Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

The ordinary fundamental theorem of calculus says that integrating a derivative undoes the differentiation. The integral of the rate of change over an interval depends only on the values of the function at the two ends of the interval. The gradient theorem is the same statement in three dimensions.

Fundamental theorem for gradients

For a scalar function $f$ with continuous first derivatives and any path $C$ from the point $a$ to the point $b$ ,

\int_{a}^{b} (\nabla f) \cdot d ℓ = f (b) - f (a) . (2.4.1)

The line integral of a gradient \nabla f along a path C from \vec a to \vec b depends only on the values of f at the two endpoints, not on the shape of the path. — **Figure 2.4.1.** The line integral of a gradient $\nabla f$ along a path $C$ from $a$ to $b$ depends only on the values of $f$ at the two endpoints, not on the shape of the path.

Why it is true

Walk along $C$ in many small steps $d ℓ$ . By Eq. (2.3.47), the quantity $\nabla f \cdot d ℓ$ is exactly the change $df$ of $f$ during one step. Adding the changes of all the steps gives the total change of $f$ between the start and the end of the walk, which is $f (b) - f (a)$ , whatever route was taken.

The gradient theorem on a contour map. The brown curves are computed contour lines (level values marked) of a “landscape” f(x,y) with a main summit of height 3.0 and a smaller hill. Path C_1 goes straight over the main summit (highest point 3.0), C_2 makes a detour to the north (highest point about 1.2), and C_3 climbs over the shoulder of the small hill (highest point about 1.5). A numerical evaluation of \int\nabla f\cdot d\vec\ell along each of them gives the same value, 1.059, which is the height difference f(\vec b)-f(\vec a)=1.077-0.018 (script data/gen-sc.py). — **Figure 2.4.2.** The gradient theorem on a contour map. The brown curves are computed contour lines (level values marked) of a “landscape” $f (x, y)$ with a main summit of height $3.0$ and a smaller hill. Path $C_{1}$ goes straight over the main summit (highest point $3.0$ ), $C_{2}$ makes a detour to the north (highest point about $1.2$ ), and $C_{3}$ climbs over the shoulder of the small hill (highest point about $1.5$ ). A numerical evaluation of $\int \nabla f \cdot d ℓ$ along each of them gives the same value, $1.059$ , which is the height difference $f (b) - f (a) = 1.077 - 0.018$ (script data/gen-sc.py).

Meaning and consequences

Path independence. The integral (2.4.1) is the same for every path joining $a$ to $b$ (Figure 2.4.1). Think of $f$ as the height of a landscape: however you walk from $a$ to $b$ , the total climb (counting descents as negative climbs) is the difference in altitude (Figure 2.4.2).
Zero circulation. If the path is closed ( $b = a$ ), $\oint_{C} \nabla f \cdot d ℓ = 0$ . A field that is the gradient of a scalar has no circulation around any loop: a round trip always brings you back to the altitude where you started.
Electrostatics. We shall see in Section 3.3 that the electrostatic field is the (negative) gradient of a scalar potential, $E = - \nabla V$ . The theorem then states that the work done by the field on a charge moved between two points does not depend on the path, and that $V (b) - V (a) = - \int_{a}^{b} E \cdot d ℓ$ .

In plain words

For a gradient field, the line integral is a "height difference": only the start and the end matter. If you know $f$ , you never need to integrate $\nabla f$ along a path. Just subtract.

Example 2.4.1Verifying the gradient theorem

Quick check

Let $f = x y z$ . What is $\int \nabla f \cdot d ℓ$ along a complicated spiral from the origin to the point $(1, 2, 3)$ ? And around a closed loop?

$f (1, 2, 3) - f (0, 0, 0) = 6 - 0 = 6$ , whatever the shape of the spiral. Around any closed loop the result is $0$ .

2.4.1 The fundamental theorem for gradients

Why it is true

Meaning and consequences

In this part

Examples · 1