Foundations — Fundamental theorem for line integrals

This page assumes nothing. Every symbol the parent note (Fundamental theorem for line integrals) throws at you — $\mathbf{F}$, $\nabla f$, $d\mathbf{r}$, the dot, the integral ${\int }_C$ — is built here from a blank page. Read top to bottom; each block earns the next.

0. The picture we keep returning to

Imagine a warm, hilly landscape at sunset. At every point on the ground you can ask: how high am I? That single number, height, is our first object.

1. A point in the plane: $(x,y)$

• Plain words: an address on the ground.
• Picture: a single dot on the peach floor of the landscape.
• Why the topic needs it: the endpoints $A$ and $B$ in $f(B)-f(A)$ are just two such addresses.

We will meet $(x,y,z)$ later — same idea with a third number for "up".

2. A scalar function: $f(x,y)$ — the height map

• Plain words: a machine: feed in an address, get out a height.
• Picture: the surface of the hill — but painted as a flat map where colour = height.
• Why the topic needs it: this $f$ is the potential function. FTLI's whole payoff is that the answer is $f(B)-f(A)$, two readings of this machine.

The curved lines in that figure are contour lines (or level curves): all the addresses at the same height, exactly like on a real hiking map. Where they crowd together, the hill is steep.

3. Partial derivatives: $f_x$ and $f_y$

Before the arrows, we need to measure steepness in one direction at a time.

• Plain words: "If I nudge $x$ a tiny bit and freeze $y$, how fast does the height change?"
• Why this tool and not the ordinary derivative? An ordinary derivative $\frac{dg}{dx}$ needs a function of one variable. Our $f$ has two inputs. The partial derivative is the honest way to ask "change in one input only" — we simply pretend the other input is a constant.

4. The gradient: $\nabla f$ — the uphill arrow

Now we bundle both slopes into a single object that points the way up.

• Plain words: at every point, an arrow whose direction is straight uphill and whose length is how steep the hill is there.
• Picture: short arrows scattered over the map, all leaning toward higher ground, longest where contours crowd.
• Why the topic needs it: FTLI is only about fields of the special form $\mathbf{F}=\nabla f$. The whole theorem is "when the field is a gradient, the integral is easy."

5. A vector: bold $\mathbf{v}$, arrows, and length

The gradient is a vector, so let us pin that word down.

• Plain words: "go this way, this far."
• Picture: a single arrow floating on the map.
• Why the topic needs it: the field $\mathbf{F}$, the gradient $\nabla f$, the position $\mathbf{r}(t)$ and the velocity ${\mathbf{r}}^{\prime }(t)$ are all vectors. Bold = arrow, plain = number.

6. A vector field: $\mathbf{F}(x,y)$

• Plain words: wind on a weather map — every spot has its own gust.
• Picture: a whole meadow of arrows.
• Why the topic needs it: ${\int }_C\mathbf{F}\cdot d\mathbf{r}$ measures how much a field "pushes you along" a path. When $\mathbf{F}$ happens to equal $\nabla f$, we call it conservative (see Conservative vector fields) and FTLI applies.

7. A parametrised curve: $\mathbf{r}(t)$

We need to describe the trail we walk.

• Plain words: a movie of a hiker; $t$ is the clock.
• Picture: a curve on the map with a moving dot and a timestamp.
• Why the topic needs it: the parent's Step 1 turns the line integral into an ordinary integral in $t$ using exactly this $\mathbf{r}(t)$.

The tiny displacement in the parent note, $d\mathbf{r}={\mathbf{r}}^{\prime }(t)dt$, is just "velocity $\times$ tiny time = tiny step."

8. The dot product: $\mathbf{a}\cdot \mathbf{b}$

The symbol between $\nabla f$ and $d\mathbf{r}$ is a dot, and it is not ordinary multiplication.

• Plain words: big and positive when the arrows agree, zero when they are perpendicular, negative when they oppose.
• Why this tool and not multiplication? You cannot multiply two arrows like numbers. But you can ask "how much of the field pushes me forward along my step?" That question — field component along motion — is precisely the dot product. That is why the line integrand is $\mathbf{F}\cdot d\mathbf{r}$ and not $\mathbf{F}d\mathbf{r}$.

9. The integral sign: $\int$ and ${\int }_C$

• Plain words: a grand total of tiny contributions.
• The line integral ${\int }_C\mathbf{F}\cdot d\mathbf{r}$: at each tiny step along the path $C$, take the push-along-the-step number $\mathbf{F}\cdot d\mathbf{r}$, and total them. See Line integrals of vector fields.
• The loop ${\oint }_C$: the same, with a little circle warning you the path is closed (ends where it began).

10. The Fundamental Theorem of Calculus (the 1D ancestor)

• Plain words: the whole equals (final value) − (starting value).
• Why the topic needs it: FTLI is this exact statement, dressed up. The parent proof reduces the line integral to ${\int }_a^b\frac{dg}{dt}dt$ and then quotes this. See Fundamental Theorem of Calculus.

11. Putting the symbols in one sentence

Now every piece of the parent's headline formula is defined: $\underset{\text{§9 sum along path}}{\underbrace{{\int }_C}}\underset{\text{§4 uphill arrows}}{\underbrace{\nabla f}}\cdot \underset{\text{§7 tiny step}}{\underbrace{d\mathbf{r}}}=\underset{\text{§2 two height readings}}{\underbrace{f(B)-f(A)}}.$

Read aloud: "Adding up the uphill-push along every tiny step of the trail equals the height at the end minus the height at the start."

Prerequisite map

Equipment checklist

Test yourself — cover the right side and answer before revealing.

Connections

• Fundamental theorem for line integrals — the parent topic these foundations feed.
• Gradient — the uphill-arrow operator $\nabla$ built in §4.
• Line integrals of vector fields — the sum-along-a-path in §9.
• Conservative vector fields — fields of the special form $\mathbf{F}=\nabla f$.
• Potential functions — the height map $f$ of §2.
• Fundamental Theorem of Calculus — the 1D ancestor of §10.
• Green's theorem — where these same $P,Q$ reappear.
• Curl — the 3D test built from partials like §3.