Calculus III: Potential Functions

In multivariable calculus, the concept of a potential function is a powerful tool that simplifies complex line integral calculations into simple evaluations of a scalar function. This framework is indispensable in physics and engineering, where it elegantly describes conservative force fields like gravity and electrostatics, allowing you to analyze work and energy without tracing complicated paths. Mastering potential functions transforms your approach to vector fields, revealing an underlying simplicity governed by path independence and a single, unifying scalar field.

Conservative Vector Fields and the Curl Test

A vector field $\vec{F}(x,y,z)=⟨P,Q,R⟩$ is called conservative if it is the gradient of some scalar function $f$. That is, $\vec{F}=\nabla f=⟨f_x,f_y,f_z⟩$. The function $f$ is the potential function for $\vec{F}$. The key property of a conservative field is that the work done by $\vec{F}$ on a particle moving between two points is independent of the path taken; it depends only on the starting and ending points.

But how can you quickly determine if a given vector field is conservative? For sufficiently nice fields (where component functions have continuous first partial derivatives), a powerful test exists: $\vec{F}$ is conservative on a simply connected region if and only if its curl is zero. The curl of a three-dimensional field is given by $\nabla \times \vec{F}$. In component form, the condition $\nabla \times \vec{F}=\vec{0}$ translates to three simple partial derivative checks: $$\frac{\partial R}{\partial y}=\frac{\partial Q}{\partial z},\frac{\partial P}{\partial z}=\frac{\partial R}{\partial x},\frac{\partial Q}{\partial x}=\frac{\partial P}{\partial y}$$ If your field is two-dimensional, $\vec{F}(x,y)=⟨P,Q⟩$, the condition simplifies to checking that $\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}=0$. For example, consider $\vec{F}=⟨2x+y^2,2xy⟩$. Here, $P=2x+y^2$ and $Q=2xy$. Computing the partials gives $\frac{\partial Q}{\partial x}=2y$ and $\frac{\partial P}{\partial y}=2y$. Since they are equal, the field is conservative.

Finding a Potential Function by Integration

Once you've verified a field is conservative, the next step is to find its potential function $f(x,y,z)$. This is done through a systematic process of partial integration and comparison.

Step-by-Step Procedure:

1. Integrate $P(x,y,z)$ with respect to $x$. This yields $f(x,y,z)=\int Pdx=F_1(x,y,z)+g(y,z)$, where $g(y,z)$ is an arbitrary "constant" of integration that can depend on $y$ and $z$.
2. Differentiate this result with respect to $y$ and set it equal to $Q$: $f_y=\frac{\partial }{\partial y}[F_1]+g_y(y,z)=Q$. Solve for $g_y(y,z)$ and integrate with respect to $y$ to find $g(y,z)$ up to a function $h(z)$.
3. Update your expression for $f(x,y,z)$ with the new information for $g(y,z)$.
4. Finally, differentiate this updated $f$ with respect to $z$ and set it equal to $R$: $f_z=R$. This will allow you to solve for $h^{\prime }(z)$ and, by integration, $h(z)$.

Let's apply this to the example $\vec{F}=⟨2x+y^2,2xy⟩$.

• Step 1: $f(x,y)=\int (2x+y^2)dx=x^2+x{y}^2+g(y)$.
• Step 2: $f_y=2xy+g^{\prime }(y)$. Set this equal to $Q=2xy$. Thus, $2xy+g^{\prime }(y)=2xy$, so $g^{\prime }(y)=0$. Integrate: $g(y)=C$.
• Step 3: Therefore, the potential function is $f(x,y)=x^2+x{y}^2+C$.

Path Independence and the Fundamental Theorem for Line Integrals

The existence of a potential function is equivalent to path independence for line integrals of the vector field. If $\vec{F}$ is conservative with potential $f$, then for any piecewise-smooth curve $C$ from point $A$ to point $B$, $${\int }_C\vec{F}\cdot d\vec{r}=f(B)-f(A)$$ This is the Fundamental Theorem for Line Integrals. It eliminates the need to parameterize the path $C$, compute the dot product, and evaluate a single-variable integral. You simply evaluate the potential function at the endpoints and subtract.

