The Central Idea
A vector field is called conservative if it can be written as the gradient of a scalar potential:
\[ \boxed{ \mathbf{F}=\nabla \phi }. \]The scalar field \(\phi\) is called a potential function for \(\mathbf{F}\).
In many physical settings, a sign convention is instead written
\[ \boxed{ \mathbf{F}=-\nabla U }, \]where \(U\) is potential energy.
Why Potentials Matter
A general vector field may require us to track several components at once. A potential replaces that structure with one scalar function.
Instead of studying
\[ \mathbf{F}(x,y) = \langle P(x,y),Q(x,y)\rangle, \]we may be able to study a scalar function
\[ \phi(x,y) \]whose gradient reproduces the field.
This can simplify geometric interpretation and line-integral calculations.
A First Conservative Field
Let
\[ \phi(x,y)=x^2+y^2. \]Then
\[ \nabla\phi = \langle 2x,2y\rangle. \]Therefore, the vector field
\[ \boxed{ \mathbf{F}(x,y)=\langle 2x,2y\rangle } \]is conservative.
Potential Functions Are Not Unique
If
\[ \mathbf{F}=\nabla\phi, \]then adding a constant to \(\phi\) does not change its gradient:
\[ \nabla(\phi+C)=\nabla\phi. \]Therefore,
\[ \boxed{ \phi+C } \]is also a potential for the same field.
Only Potential Differences Matter
Because potentials are determined only up to an additive constant, their absolute zero level can often be chosen for convenience.
The quantity that remains unchanged under
\[ \phi\mapsto\phi+C \]is the difference
\[ \boxed{ \phi(B)-\phi(A) }. \]Potential differences are therefore more fundamental than arbitrary absolute labels.
Conservative Fields and Work
If a force field \(\mathbf{F}\) acts along a path \(C\), the work is represented by the line integral
\[ W = \int_C \mathbf{F}\cdot d\mathbf{r}. \]If
\[ \mathbf{F}=\nabla\phi, \]then the work depends only on the endpoints:
\[ \boxed{ \int_C \mathbf{F}\cdot d\mathbf{r} = \phi(B)-\phi(A) }. \]The Fundamental Theorem for Line Integrals
The endpoint rule
\[ \int_C \nabla\phi\cdot d\mathbf{r} = \phi(B)-\phi(A) \]is often called the Fundamental Theorem for Line Integrals.
It is the multivariable analogue of the one-variable relationship
\[ \int_a^b f'(x)\,dx = f(b)-f(a). \]Why the Endpoint Rule Works
Let a path be
\[ \mathbf{r}(t), \qquad a\leq t\leq b. \]Then
\[ \frac{d}{dt} \phi(\mathbf{r}(t)) = \nabla\phi(\mathbf{r}(t)) \cdot \mathbf{r}'(t). \]Integrating with respect to \(t\),
\[ \int_a^b \nabla\phi(\mathbf{r}(t)) \cdot \mathbf{r}'(t)\,dt = \phi(\mathbf{r}(b))-\phi(\mathbf{r}(a)). \]Path Independence
A vector field has path-independent line integrals when the value of
\[ \int_C\mathbf{F}\cdot d\mathbf{r} \]depends only on the starting and ending points.
Conservative fields have this property on appropriate domains.
Thus different routes between the same endpoints produce the same net work.
Example: Same Endpoints, Different Paths
Suppose
\[ \mathbf{F}=\nabla\phi \]and two paths \(C_1\) and \(C_2\) both begin at \(A\) and end at \(B\).
Then
\[ \int_{C_1}\mathbf{F}\cdot d\mathbf{r} = \phi(B)-\phi(A), \]and
\[ \int_{C_2}\mathbf{F}\cdot d\mathbf{r} = \phi(B)-\phi(A). \]Therefore,
\[ \boxed{ \int_{C_1}\mathbf{F}\cdot d\mathbf{r} = \int_{C_2}\mathbf{F}\cdot d\mathbf{r} }. \]Closed-Loop Integrals
If a conservative field is integrated around a closed path, the starting and ending points are the same.
Therefore,
\[ \phi(B)-\phi(A)=0. \]Hence,
\[ \boxed{ \oint_C\mathbf{F}\cdot d\mathbf{r}=0 }. \]Closed Loops and Conservation
Returning to the starting point restores the same potential value.
