Lecture notes · Calculus II · Lecture 19

Vector Fields

MIT 18.02 Multivariable Calculus, lecture 19, explained in plain English

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Part 1 · 0:00–10:00

This part introduces vector fields, which give an arrow (a vector) at every point in a plane, and shows how to picture a few simple ones.

Key ideas

A vector field gives a vector at every point in a plane. The vector's parts depend on where you are, so each x and y position gets its own arrow. 1:30
Real examples include wind speed and direction on a weather map, and gravity pulling objects toward the ground. 3:50
You cannot draw an arrow at every point, so you pick a few points and draw arrows there. Then you have enough to see the overall pattern. 2:17
If a vector field is the same everywhere, like 2i + j, it is just one arrow copied across the plane. 5:38
For x i, the arrows point sideways, stay zero along the y-axis, get longer as x grows, and point left when x is negative. For x i + y j, the arrows point away from the origin and get longer farther out. 9:20

Words to know

vector field: A rule that gives an arrow (a vector) to every point in a flat space, where the arrow can change from point to point.
component: One of the parts of a vector, such as its sideways part or its up-and-down part.
magnitude: How long a vector is, or how strong it is, without caring which way it points.
radially: In a straight line going out from a center point, like the spokes of a wheel.

Example

Picture a weather map with small arrows scattered across it. Each arrow shows which way the wind blows at that spot and how fast, and the arrows change as you move across the map. That is a vector field you can see.

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Part 2 · 10:00–20:00

This part shows how a vector field can describe motion like fluid flow or rotation, and how to add up the work a changing force does along a curved path.

Key ideas

A vector field gives an arrow at every point. One example points away from the origin, and its length grows with the distance from the origin, like a source pouring fluid out. 10:00
The field <-y, x> points perpendicular to the position vector at each point. It makes fluid spin around the origin, with zero speed at the origin and faster speed farther out. 10:45
This spinning field is the velocity field for uniform rotation. A particle going around once travels 2π, since a circle's length is 2π times its radius. 13:01
Work is the force dotted with the small displacement vector. If the force changes along a curved path, you cut the path into tiny pieces, add up the small work amounts, and that sum becomes an integral. 19:15
The total work along a curve C is called a line integral. It is written as an integral over the trajectory. 19:59

Words to know

vector field: A rule that gives an arrow (a vector) at every point in space, such as the speed and direction of fluid at each spot.
dot product: A way to multiply two arrows that gives one number. It tells you how much the two arrows point the same way.
line integral: A total built by adding up many small contributions along a curved path, such as the total work done by a changing force.
angular velocity: How fast something turns around a center, measured in radians per second.

Example

Think of pushing a shopping cart along a winding aisle where you push harder in some spots and softer in others. To find the total effort, you would split the path into small steps, multiply the push at each step by how far you moved in that direction, and add everything up. That adding-up process is what a line integral does.

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Part 3 · 20:00–30:00

This part shows how to calculate the work a force does along a curved path by splitting the path into tiny time steps and adding up the results.

Key ideas

Work along a path C is written as the integral along C of F dot dr. The idea is to add up many tiny pieces, where each piece is the force dotted with a tiny movement along the path. 20:26
To actually compute it, cut the path into tiny time intervals. During each tiny interval, the movement is the velocity times the length of that time. 21:36
Because velocity is how position changes over time, we can replace the tiny movement with velocity times dt. Then we integrate F dot velocity over time, from the start time to the end time. 22:36
In the example, the particle moves along x = t, y = t squared from t = 0 to t = 1. Plugging in the force and velocity and integrating gives a total work of one-third. 27:06
The answer depends on the path, not just the start and end points. So you must use the exact parametric equations of the curve you are given. 29:57

Words to know

line integral: A total built by adding up many tiny pieces along a path, not just over a straight line of numbers.
dot product: A way to combine two vectors: multiply their matching parts and add the results. It tells you how much two arrows point the same way.
velocity vector: An arrow that shows both how fast something is moving and in which direction.
parametric equation: A way to describe a curve where both x and y are given as formulas of one shared variable, such as time t.

Example

Imagine pushing a box along a sidewalk. Only the part of your push that points in the direction the box moves does useful work. If you push sideways, you do no work at all, even though you are pushing hard.

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Part 4 · 30:00–40:00

This part shows how to compute a line integral by describing the curve with one variable and turning everything into an ordinary single-variable integral.

Key ideas

A force times a tiny step along the path, written F dot dr, can be rewritten as M dx plus N dy when the force has components M and N. This is just a different notation for the same thing. 32:45
The tricky part is that x and y are linked along the curve, so you cannot treat them as separate. You have to describe the whole curve with one parameter, such as t. 33:27
To evaluate, write x and y in terms of one variable, then replace dx and dy using derivatives. For example, if x is t and y is t squared, then dx is dt and dy is 2t dt. 36:34
In the worked example, the integral from t equals 0 to t equals 1 gives 1/3. The lesson says the answer depends on the path itself, not on which parameter you chose to describe it. 39:28

Words to know

line integral: A sum of a force or field along a specific path, added up one tiny step at a time.
differential: A small change in a variable, such as dx, which is a tiny step in x.
parameter: A single variable, like t, that controls where you are on a curve, so the whole curve can be described with just that one value.
parametric equation: A pair of equations that give x and y each in terms of the same parameter, such as x equals t and y equals t squared.

Example

Imagine walking a winding trail where you only track one number, like how many minutes you have been walking. Once every position on the trail is tied to that one number, you can measure the trip with a single simple sum. The lesson does the same thing: it ties x and y to one variable so the integral becomes a regular one.

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Part 5 · 40:00–51:08

This part shows how to find the work done along a curved path, and how a bit of geometry can save you from heavy calculation.

Key ideas

To compute the work along a path, pick the simplest way to describe the curve, such as using x itself as the variable. Avoid messy choices like sines when a simple one works. 40:19
A tiny step along a path points in the same direction as the tangent, which is the line that just touches the curve. Its length is the small arc length, called ds. 42:34
Because of this, the work can be written as the force's part along the tangent, times the arc length. Only the force pushing along the path counts. 44:28
If the force is always at a right angle to the motion, the work is zero, and you can see that without any calculation. 46:54
If the force points the same way as the motion and has constant size a, the work is a times the path length. For a circle of radius a, that is 2 pi a squared. 49:52

Words to know

line integral: A way of adding up a force along a curved path, piece by piece, to get the total work.
parameterize: To describe a curve using one simple variable, such as t or theta, so every point is written in terms of that variable.
tangent: A line or direction that just touches a curve at one point and points the same way the curve is going there.
arc length: The distance measured along a curved path, like the length of a string laid along the curve.

Example

Imagine pushing a shopping cart around a store aisle. Only the part of your push that points in the direction the cart is moving adds to the work you do; a push straight sideways does nothing to move it forward. The geometric method in this lesson works the same way, by keeping only the force along the path.

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

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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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