Lecture notes · Calculus II · Lecture 23

Flux

MIT 18.02 Multivariable Calculus, lecture 23, explained in plain English

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Part 1 · 0:00–10:00

This part introduces flux, a way to measure how much of a vector field flows across a curve in the plane, and explains it using the idea of fluid passing through a membrane.

Key ideas

Flux is a kind of line integral. You break the curve into tiny pieces, take the dot product of the field with the unit normal vector on each piece, multiply by the piece's length, and add them up. 1:10
The normal vector points straight out from the curve, perpendicular to it, with length one. By convention, it points to the right as you travel along the curve. 3:28
Work adds up the part of the field that goes along the curve. Flux adds up the part of the field that crosses the curve. The math is much the same, but the meaning is different. 6:02
If the vector field is a velocity field, meaning it describes how a fluid is moving, then flux measures how much fluid passes through the curve per unit time. 8:31
To see why, zoom in on a tiny piece of the curve. Over a short piece, the flow looks almost uniform, so the amount crossing depends on how fast and in what direction the fluid moves compared to the curve. 9:21

Words to know

flux: How much of something flows across a line or curve, counted per unit time when the field is a flow.
vector field: A rule that gives an arrow, with a direction and a size, at every point in a space. Think of arrows showing how things move at each spot.
unit normal vector: An arrow that sticks straight out from a curve, pointing at a right angle to it, and has a length of exactly one.
line integral: A way of adding up values along a curve by cutting the curve into tiny pieces, measuring each piece, and summing the results.

Example

Imagine a river with a dam across it that has small holes, so water can still pass through. Flux is like counting how much water goes through the dam each second. Water flowing along the dam's surface does not count, only water crossing through it.

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Part 2 · 10:00–20:00

This part explains flux as the net amount of stuff that flows across a curve per unit time, and shows how to calculate it for a simple circle.

Key ideas

Flux measures how much material crosses a curve in a small amount of time. Picture a thin slice of curve: the material that crosses it in one tiny moment fills a parallelogram. 10:21
The parallelogram's area is base times height. The base is the small piece of curve, and the height is the part of the flow that points straight across the curve, which is F dot n. 12:35
Flow that crosses in the direction of the normal vector counts as positive, and flow in the opposite direction counts as negative. The total is the net flow. 14:00
For a closed curve, the usual choice is to count the flow going out of the region the curve encloses. 17:55
For a circle with the field xi + yj, F dot n equals a at every point, so the flux is just a times the distance around the circle. 19:46

Words to know

flux: The net amount of something, like water or matter, that flows across a line or curve in a set amount of time.
normal vector: An arrow that points straight out from a curve or surface, at a right angle to it.
vector field: A rule that gives an arrow, with a direction and a strength, at every point in space, like the speed and direction of a flowing river at each spot.
line integral: A way of adding up a quantity all along a curve, by splitting the curve into tiny pieces and summing them.

Example

Imagine water flowing through a screen with holes in it. Only the part of the water that pushes straight through the screen counts toward the flow, while water that just slides sideways along it does not. Flux works the same way: only the part of the field pointing straight across the curve matters.

3 checked practice questions for this part are in the app.

Part 3 · 20:00–30:00

This part shows how to calculate flux along a curve using x and y components, instead of drawing normal vectors.

Key ideas

If the vector field is tangent to the curve, it has no normal part, so nothing flows across the curve and the flux is zero. 21:05
For a simple field you can often guess whether the flux is positive, negative, or zero by asking which way the flow crosses the curve. 21:51
For complicated curves, finding the normal vector by hand is hard, so it is better to calculate in coordinates. 22:32
The normal direction n ds is the tangent direction T ds turned 90 degrees clockwise, which gives a rule using the pieces dx and dy. 24:49
The flux becomes a line integral of negative Q dx plus P dy, and you finish it by writing x and y in terms of one variable and doing a normal single integral. 28:05

Words to know

flux: How much of a flow passes across a curve, like how much water crosses a line you drew in a river.
normal vector: An arrow that points straight out from a curve, at right angles to it.
tangent: Running along a curve, touching it in the direction it goes, not crossing it.
line integral: A sum of values taken along a curve, instead of over a flat area or a single number line.

Example

Imagine a river flowing past a fence. If the water runs parallel to the fence, no water crosses it, so the flux is zero. If the water pushes straight through the fence, you can see the flow crossing it and judge the sign of the flux.

3 checked practice questions for this part are in the app.

Part 4 · 30:00–40:00

This part shows how to measure flux with a line integral, and how Green's theorem turns a flux around a closed curve into a double integral.

Key ideas

To find flux instead of work, you change one thing. You use minus N dx plus M dy in place of M dx plus N dy. The lecturer says you just have to memorize this formula. 30:27
If the curve is closed, you can skip the line integral. A version of Green's theorem for flux turns it into a double integral over the region inside the curve. 31:35
The divergence of a vector field with parts P and Q is P sub x plus Q sub y. You take the partial of P with respect to x, take the partial of Q with respect to y, and add them. There are no signs to switch. 34:08
When the curve goes counterclockwise, the normal vector points out of the region. So a part of the curve where the field flows outward adds positive flux. 35:10
The flux version is called Green's theorem in normal form. It is the same theorem as the work version, just written with different letters. The two versions are used for different jobs: work along a closed curve, or flux out of a region. 39:45

Words to know

flux: How much of a flow passes through a curve or surface. Think of how much water crosses a fence, not how fast it moves along it.
divergence: A number that tells you how much a field spreads out at each point. It is found by adding the two partial derivatives P sub x and Q sub y.
Green's theorem: A rule that says a line integral around a closed curve equals a double integral over the region inside that curve. It lets you swap a hard calculation for an easier one.
normal component: The part of a vector that points straight out from the curve, at a right angle to it. Flux uses this part, while work uses the part that follows the curve.

Example

Imagine a garden hose spraying water through a fence. The flux is how much water crosses the fence, and only the part pushing straight through counts. Water running along the fence does not add to the flux.

3 checked practice questions for this part are in the app.

Part 5 · 40:00–50:13

This part shows how Green's theorem turns a flux problem into a double integral of divergence, and explains what divergence measures.

Key ideas

For the field F = xi + yj, the divergence is found by taking the partial derivative of each part and adding them: 1 + 1 = 2. Divergence is a number that comes from a vector field. 43:43
Green's theorem says the flux out of a region equals the double integral of the divergence over that region. For this field, the flux out of a circle is twice its area, which is 2πa², the same answer as the direct calculation. 44:25
The result holds for any closed curve, not just a circle centered at the origin. Moving the circle somewhere else in the plane does not change the flux. 45:51
Divergence measures how much a flow is spreading out or expanding. It ignores pure sliding motion (translation) and pure spinning motion (rotation). 47:20
Another way to think of divergence is as a source rate: how much fluid is being added to the system per unit time per unit area. Positive divergence means fluid is being pumped in everywhere. 48:46

Words to know

Flux: The total amount of something, like water or air, that flows out through a boundary.
Divergence: A number that tells you how much a flow is spreading out at a point. Positive means spreading out, negative means squeezing in.
Double integral: A way to add up a quantity over a flat two-dimensional region, like finding a total over a whole area.
Green's theorem: A rule that says the flux out of a closed curve equals the double integral of the divergence over the region inside it.

Example

Think of a balloon being blown up. As it grows, the air inside spreads out to fill more space, so the flow has positive divergence. If you let the air out, the space shrinks and the divergence is negative.

3 checked practice questions for this part are in the app.

← Lecture 19: Vector Fields · Lecture 28: Divergence Theorem →

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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