Therefore, a conservative field performs zero net work around a complete closed loop.
This is one of the defining geometric signatures of conservative behavior.
Potential Energy Convention
In mechanics, force is often related to potential energy \(U\) by
\[ \boxed{ \mathbf{F}=-\nabla U }. \]The negative sign means the force points toward decreasing potential energy.
Therefore, the work done by the force from \(A\) to \(B\) is
\[ \boxed{ W = U(A)-U(B) }. \]Why the Negative Sign Appears
If
\[ \mathbf{F}=-\nabla U, \]then the force points opposite the direction in which \(U\) increases most rapidly.
In many physical systems, spontaneous force-driven motion tends locally toward lower potential energy.
Equipotential Sets
A level set of a potential function is called an equipotential set.
In two dimensions,
\[ \phi(x,y)=c \]gives an equipotential curve.
In three dimensions,
\[ \phi(x,y,z)=c \]gives an equipotential surface.
The Field Is Perpendicular to Equipotentials
If
\[ \mathbf{F}=\nabla\phi, \]then the field points perpendicular to the level sets of \(\phi\).
If instead
\[ \mathbf{F}=-\nabla U, \]the field still remains perpendicular to equipotential sets, but points toward decreasing \(U\).
No Potential Change along an Equipotential
Moving tangent to an equipotential curve gives
\[ d\phi=0 \]to first order.
Since the field is normal to the equipotential, motion tangent to the equipotential has zero dot product with the gradient:
\[ \nabla\phi\cdot\mathbf{t}=0. \]No Conservative Work along an Equipotential
If a displacement remains tangent to an equipotential surface, then
\[ d\phi=0. \]Therefore,
\[ \mathbf{F}\cdot d\mathbf{r}=0 \]for \(\mathbf{F}=\nabla\phi\).
A conservative field does no differential work along a direction tangent to its equipotential set.
Finding a Potential from a Vector Field
Suppose
\[ \mathbf{F}(x,y) = \langle P(x,y),Q(x,y)\rangle. \]To find a potential \(\phi\), we seek
\[ \phi_x=P \]and
\[ \phi_y=Q. \]The two component equations must be compatible with one scalar function.
Example: Recovering a Potential
Let
\[ \mathbf{F}(x,y) = \langle 2x,2y\rangle. \]We need
\[ \phi_x=2x. \]Integrating with respect to \(x\),
\[ \phi=x^2+g(y), \]where \(g(y)\) is still unknown.
Complete the Potential
Differentiate
\[ \phi=x^2+g(y) \]with respect to \(y\):
\[ \phi_y=g'(y). \]Since the field requires
\[ \phi_y=2y, \]we get
\[ g'(y)=2y. \]Therefore,
\[ g(y)=y^2+C. \]The Recovered Potential
Combining the pieces gives
\[ \boxed{ \phi(x,y)=x^2+y^2+C }. \]Checking,
\[ \nabla\phi = \langle 2x,2y\rangle, \]which matches the original field.
The “Constant” after Partial Integration
When integrating
\[ \phi_x=P(x,y) \]with respect to \(x\), the integration constant may depend on \(y\):
\[ \phi(x,y) = \int P(x,y)\,dx+g(y). \]This is essential because differentiation with respect to \(x\) treats any function of \(y\) as constant.
Three-Dimensional Potential Recovery
For
\[ \mathbf{F} = \langle P,Q,R\rangle, \]a potential must satisfy
\[ \phi_x=P, \qquad \phi_y=Q, \qquad \phi_z=R. \]Recovering \(\phi\) requires reconciling all three component equations.
A Compatibility Test in Two Dimensions
Suppose
\[ \mathbf{F}(x,y)=\langle P(x,y),Q(x,y)\rangle \]and assume the required derivatives are continuous.
If
\[ \mathbf{F}=\nabla\phi, \]then
\[ P_y = \phi_{xy} \]and
\[ Q_x = \phi_{yx}. \]Under the usual smoothness assumptions,
\[ \boxed{ P_y=Q_x }. \]Why Equality of Cross Partials Matters
If
\[ P_y\neq Q_x, \]then no sufficiently smooth scalar potential can satisfy
\[ \phi_x=P, \qquad \phi_y=Q \]on that region.
