Fun

Shine a light down onto one arrow and see what shadow the other casts. That shadow is the whole idea.

One vector carrying two facts at once: which plane a pair of arrows spans, and how much area they enclose in it.

The shortest way from a point to a plane is always the perpendicular one — here it is, drawn.

Polar, cylindrical and spherical. Watch the little box of volume shrink as it approaches the axis, and see where the extra factors come from.

Slice a surface at every height and watch its topographic map appear. Crowded lines mean steep ground.

The whole zoo — ellipsoids, saddles, cones and both hyperboloids — and the traces that tell them apart.

Walk into the same point from different directions and arrive at different answers. Then there is no limit.

Every smooth surface is flat if you look closely enough. That flat thing is the tangent plane.

Point a compass needle anywhere on a contour map and watch the slope in that direction respond. One direction always wins.

Walk along a constraint until a level curve just barely touches it. That kiss is the answer.

Step downhill, over and over, and hope you picked a step size that doesn't throw you off the mountain.

Flat spots come in three flavours, and one determinant tells them apart. Watch a peak and a saddle collide and annihilate.

Follow the tangent to where it hits zero, and repeat. Which root you land on depends on where you start — sometimes wildly.

Ride along a curve carrying three perpendicular arrows. Curvature is the circle that fits best.

Two dots race the same path, one at constant speed. The difference between them is what the arclength parameter fixes.

The same question twice: gravity that pulls straight down, and gravity that pulls toward a point.

Iron filings, a weather map, a leaf on a stream. Drop particles into a field, then measure how much it spreads and how much it spins.

Drag a path through a field and watch the work accumulate. Walk it backwards and every bit of it changes sign.

Some fields don't care which way you went, only where you ended up. Most do care — and one famous one only pretends not to.

Stand a fence along a winding path, as tall as the function is high. Its area is the scalar line integral.

Sweep a region with vertical strips, then with horizontal ones. Same region, two very different sets of limits.

Stack boxes under a surface until they stop being boxes. That limit is the double integral.

A solid casts three shadows, and each one is the outer region of a different order of integration.

Bend a grid and watch a tiny square stretch. The area it gains is exactly the factor the integral needs.

Bend a flat sheet of graph paper into a shape. Where the sheet stretches, every square on it has to be counted for more.

Throw darts at a shape and count the hits. Hopeless in one dimension, unbeatable in twenty.

Where does a shape balance, and how hard is it to spin? Two integrals of the same lamina, answering different questions.

Tile a region with tiny spinning cells. Every interior edge cancels against its neighbour, and only the boundary survives.

How much of a flow actually gets through a surface — and why you have to decide which side counts as out before the question means anything.

Stretch any surface you like across the same wire loop. The spin caught by all of them is identical.

Shrink a closed surface toward a point and the flux per unit volume settles onto a single number: the divergence there.

Every path from the top of the diagram to the bottom is one term. Collect them all and you have the chain rule.

Grad, curl and div are the same operator wearing three hats — and the four big theorems are one theorem.