Foundations — Line integrals — scalar and vector, work done

Before you can read the parent note, you need to be fluent in a handful of symbols. This page builds every single one from nothing, in the order they depend on each other. If a symbol below looks scary, that is a sign the parent note assumed it — so we earn it here first.

1. The number line and the ordinary integral ${\int }_a^{b}f(x)dx$

Let us start from the thing you already trust.

• Plain words: total accumulated amount of $f$ across the interval.
• The picture: area under a curve, built from skinny rectangles (look at the blue strips in the figure).
• Why the topic needs it: a line integral is exactly this, except the "interval" is bent into a curve in space. Every line integral we compute will, at the last step, become one of these.

The two pieces to notice: $f(x)$ is the height, and $dx$ is the tiny width. The whole game of line integrals is: what plays the role of the tiny width when the road is curved?

2. Points and vectors in the plane — $(x,y)$ and ${\mathbb{R}}^n$

• Plain words: a location, or equivalently an arrow pointing to that location.
• The picture: a dot with an arrow reaching it from the corner $(0,0)$.
• Why the topic needs it: curves live in this space, and forces are arrows attached to each point of it.

3. The parameter $t$ and the parametrised curve $\mathbf{r}(t)$

Here is the trick that makes everything work.

• Plain words: $t$ is like a clock; $\mathbf{r}(t)$ is where a walker stands at time $t$.
• The picture: a moving dot tracing a bent path, with a few time-stamps written along it.
• Why the topic needs it: the curve is 1-dimensional — you only need one number to say where you are on it. That one number is $t$, and it is what lets us collapse a curve integral into a plain interval integral (§1).

See Parametric curves for the full toolkit; here we only need "one dial $t$ moves the dot".

4. Velocity ${\mathbf{r}}^{\prime }(t)$ — the direction and rate of the walk

• Plain words: which way, and how quickly, the dot is moving right now.
• The picture: a short arrow riding along the curve, tangent to it (look at the orange arrows).
• WHY this tool and not another? To measure a tiny step on the curve we need to know the direction of motion. The derivative is precisely the machine that reports "instantaneous direction and rate", so it is the right (and only) tool for the job.

The tiny displacement over a short time $\Delta t$ is $\Delta \mathbf{r}\approx {\mathbf{r}}^{\prime }(t)\Delta t,$ i.e. (direction-and-rate) × (how long we moved) = (actual little jump). This single line is the seed of both line integrals.

5. Length of an arrow — $|\mathbf{v}|$ and speed $|{\mathbf{r}}^{\prime }(t)|$

• Plain words: how long the arrow is, ignoring direction — always $\ge 0$.
• The picture: the diagonal of a box whose sides are $v_1$ and $v_2$.
• Why the topic needs it: applied to the velocity, $|{\mathbf{r}}^{\prime }(t)|$ is the speed — and speed is exactly what converts a step in time into a step in length: $\Delta s\approx |{\mathbf{r}}^{\prime }(t)|\Delta t.$

Because a square root is never negative, $ds\ge 0$: it has no direction. Remember this — it is why the scalar integral doesn't care which way you walk. See also Arc length, which is nothing but ${\int }_{C}1ds$.

6. The dot product $\mathbf{a}\cdot \mathbf{b}$ — "how much of one arrow lies along another"

• Plain words: measures how much two arrows agree in direction. Same way → positive; perpendicular → zero; opposite → negative.
• The picture: drop a shadow of $\mathbf{a}$ onto the line of $\mathbf{b}$; the dot product tracks the length of that shadow (times $|\mathbf{b}|$).
• WHY this tool and not another? Work only counts the part of a force that points along the motion. The dot product is the exact machine that extracts "the along-part", so it is unavoidable here.

Full details live in Dot product; here we only need "it keeps the along-part and throws away the sideways part".

7. The force field $\mathbf{F}$ and its components $(P,Q)$

• Plain words: a map of pushes — at every spot, an arrow tells you which way and how hard.
• The picture: a grid of little arrows (a "wind map"), each varying from place to place.
• Why the topic needs it: this is the "wind" the walker feels. The vector line integral adds up its forward-push all along the trip.

8. The unit tangent $\mathbf{T}$ — a pure direction of travel

• Plain words: "which way am I facing", stripped of "how fast".
• The picture: the orange velocity arrow, shrunk or stretched to length one.
• Why the topic needs it: it lets us rewrite the vector integral as a scalar one, $\mathbf{F}\cdot d\mathbf{r}=(\mathbf{F}\cdot \mathbf{T})ds$ — "forward-part of the force" times "tiny length". This bridges the two flavours of integral. (Undefined when $|{\mathbf{r}}^{\prime }(t)|=0$, i.e. when the walker momentarily stops — a degenerate case to avoid.)

9. The whole vocabulary in one glance

Every symbol above, and where it flows to:

Read it top to bottom: the ordinary integral plus the idea of a point give you a parametrised curve; differentiating it gives velocity; the length of velocity gives the step size $ds$; the dot product plus a force field give work.

Equipment checklist

Cover the right side and test yourself. If any answer is shaky, reread its section.

Connections

• Parametric curves — the source of $\mathbf{r}(t)$ and $t$.
• Arc length — the case ${\int }_{C}1ds$ built entirely on $ds=|{\mathbf{r}}^{\prime }|dt$.
• Dot product — the engine of §6 and §8.
• Line integrals — scalar and vector, work done — the parent topic this page feeds.