Therefore, unequal cross partials rule out conservativeness there.
Equality of Cross Partials Is Not Always Enough by Itself
The condition
\[ P_y=Q_x \]is necessary for a smooth conservative planar field.
On sufficiently well-behaved domains, such as simply connected regions, it is also enough under the usual continuity assumptions.
Domain geometry therefore matters.
Why the Domain Matters
A vector field may have zero local rotational measure throughout its domain yet still fail to possess a single-valued global potential if the domain contains certain holes.
Conservativeness is therefore not determined only by local derivative formulas.
Global topology can matter.
Simply Connected Regions
Informally, a region is simply connected if it has no holes that obstruct the continuous shrinking of closed loops to a point within the region.
On such regions, familiar derivative tests for conservative fields work especially cleanly under standard smoothness assumptions.
A Nonconservative Example
Consider
\[ \mathbf{F}(x,y)=\langle -y,x\rangle. \]Then
\[ P_y=-1 \]while
\[ Q_x=1. \]Since
\[ P_y\neq Q_x, \]this field is not conservative on the plane.
A Conservative Example with Mixed Terms
Let
\[ \mathbf{F}(x,y) = \langle 2xy+y,\ x^2+x+3y^2\rangle. \]Compute
\[ P_y=2x+1 \]and
\[ Q_x=2x+1. \]The cross partials agree, so on a suitable domain the field is a candidate for a conservative field.
Recovering the Mixed-Term Potential
Start with
\[ \phi_x=2xy+y. \]Integrate with respect to \(x\):
\[ \phi=x^2y+xy+g(y). \]Differentiate with respect to \(y\):
\[ \phi_y=x^2+x+g'(y). \]Finish the Mixed-Term Potential
Match
\[ \phi_y = x^2+x+3y^2. \]Therefore,
\[ g'(y)=3y^2. \]Integrating,
\[ g(y)=y^3+C. \]Hence,
\[ \boxed{ \phi(x,y)=x^2y+xy+y^3+C }. \]Checking a Potential
The safest verification is to differentiate the proposed potential.
If
\[ \phi(x,y)=x^2y+xy+y^3, \]then
\[ \phi_x=2xy+y \]and
\[ \phi_y=x^2+x+3y^2. \]Therefore,
\[ \nabla\phi = \mathbf{F}. \]Work from a Potential Difference
Suppose
\[ \mathbf{F}=\nabla\phi \]with
\[ \phi(x,y)=x^2+y^2. \]Find the work from
\[ A=(1,1) \]to
\[ B=(3,4). \]Compute
\[ \phi(B)=25, \qquad \phi(A)=2. \]Therefore,
\[ \boxed{W=25-2=23}. \]The Path Is Irrelevant Once Conservativeness Is Known
In the previous example, the path from \((1,1)\) to \((3,4)\) could be:
- a straight segment;
- a piecewise path;
- a smooth curved arc;
- any other allowed path inside the domain.
The line integral remains \(23\) because only the endpoints matter.
Potential Difference along a Closed Loop
If a path begins and ends at the same point \(A\), then
\[ \phi(A)-\phi(A)=0. \]Therefore,
\[ \boxed{ \oint_C\nabla\phi\cdot d\mathbf{r}=0 }. \]This remains true even if the path is long or geometrically complicated, provided it stays inside the region where the potential applies.
Conservative Fields and Energy
In mechanics, conservative forces allow energy to be transferred between kinetic and potential forms while preserving total mechanical energy when no nonconservative work is present.
This is one reason potential functions are central in physical modeling.
The field geometry and the energy accounting are two expressions of the same underlying structure.
Gravity as a Potential Field
Gravitational fields are important examples of conservative fields in appropriate regions.
Near Earth's surface, a simplified gravitational potential energy can be written
\[ U=mgh. \]The associated gravitational force points toward lower potential energy.
Electric Potential
Electrostatic electric fields are also related to scalar electric potentials:
\[ \boxed{ \mathbf{E}=-\nabla V }. \]Equipotential surfaces are therefore perpendicular to the electric field wherever the field is nonzero.
Potential Landscapes
A potential function can be visualized as a landscape.
Peaks correspond to large potential values.
Basins correspond to smaller values.
