Foundations — Heaviside step function and Dirac delta function

This page assumes you know nothing. Before you can read the parent note comfortably, every squiggle it uses must first become a picture. We build them one at a time, each resting on the last.

0. What a "function of time" even is

Why we need it. Everything on the parent page is a function of time: a switch state, a push, the wiggle of a spring. If "graph = time across, value up" is not automatic for you, nothing later will land. Look at the axes in every figure below — time always runs left to right.

1. The "before/after" idea: piecewise

Why the topic needs it. Real inputs change their behaviour at specific moments — a switch does nothing, then does something. Later we will meet special "before/after" functions (a switch, a kick), but the general idea is just this: split time at a marker and use a different rule on each side.

Here is the simplest possible picture — two flat segments with a jump at the marker $a$:

2. The Laplace variable $s$ — why we invent it

Before any integral, we must earn the letter $s$, because it appears inside everything below.

Now that $s$ has a meaning, the integral in §4 will read cleanly.

3. The area-under-a-graph: the integral $\int$

The picture: chop the region under the curve into thin vertical strips of width $dt$ and height $f(t)$; each strip has area $f(t)dt$; the $\int$ adds them all up.

Why the topic needs it — two reasons.

• The Dirac delta (coming in §7) is defined only by what its area does — "total push = area under the spike = $1$". No integral, no delta.
• The Laplace transform (§4) is an integral.

4. The exponential $e^{-st}$

The picture below shows three decay curves: a slide that drops steeply then flattens, never quite touching the axis, steeper for larger $s$.

Why the topic needs it. $e^{-st}$ is the "weighting" inside the Laplace transform. It tames functions so their infinite-time area is finite — that is exactly why the integral in §3 converges only for $s>0$.

5. The Laplace transform $\mathcal{L}\{\cdot \}$ and $F(s)$

The picture: two side-by-side worlds. On the left, a graph in time $t$. A tunnel labelled $\mathcal{L}$ carries it to the right, where it becomes a graph in $s$.

Why the topic needs it. Differential equations are hard in the time world. The Laplace transform turns calculus (derivatives, delays) into algebra (multiply, divide) in the $s$-world. You solve the easy algebra, then tunnel back. See Laplace Transform — Definition and Existence for when the integral is even allowed to exist, and Solving IVPs with Laplace Transforms for the round-trip in action.

6. The delay stamp $e^{-as}F(s)$ (why shifting time = multiplying)

The picture: the same wiggle, slid rightward on the time axis; in the $s$-world only a constant multiplier $e^{-as}$ appears.

Why the mandatory gate $u(t-a)$? Because "play it late" must be silent before time $a$. Drop the gate and your answer wrongly rings before the event — the parent page's most-warned mistake. (The gate $u$ is defined next, in §7.)

7. The derivative $\frac{d}{dt}$ and "slope"

The picture: zoom into the graph at a point; the derivative is the tilt of the tiny line you see.

Why the topic needs it. It is the hinge that turns a switch into a kick — but to see that we first need the switch and the kick themselves, which we now define.

8. Symbols specific to steps & impulses

9. The unifying fact: $\delta$ is the slope of $u$

Now that $u$, $\delta$, and $\frac{d}{dt}$ all exist, we can join them. The claim is: $\frac{d}{dt}u(t-a)=\delta (t-a).$ Let us build it slowly instead of asserting it, using a finite "honest" version and a limit.

Step 1 — WHAT. The ramp's total climb is $1$, so its slope-box has area $\text{width}\times \text{height}=\epsilon \cdot \frac{1}{\epsilon }=1$. Whatever $\epsilon$ is, the box's area stays exactly $1$.

Step 2 — WHY. Making the switch sharper means shrinking $\epsilon$. As $\epsilon \to 0^{+}$ the ramp becomes the true instantaneous cliff of $u$, and its slope-box becomes taller ($1/\epsilon \to \infty$) and thinner (width $\to 0$) while its area is pinned at $1$.

Step 3 — WHAT IT LOOKS LIKE. That limiting object — zero everywhere, infinite at the one instant, area exactly $1$ — is precisely the amber spike $\delta (t-a)$ from §8. So the slope of the step is the delta. The figure below walks $\epsilon$ from wide to narrow so you can watch the box grow into the spike.

10. How these tools chain together for the topic

Two more names to file away before the map: Piecewise and Periodic Forcing Functions is what §1 + §8 build up to (writing messy inputs with steps), and Impulse Response and Convolution is what §8's $\delta$ unlocks (the system's answer to a single kick).

Read the map top-to-bottom as "what feeds what": the raw ideas (function, integral, exponential, derivative) sit near the top; each arrow means "you need the tail before the head"; everything funnels down into solving IVPs with steps and impulses, which is the parent topic's goal.

Equipment checklist

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