Gradient field

A gradient field is a vector field $\mathbf{F}$ that arises as the gradient of some scalar function $\varphi$:

$\mathbf{F}=\nabla \varphi .$

The function $\varphi$ is called a potential function for $\mathbf{F}$. A gradient field is also called a conservative field.

Lots of physical fields are gradient fields: gravity, electrostatic fields in regions without time-varying magnetic flux, and so on. Having a potential is what makes them tractable.

Why gradient fields are special

Three equivalent characterizations on a connected open region $\Omega$:

1. $\mathbf{F}=\nabla \varphi$ for some scalar $\varphi$.
2. Line integrals ${\int }_C\mathbf{F}\cdot d\mathbf{r}$ are path-independent: they depend only on endpoints.
3. Every closed-loop circulation $\oint \mathbf{F}\cdot d\mathbf{r}$ is zero.

The full chain of implications is in Conservative vector field. One geometric property (closed-loop circulation = 0) is equivalent to one algebraic property (a scalar potential exists).

The Fundamental Theorem of Line Integrals

If $\mathbf{F}=\nabla \varphi$ and $C$ goes from point $A$ to point $B$, then

${\int }_C\mathbf{F}\cdot d\mathbf{r}=\varphi (B)-\varphi (A).$

Evaluate the potential at the endpoints and subtract. No parameterization needed. See Fundamental theorem of line integrals.

The cross-partial test

A necessary condition for $\mathbf{F}=⟨P,Q,R⟩$ to be a gradient field: the mixed partials of $\varphi$ must agree (by Clairaut’s theorem), which translates into

$\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x},\frac{\partial Q}{\partial z}=\frac{\partial R}{\partial y},\frac{\partial R}{\partial x}=\frac{\partial P}{\partial z}.$

In 2D only the first condition: $P_y=Q_x$.

Compactly: $\nabla \times \mathbf{F}=0$. The curl of a gradient is zero.

On simply connected domains the converse holds too: curl-free implies gradient. On non-simply-connected domains curl-free doesn’t suffice; the 2D rotational field on ${\mathbb{R}}^2\setminus \{0\}$ is the classical counterexample.

Not every vector field is a gradient

$\mathbf{F}=⟨-y,x⟩$ (the 2D rotational field, defined globally) has $P_y=-1\ne 1=Q_x$, so it’s not a gradient field on any region. Its circulation around the unit circle is $2\pi$, not zero, so it’s non-conservative.

The sharper case: $\mathbf{F}=⟨-y/(x^2+y^2),x/(x^2+y^2)⟩$ satisfies $P_y=Q_x$ everywhere on its domain ${\mathbb{R}}^2\setminus \{0\}$ but is still not a gradient field there, because the domain isn’t simply connected. This is the standard topological-obstruction example, and the 2D prototype of what becomes the Residue theorem in complex analysis.

Finding a potential

Algorithm in 2D, given $\mathbf{F}=⟨P,Q⟩$ with $P_y=Q_x$:

1. Integrate ${\varphi }_x=P$ with respect to $x$, holding $y$ constant: $\varphi (x,y)=\int Pdx+g(y)$.
2. Differentiate this $\varphi$ with respect to $y$ and set equal to $Q$. This gives an ODE for $g(y)$.
3. Integrate to find $g(y)$, with an arbitrary constant we can take to be $0$.

In 3D, three steps instead of two — same idea with one more integration. See Conservative vector field for worked examples.