If a force is
\[ -\nabla U, \]it points locally downhill on this potential landscape.
Equipotential Spacing and Field Strength
Since
\[ \|\mathbf{F}\| = \|\nabla\phi\| \]for
\[ \mathbf{F}=\nabla\phi, \]closely spaced equipotential contours indicate larger field magnitude when the potential intervals are equal.
Widely spaced equipotentials indicate weaker local change in the potential.
Direction from Equipotential Maps
To infer the direction of
\[ \nabla\phi, \]move perpendicular to the equipotential curves toward larger labels.
To infer the direction of
\[ -\nabla\phi, \]move perpendicular toward smaller labels.
Potential and Flow Lines
A gradient field
\[ \mathbf{F}=\nabla\phi \]has flow lines that locally cross equipotential sets toward increasing potential.
A negative-gradient field
\[ \mathbf{F}=-\nabla\phi \]has flow lines that cross them toward decreasing potential.
Potential Change along a Gradient Flow
Suppose
\[ \mathbf{r}'(t)=\nabla\phi. \]Then
\[ \frac{d}{dt}\phi(\mathbf{r}(t)) = \nabla\phi\cdot\nabla\phi = \|\nabla\phi\|^2. \]Therefore,
\[ \boxed{ \frac{d\phi}{dt}\geq0 }. \]Potential never decreases along a gradient flow.
Potential Change along Negative Gradient Flow
If
\[ \mathbf{r}'(t)=-\nabla\phi, \]then
\[ \frac{d}{dt}\phi(\mathbf{r}(t)) = -\|\nabla\phi\|^2. \]Therefore,
\[ \boxed{ \frac{d\phi}{dt}\leq0 }. \]Potential never increases along a negative-gradient flow.
Critical Points of a Potential
If
\[ \nabla\phi=\mathbf{0}, \]then the associated gradient field vanishes.
These points are critical points of the scalar potential and equilibria of the gradient flow.
Minima, Maxima, and Saddles
A critical point of a potential may be:
- a local minimum;
- a local maximum;
- a saddle point;
- or a more degenerate stationary point.
The condition
\[ \nabla\phi=\mathbf{0} \]identifies the candidate point but does not classify it by itself.
A Conservative Field Can Be Reversed
If
\[ \mathbf{F}=\nabla\phi, \]then
\[ -\mathbf{F} = \nabla(-\phi). \]Therefore, reversing a conservative field produces another conservative field.
Sums of Conservative Fields
If
\[ \mathbf{F}=\nabla\phi \]and
\[ \mathbf{G}=\nabla\psi, \]then
\[ \mathbf{F}+\mathbf{G} = \nabla(\phi+\psi). \]Therefore, the sum of conservative fields is conservative on the common domain.
Constant Multiples of Conservative Fields
If
\[ \mathbf{F}=\nabla\phi, \]then for a constant \(c\),
\[ c\mathbf{F} = \nabla(c\phi). \]Constant scaling preserves conservativeness.
Potential Differences Are Additive
For points \(A\), \(B\), and \(C\),
\[ \phi(C)-\phi(A) = [\phi(B)-\phi(A)] + [\phi(C)-\phi(B)]. \]Therefore, work in a conservative field naturally adds across successive path segments.
Conservative Fields and Closed Curves
If every closed-loop line integral vanishes on a suitable connected domain, then the field exhibits the path-independence behavior associated with a conservative field.
This provides another way to characterize conservativeness under standard hypotheses.
Several Equivalent Ideas
On an appropriate domain and under standard smoothness assumptions, the following ideas are closely connected:
- the field has a scalar potential;
- line integrals are path independent;
- closed-loop line integrals vanish;
- the relevant cross-partial or curl condition vanishes.
The precise equivalences depend on the domain and regularity assumptions.
Common Error: Forgetting the Sign Convention
Mathematics often defines a conservative field as
\[ \mathbf{F}=\nabla\phi. \]Physics often writes force as
\[ \mathbf{F}=-\nabla U. \]These are not contradictory. They use different choices for the scalar potential.
Common Error: Treating the Potential as Unique
If
\[ \phi \]is a potential, then
\[ \phi+C \]gives the same vector field.
