Lecture notes · Calculus II · Lecture 28

Divergence Theorem

MIT 18.02 Multivariable Calculus, lecture 28, explained in plain English

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Part 1 · 0:00–10:00

This part explains where the formula for a surface's area element and normal direction comes from, for a surface written as z equals a function of x and y.

Key ideas

Flux is the total flow of a vector field through a surface. It is found with a double integral of F dot n dS, where n is the unit normal (a unit-length arrow pointing straight out from the surface). 0:23
The lecture says it will explain the formula for n dS when the surface is a graph, meaning z is a function of x and y. It will not just give the formula without reasons. 1:29
To find n dS, zoom in on a tiny rectangle of the x-y plane. The surface above it is so small that it looks like a flat parallelogram. 4:03
The cross product of the two side vectors gives both the area of the parallelogram and its normal direction at the same time. That is why n dS is an easy formula to use. 4:25
The normal can point either way, depending on whether you take U cross V or V cross U. The lecture says the sign will be sorted out later. 6:15

Words to know

flux: How much of a flow goes through a surface, like water passing through a net.
unit normal: An arrow of length one that sticks straight out from a surface, at a right angle to it.
cross product: A way to multiply two arrows (vectors) in 3D that gives a new arrow at a right angle to both. Its length matches the area of the parallelogram they form.
partial derivative: How fast a function changes when you change only one input and hold the others fixed. Here f sub x means how z changes as x changes.

Example

Imagine a small flat tile sitting on a gently sloping hill. If the tile is tiny, it matches the hill's surface well, so its area and the direction it faces tell you what the hill's surface is doing at that spot. This is only a picture to help you, not a formula from the lecture.

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

Part 2 · 10:00–20:00

This part shows how to find the flux of a vector field through a curved surface, using a formula for the surface's normal vector and a worked example with a paraboloid.

Key ideas

When a surface is given as z = f(x, y), there is a formula for n dS, which combines the normal vector (the arrow standing straight out from the surface) with the small piece of area. The formula is (negative f_x, negative f_y, 1) dx dy. 10:20
You choose which way the normal vector points. Using the plus-sign version makes it point up, and the opposite signs make it point down. The sign of the answer depends on this choice. 10:43
To compute the flux, you put the surface's equation into the integral to get rid of z. Then you integrate over the region below the surface, which is its shadow on the flat x-y plane. 16:15
The surface gives you the limits of the integral and the n dS part. The vector field is the only other piece, so you can reuse the same surface with a different vector field. 19:10

Words to know

normal vector: an arrow that points straight out from a surface, like a stick poking out of a table.
flux: how much of a vector field flows through a surface, like how much water passes through a net.
paraboloid: a bowl-shaped surface, like the curve of a satellite dish, given by an equation like z = x^2 + y^2.
partial derivative: the slope of a function when you change only one variable and hold the others still.

Example

Picture water flowing up through a bowl-shaped net. The flux measures how much water passes through the net. If you decide that the net's 'front' faces up, water flowing upward counts as positive.

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

Part 3 · 20:00–30:00

This part shows how to set up a flux integral on a curved surface when you describe it with two parameter variables, u and v, instead of writing z as a function of x and y.

Key ideas

A surface can be described by a position vector that depends on two parameters, u and v. Since a surface has only two independent directions to move in, two variables are enough. 21:56
To find the small surface piece n dS, change u a little while keeping v fixed, and change v a little while keeping u fixed. This makes a tiny parallelogram on the surface. 23:25
The two sides of that tiny parallelogram are the partial derivatives of the position vector with respect to u and v, each multiplied by its small change. Their cross product gives the area and the normal direction. 24:33
So n dS equals the cross product of the two partial derivatives, times du dv, up to a sign that depends on which way you choose the normal vector. 26:56
Checking with easy cases helps. If u and v are just x and y, you get the same formula as before. A sphere parametrized by angles gives the same n dS you already know. 27:47

Words to know

parametric equation: a way of describing a shape where x, y, and z are each given by a formula that uses other variables, called parameters.
parameter: an extra variable, like u or v, that helps you describe a surface or curve. You can name it anything you like.
cross product: a way of multiplying two vectors to get a new vector that is perpendicular to both. It gives the area and direction of the small piece of surface.
normal vector: an arrow that points straight out from a surface, at right angles to it.

Example

Think of a sheet of fabric draped over a ball. You can mark its position with two numbers, like how far you went east and how far you went north on the fabric. The two directions you can move in on the cloth are like u and v, and the small patch you cut out from those two directions tells you its size and which way it faces.

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

Part 4 · 30:00–40:00

This part shows how to find the surface element n dS for a slanted surface by using a normal vector and projecting the surface onto a flat plane.

Key ideas

A normal vector is an arrow that sticks straight out from a surface. It can be any length, so you do not need it to be a unit vector (length one). 30:00
If a surface is given as g(x, y, z) = 0, the gradient of g points perpendicular to the surface, so it can serve as the normal vector. 31:28
A small patch on a slanted surface looks smaller when you shine light straight down on it. The flat shadow area dA equals dS times the cosine of the angle alpha between the surface and the horizontal. 35:19
The cosine of that angle can be found with a dot product: take N dot k and divide by the lengths of N and k. Since k has length one, this gives cos alpha = (N dot k) / length of N. 36:50
The final formula is n dS = plus or minus N over (N dot k) times dx dy. The plus or minus sign is there because the unit normal might point either way. 39:07

Words to know

normal vector: An arrow that points straight out from a surface, like a pole standing straight up from a table.
gradient: A vector that shows the direction in which a function grows fastest. Here it is used as a normal vector for a surface given by g = 0.
cosine: A number from the angle between two directions. A flat surface has cosine one, and a steep surface has a smaller cosine.
projection: The shadow a shape makes on a flat plane when light shines straight onto it.

Example

Think of a sloped roof and the flat floor plan of the house. A tile on the roof has more area than its shadow on the ground. The steeper the roof, the smaller the shadow compared to the tile. The cosine of the roof's angle tells you how much smaller.

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

Part 5 · 40:00–49:14

This part introduces the divergence theorem, which lets you swap a hard surface integral over a closed surface for an easier triple integral over the space inside it.

Key ideas

Computing surface integrals can be hard, so the divergence theorem offers a shortcut: instead of working over the surface, you work over the region of space inside it. 42:34
The theorem only works for a closed surface, one with no gaps or edges that completely encloses a region. The normal vector must point outward. 43:52
The divergence of a 3D vector field with parts P, Q, and R is found by adding the x-derivative of P, the y-derivative of Q, and the z-derivative of R. 45:26
For the field zk through a sphere, the divergence is just 1, so the triple integral of 1 over the ball is simply the volume of the ball, 4/3 pi a cubed. 47:48
Physically, divergence measures sources and sinks inside a region. The total flow out of the region equals the sources minus the sinks inside it. 48:49

Words to know

closed surface: A surface with no gaps or edges that completely wraps around a region of space, like the skin of a ball.
divergence: A number that tells you how much a flow is spreading out at a point, or how many sources or sinks sit there.
flux: The total amount of something, like water, that passes through a surface.
gradient: A vector made of all the partial derivatives of a function, which points in the direction the function increases fastest.

Example

Imagine a closed balloon full of water with small taps that add water (sources) and small drains that remove water (sinks). The total water leaving through the balloon's skin is the number of taps minus the number of drains inside. You can count what is inside instead of measuring every drop at the skin.

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

← Lecture 23: Flux · Lecture 31: Stokes' 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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