=h'(y).
\]
But \(\phi_y=Q=2y\), so
\[
h'(y)=2y.
\]
Therefore,
\[
h(y)=y^2.
\]
A potential function is:
\[
\phi(x,y)=x^2+y^2+C.
\]
6. Example: a nonconservative field
Consider the rotation field
\[
\mathbf F=\langle -y,x\rangle.
\]
Here
\[
P=-y,
\qquad
Q=x.
\]
Compute:
\[
P_y=-1,
\qquad
Q_x=1.
\]
Since \(P_y\ne Q_x\), the field is not conservative.
The scalar curl is:
\[
Q_x-P_y
=
1-(-1)
=
2.
\]
This nonzero curl indicates local rotation.
7. How to find a potential function in 2D
Suppose \(\mathbf F=\langle P,Q\rangle\). To find \(\phi\):
- Integrate \(P\) with respect to \(x\).
- Add an unknown function \(h(y)\).
- Differentiate the result with respect to \(y\).
- Set that derivative equal to \(Q\).
- Solve for \(h(y)\).
\[
\phi(x,y)=\int P(x,y)\,dx+h(y).
\]
8. How to find a potential function in 3D
Suppose \(\mathbf F=\langle P,Q,R\rangle\). To find \(\phi\):
- Integrate \(P\) with respect to \(x\).
- Use \(Q\) to determine the missing \(y\)-dependent part.
- Use \(R\) to determine the missing \(z\)-dependent part.
\[
\phi_x=P,
\qquad
\phi_y=Q,
\qquad
\phi_z=R.
\]
A correct potential must satisfy all three equations.
9. Domain warning
The curl-zero test usually requires the domain to be simply connected.
This means the region should not have holes.
For example, a field can have zero curl on a punctured domain but still fail to be globally
conservative because the domain has a hole. This is why domain information matters in vector calculus.
11. Common mistakes
- Checking only one point: a field must pass the curl condition on a region, not just at one point.
- Ignoring the domain: curl zero is not always enough if the domain has holes.
- Mixing up \(P_y\) and \(Q_x\): for \(\langle P,Q\rangle\), compare \(P_y\) and \(Q_x\).
- Forgetting the \(h(y)\) or \(g(y,z)\) term: when integrating with respect to one variable, add the missing function of the other variables.
- Assuming every zero-curl field has an easy potential: a potential may exist but still be hard to find symbolically.
- Confusing potential with line integral: the potential is a function; the line integral is a number.