A correct potential is therefore usually stated
\[ \boxed{\phi+C}. \]Common Error: Using an Ordinary Constant Too Early
When integrating
\[ \phi_x=P(x,y), \]the missing term should be written as a function of the other variable:
\[ g(y), \]not merely a constant \(C\).
Only after all component conditions are reconciled does the final arbitrary constant appear.
Common Error: Checking Only One Component
A proposed potential must reproduce every component of the vector field.
If
\[ \nabla\phi = \langle \phi_x,\phi_y\rangle, \]then both
\[ \phi_x=P \]and
\[ \phi_y=Q \]must hold.
Common Error: Assuming Cross-Partial Equality Always Proves Conservativeness
The condition
\[ P_y=Q_x \]is powerful, but the domain also matters.
On a domain with holes, local derivative conditions may not guarantee a globally defined single-valued potential.
Common Error: Confusing Potential with Potential Energy
A mathematical potential \(\phi\) is any scalar function whose gradient gives the vector field.
Potential energy \(U\) is a specific physical scalar quantity.
Depending on convention,
\[ \mathbf{F}=-\nabla U. \]The terms are related but not interchangeable without context.
Common Error: Ignoring the Path Domain
The endpoint formula applies when the path stays inside a region where the potential is defined.
If the field or potential has singularities or excluded regions, a proposed path must respect those domain restrictions.
A Reliable Procedure for Testing a Planar Field
- Write \(\mathbf{F}=\langle P,Q\rangle\).
- Identify the domain.
- Compute \(P_y\) and \(Q_x\).
- If they differ, the field is not conservative there.
- If they agree, inspect whether the domain conditions support the usual conservativeness test.
- Recover a potential when needed.
- Differentiate the potential to verify the field.
A Reliable Procedure for Finding a Potential
- Set \(\phi_x=P\).
- Integrate \(P\) with respect to \(x\).
- Add an unknown function of the remaining variables.
- Differentiate the result with respect to the next coordinate.
- Match that derivative to the corresponding field component.
- Continue until all components agree.
- Add the final arbitrary constant.
- Differentiate the completed potential to verify.
A Reliable Procedure for Work in a Conservative Field
- Confirm or establish that the field is conservative.
- Find a potential \(\phi\).
- Identify the initial point \(A\).
- Identify the final point \(B\).
- Compute \(\phi(B)-\phi(A)\).
- Do not parameterize the full path unless the problem specifically requires it.
Example: Full Conservative-Field Workflow
Let
\[ \mathbf{F}(x,y) = \langle 2xy+3,\ x^2+4y\rangle. \]Compute the cross partials:
\[ P_y=2x, \] \[ Q_x=2x. \]They agree.
On a suitable simply connected domain such as all of \(\mathbb{R}^2\), the field is conservative.
Recover the Potential
Begin with
\[ \phi_x=2xy+3. \]Integrate with respect to \(x\):
\[ \phi=x^2y+3x+g(y). \]Differentiate with respect to \(y\):
\[ \phi_y=x^2+g'(y). \]Match with
\[ x^2+4y. \]Therefore,
\[ g'(y)=4y, \]so
\[ g(y)=2y^2+C. \]Completed Potential
The potential is
\[ \boxed{ \phi(x,y)=x^2y+3x+2y^2+C }. \]Differentiate:
\[ \nabla\phi = \langle 2xy+3,\ x^2+4y\rangle. \]This reproduces the original field.
Example: Work between Two Points
Using
\[ \phi(x,y)=x^2y+3x+2y^2, \]find the work from
\[ A=(0,0) \]to
\[ B=(1,2). \]Evaluate:
\[ \phi(1,2) = 1^2(2)+3(1)+2(2^2) = 2+3+8 = 13, \] \[ \phi(0,0)=0. \]Therefore,
\[ \boxed{W=13}. \]Practice 1: Identify a Potential
Let
\[ \phi(x,y)=x^2+3y^2. \]Find the conservative field \(\nabla\phi\).
Practice 2: Recover a Potential
Find a potential for
\[ \mathbf{F}(x,y)=\langle 2x,4y\rangle. \]Practice 3: Test Conservativeness
Determine whether
\[ \mathbf{F}(x,y)=\langle y,x\rangle \]passes the planar cross-partial test.