Consider the conservative field $\vec{F}=⟨y,x⟩$ with potential $f(x,y)=xy$. To compute the work done along any path from $(1,2)$ to $(4,-1)$, you simply calculate $f(4,-1)-f(1,2)=(4\cdot -1)-(1\cdot 2)=-4-2=-6$. This is dramatically simpler than integrating along a specific curve like a parabola or a broken line segment.

Applications to Gravitational and Electric Potential Fields

The theory of potential functions is not merely mathematical abstraction; it is the language of fundamental physical forces. Both Newtonian gravitational fields and electrostatic fields (in the absence of time-varying magnetic fields) are conservative.

Gravitational Potential: For a point mass $M$ at the origin, the gravitational force on a test mass $m$ is ${\vec{F}}_g=-\frac{GMm}{r^2}\hat{r}$, where $\hat{r}$ is the unit radial vector. This field is conservative. Its potential function, gravitational potential energy, is $U_g(r)=-\frac{GMm}{r}+C$. The work done by gravity when moving mass $m$ from radius $r_1$ to $r_2$ is simply $U_g(r_2)-U_g(r_1)$, confirming that work is path-independent and depends only on radial distances.

Electric Potential: For a point charge $Q$ at the origin, the electrostatic force on a test charge $q$ is ${\vec{F}}_e=\frac{1}{4\pi {ϵ}_0}\frac{Qq}{r^2}\hat{r}$. This Coulomb force is also conservative. Its corresponding scalar field is the electric potential (voltage), $V(r)=\frac{1}{4\pi {ϵ}_0}\frac{Q}{r}+C$. The potential energy of the test charge is $U_e=qV(r)$. The work done by the electric field when moving the charge is again the difference in potential energy. In circuit theory and electromagnetics, we constantly work with this scalar potential $V$ because it is much simpler to handle than the vector electric field $\vec{E}$ itself, which is related by $\vec{E}=-\nabla V$.

Common Pitfalls

1. Applying the curl test without checking the region. A vector field can have zero curl everywhere it's defined but still not be conservative if its domain is not simply connected. The classic example is the two-dimensional field $\vec{F}=⟨\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2}⟩$. Its curl is zero everywhere except the origin, but its domain is punctured (not simply connected). A line integral around a closed loop encircling the origin is not zero, proving the field is not conservative on its entire domain. Always consider the topology of the region.
2. Incorrectly handling the "constant" of integration. When finding a potential function, the most common error is forgetting that the constant from the first integration ($g(y,z)$) can be a function of the other variables. Treat it as an unknown function and use the subsequent differentiation steps with respect to $y$ and $z$ to solve for it systematically.
3. Using the Fundamental Theorem on non-conservative fields. The theorem ${\int }_C\vec{F}\cdot d\vec{r}=f(B)-f(A)$ applies only if $\vec{F}$ is conservative. Applying it to a field that hasn't been verified as conservative will lead to incorrect results. Always perform the curl test (while considering the region) first.
4. Confusing potential with potential energy. In physics applications, the potential $V$ (e.g., electric potential in volts) is the potential function per unit charge or mass. The potential energy $U$ is the product of the charge or mass and the potential ($U=qV$ or $U=m\Phi$). Be clear about which scalar quantity you are calculating or using in the Fundamental Theorem.

Summary

• A conservative vector field $\vec{F}$ is the gradient of a potential function $f$ ($\vec{F}=\nabla f$). On a simply connected region, this is equivalent to the condition $\nabla \times \vec{F}=\vec{0}$.
• You find a potential function through a method of partial integration, carefully solving for arbitrary functions of the other variables at each step.
• For conservative fields, line integrals are path independent. The work done moving between two points depends only on the values of the potential function at those points.
• The Fundamental Theorem for Line Integrals formalizes this: ${\int }_C\vec{F}\cdot d\vec{r}=f(B)-f(A)$, providing a massive simplification for calculating work.
• This mathematical framework is essential for modeling gravitational and electrostatic fields, where the scalar potential (energy or voltage) is a fundamental, measurable quantity.