Lecture notes · Calculus II · Lecture 31

Stokes' Theorem

MIT 18.02 Multivariable Calculus, lecture 31, explained in plain English

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Part 1 · 0:00–10:00

This part introduces curl, a way to measure how much a flow spins, and Stokes' theorem, which links a closed-path line integral to a surface integral of curl.

Key ideas

Curl measures the rotation part of a flow. Think of it as how much a fluid is spinning at each point. 1:34
The direction of the curl points along the axis of rotation. Its size equals twice the angular velocity, which is how fast things spin. 2:05
A flow that only moves everything the same way, or only stretches things, has zero curl. Stretching is measured by divergence instead. 3:30
Stokes' theorem says the work done by a vector field around a closed curve equals the double integral of curl F over any surface bounded by that curve. 7:19
You can pick whatever surface you like, as long as its edge is the curve. A flat disk is often the simplest choice. 9:34

Words to know

curl: A measure of how much a vector field spins or rotates at a point. You get it by taking a special cross product with the del symbol.
divergence: A measure of how much a vector field spreads things out or stretches them at a point.
conservative: A vector field is conservative when its curl is zero. Then you can find a potential function for it, and the work does not depend on the path taken.
closed curve: A path that starts and ends at the same point, like a loop.

Example

Imagine stirring a cup of tea with a spoon. The swirl of the tea around the spoon is like curl, because it measures spinning. If the tea simply flows steadily in one direction without spinning, there is no curl there, even if the water is moving fast.

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Part 2 · 10:00–20:00

This part explains that Stokes' theorem only gives matching answers when the direction of the curve and the direction of the surface are chosen in a compatible way, and it shows two ways to check that.

Key ideas

Stokes' theorem lets you pick any surface that has the curve as its edge, then find the flux of curl F through it. The answer matches the line integral around the curve. 10:00
The catch is orientation. A surface has two sides, and a curve has two directions, so you must choose them to match. Flipping one flips the answer. 10:43
Rule one: if you walk along the curve C in its chosen direction with the surface on your left, the normal vector (the arrow that sticks straight out of the surface) points up for you. 12:15
If you reverse the direction of the curve, the surface ends up on your right, and the normal vector points down. 13:47
Rule two is the right-hand rule. Thumb along the curve, index finger into the surface, and the middle finger then points along the normal vector. 15:21

Words to know

orientation: the direction you choose to face or travel, such as which side of a surface counts as 'positive' or which way you go around a curve.
flux: how much of a vector field passes through a surface, like how much wind goes through an open window.
boundary: the edge of a surface. The curve that goes around the edge is the boundary.
normal vector: an arrow that points straight out from a surface, at a right angle to it.

Example

Imagine a flat sheet of glass with its edge marked by a line. If you walk around the edge with the glass on your left, the arrow sticking out of the glass points up toward your head. If you turn around and walk the other way, the glass is now on your right, and the arrow points down.

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Part 3 · 20:00–30:00

This part shows how to choose the direction of a surface's normal vector and its boundary curves, and how Stokes' theorem turns into Green's theorem when the surface lies flat in the x-y plane.

Key ideas

Stretching or bending a surface without tearing it does not change its orientation rules. If you flatten a cone into a disk, the normal vector still points the same way on the same side. 20:00
For a cylinder with the normal vector pointing outward, the bottom curve goes counterclockwise and the top curve goes clockwise. You can check this by walking on the outside of the surface with the surface on your left, or by using the right-hand rule. 21:03
If a problem gives you neither the curve's direction nor the surface's direction, you choose both. You must choose them in a way that matches each other. 23:00
When the surface is a flat piece of the x-y plane and the curve goes counterclockwise, the normal vector points up. Stokes' theorem then becomes Green's theorem, so Green's theorem is a special case of Stokes' theorem. 24:16
These orientation rules are conventions. They come from our choice of right-handed coordinate systems, and someone using left-handed coordinates would have the rules reversed. 28:51

Words to know

normal vector: An arrow that sticks straight out from a surface, like a pole standing up from the ground.
orientation: The chosen direction of a surface or curve. It tells you which side is the front, or which way the loop goes around.
curl: A vector that measures how much a field tends to spin or swirl around a point.
right-hand rule: A way to find a direction using your right hand. Point your fingers along the loop and your thumb shows the direction of the normal vector.