Practice 4: Nonconservative Field
Determine whether
\[ \mathbf{F}(x,y)=\langle -y,x\rangle \]passes the planar cross-partial test.
Practice 5: Work from a Potential
Let
\[ \phi(x,y)=x^2+y^2 \]and
\[ \mathbf{F}=\nabla\phi. \]Find the work from \((0,0)\) to \((3,4)\).
Practice 6: Closed Loop
If \(\mathbf{F}\) is conservative on a region containing a closed path \(C\), evaluate
\[ \oint_C\mathbf{F}\cdot d\mathbf{r}. \]Practice 7: Potential Energy Sign
If
\[ \mathbf{F}=-\nabla U, \]does the force point toward increasing or decreasing potential energy?
Practice 8: Equipotential Direction
If
\[ \mathbf{F}=\nabla\phi, \]what is the geometric relationship between \(\mathbf{F}\) and the equipotential curve \(\phi=c\)?
Practice 9: Potential Is Not Unique
If
\[ \phi(x,y)=x^2+y^2 \]is a potential, give another potential for the same field.
Practice 10: Full Potential Recovery
Find a potential for
\[ \mathbf{F}(x,y) = \langle 3x^2+2y,\ 2x+4y\rangle. \]Practice Answers
Practice 1
\[ \phi_x=2x, \qquad \phi_y=6y. \]Therefore,
\[ \boxed{ \nabla\phi = \langle 2x,6y\rangle }. \]Practice 2
Start with
\[ \phi_x=2x. \]Integrate:
\[ \phi=x^2+g(y). \]Since
\[ \phi_y=4y, \]we get
\[ g(y)=2y^2+C. \]Therefore,
\[ \boxed{ \phi=x^2+2y^2+C }. \]Practice 3
Here,
\[ P=y, \qquad Q=x. \]Therefore,
\[ P_y=1, \qquad Q_x=1. \]The field passes the planar cross-partial test.
Practice 4
Here,
\[ P=-y, \qquad Q=x. \]Therefore,
\[ P_y=-1, \qquad Q_x=1. \]Since the values differ, the field is not conservative on the plane.
Practice 5
\[ \phi(3,4)=25, \qquad \phi(0,0)=0. \]Therefore,
\[ \boxed{W=25}. \]Practice 6
The initial and final points are the same, so
\[ \boxed{ \oint_C\mathbf{F}\cdot d\mathbf{r}=0 }. \]Practice 7
Because
\[ \mathbf{F}=-\nabla U, \]the force points toward
\[ \boxed{\text{decreasing potential energy}}. \]Practice 8
The gradient is normal to the equipotential curve, so
\[ \boxed{ \mathbf{F}\perp\{\phi=c\} }. \]Practice 9
Any additive constant gives another potential. For example,
\[ \boxed{ \phi(x,y)=x^2+y^2+7 }. \]Practice 10
Start with
\[ \phi_x=3x^2+2y. \]Integrate with respect to \(x\):
\[ \phi=x^3+2xy+g(y). \]Differentiate with respect to \(y\):
\[ \phi_y=2x+g'(y). \]Match with
\[ 2x+4y. \]Thus,
\[ g'(y)=4y, \]so
\[ g(y)=2y^2+C. \]Therefore,
\[ \boxed{ \phi(x,y)=x^3+2xy+2y^2+C }. \]What to Remember
A conservative field is generated by a scalar potential:
\[ \boxed{ \mathbf{F}=\nabla\phi }. \]In many physical applications,
\[ \boxed{ \mathbf{F}=-\nabla U }. \]- potentials are determined only up to an additive constant;
- potential differences are independent of that constant;
- conservative line integrals depend only on endpoints;
- closed-loop integrals vanish for conservative fields on appropriate domains;
- equipotential sets are perpendicular to the gradient field;
- cross-partial tests help identify conservativeness in smooth planar fields;
- domain topology matters when deciding whether local tests imply a global potential;
- potential functions connect vector fields, work, energy, and geometry.
The central structural idea is: a conservative vector field can be understood through one scalar function whose differences encode the net effect between points.
Where We Go Next
Potentials explain how some vector fields derive from scalar structure and why their line integrals depend only on endpoints.
The next lesson, Flux through Curves and Surfaces, shifts attention from motion along a path to the amount of a vector field that passes through a boundary.