Example

Think of a sheet of paper with a loop drawn on it. If you curl the paper into a cylinder, the loop still goes the same way around the paper, and you can still tell which side is the outside. The rule for choosing directions stays the same no matter how you bend the sheet.

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Part 4 · 30:00–40:00

Stokes' theorem only works on well-behaved surfaces, and the lecture explains the idea behind why it is true: break any surface into tiny flat pieces and use Green's theorem on each one.

Key ideas

Stokes' theorem needs a surface whose edge is the curve C, so the surface usually cannot be closed like a ball. A closed surface has no edge to attach the curve to. 30:05
The vector field must be smooth (continuous and differentiable) on the surface. If it breaks at one point, you may need to pick a different surface that avoids that point. 31:11
The proof starts from Green's theorem, which is the special case where the curve and surface lie flat in a coordinate plane. Work, flux, and curl do not depend on the coordinate system, so the same result holds for any flat plane. 33:01
Any curved surface can be cut into tiny pieces that are almost flat. Stokes' theorem applies to each piece. 36:47
When you add up the line integrals around all the tiny pieces, the inner edges cancel out. What remains is the work along the outer curve C, which is why the big theorem holds. The lecturer admits he skipped a careful check of this step. 38:34

Words to know

flux: How much of a vector field passes through a surface, like how much water flows through a window.
curl: A measure of how much a field makes things spin or rotate at a point.
Green's theorem: An earlier result for flat regions in the x-y plane that links a line integral around the edge to a double integral over the region inside.
boundary: The edge of a surface, the curve that goes all the way around it.

Example

Think of a tiled floor covering a curved dome. Each tile is so small it looks flat, so you can measure each one with a simple flat-floor rule. Add up all the tile measurements, and the shared edges between tiles cancel out, leaving only the outer border of the dome.

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Part 5 · 40:00–48:25

The lecturer computes the work done by a vector field around a circle two ways, directly and with Stokes' theorem, and gets the same answer, pi.

Key ideas

The goal is to find the work done by the field zi + xj + yk as you go once counterclockwise around the unit circle in the x, y plane. 41:21
The direct way: describe the circle with x = cos t and y = sin t, then add up the small pieces as t goes from 0 to 2pi. The answer is pi. 43:26
Stokes' theorem says you can get the same answer by summing up the curl of the field over any surface whose edge is the circle. The lecturer picks a paraboloid instead of the easier flat disk to show the choice is flexible. 43:50
The curl of this field is (1, 1, 1), which is the same at every point. The surface normal, pointing up, is (2x, 2y, 1) times dx dy. 45:52
The flux integral becomes the integral of 2x + 2y + 1 over the unit disk. The x and y parts cancel by symmetry, leaving the area of the disk, which is pi. 47:17

Words to know

curl: A measure of how much a vector field swirls or rotates around a point.
flux: The total amount of something passing through a surface, found by adding up the field pushing through each small piece of it.
parameterize: To describe a curve or shape using one variable, like t, so that each point on it is given by a formula.
paraboloid: A bowl-shaped 3D surface, here given by the equation z = 1 - x^2 - y^2.

Example

Imagine walking once around the edge of a fenced yard and measuring how much a wind pushes you along the way. You could add that up step by step along the fence, or you could measure how the wind swirls over the grass inside the fence and add that up instead. Stokes' theorem says both ways give the same total.

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← Lecture 28: Divergence Theorem · Lecture 33: Maxwell's Equations →

Notes written by AI (Claude) from the lecture's official transcript, so check anything important against the lecture. MIT OpenCourseWare · CC BY-NC-SA 4.0 · not affiliated with or endorsed by MIT. These notes are shared under the same license.

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