Lecture notes · Calculus II · Lecture 12

Gradient

MIT 18.02 Multivariable Calculus, lecture 12, explained in plain English

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Part 1 · 0:00–10:00

This part introduces the gradient, a vector made from partial derivatives, and shows that it is always perpendicular to the level curves or surfaces of a function.

Key ideas

If a value w depends on x, y, and z, and those depend on time t, the chain rule says the change in w over time comes from how fast x, y, and z change, weighted by how sensitive w is to each one. 1:06
The gradient is a vector made by putting all the partial derivatives together as its parts. It is a shorthand, not new information. 1:52
The gradient is not one fixed vector. It gives a different vector at each point in space. 2:12
The gradient is always perpendicular to the level curve or level surface, meaning the set of points where the function has the same value. 4:03
For a flat, linear function, the level sets are planes, and the gradient is the normal vector, the arrow that sticks straight out of the plane. 6:39

Words to know

gradient: A list of all the partial derivatives of a function, written as one arrow (a vector) that points in a direction at each point.
partial derivative: How much a function changes when you nudge only one input and hold the others still.
level curve or level surface: All the points where a function gives the same answer, like a contour line on a map that marks one height.
normal vector: An arrow that sticks straight out from a flat surface, at a right angle to it.

Example

Think of a hiking map with contour lines. Each line joins places at the same height. If you want to climb fastest, you walk straight uphill, which is at a right angle to the contour line you are standing on. The gradient works the same way: it always points at a right angle to the level line through that point.

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

Part 2 · 10:00–20:00

This part shows why the gradient vector must stand straight out from a level surface, by looking at how a point moving on that surface changes the function's value.

Key ideas

A level surface is a set of points where a function keeps the same value, c. If you move along it, the function does not change. 11:44
A moving point on a level surface has a velocity vector that is tangent to the surface, which means it just skims along the surface. 13:32
The chain rule says the rate of change of w along a path is the dot product of the gradient and the velocity. Since w stays at c, that rate is zero, so the gradient and the velocity are perpendicular. 15:28
Any direction in the tangent plane can be a velocity of some path on the surface, so the gradient is perpendicular to every tangent direction. That makes it perpendicular to the whole surface at that point. 19:28
The lecturer says the gradient always points toward higher values of the function, but the proof of that direction is left for later. 10:43

Words to know

gradient: A vector that collects all the rates of change of a function, one for each variable. It is written with a symbol that looks like an upside-down triangle.
level surface: The set of all points where a function (called w here) has the same fixed value c. In two variables, this is called a level curve.
tangent: A direction that just touches a surface or curve at one point without cutting into it. A tangent plane is a flat sheet that touches the surface at that point.
perpendicular: At a right angle to each other, like the corner where a wall meets the floor.

Example

Imagine walking along a path that goes around a hill at the same height above sea level. Your height never changes as you walk, so the height is constant along that path. The steepest way up the hill points straight up the slope, at a right angle to that flat path.

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

Part 3 · 20:00–30:00

This part shows how to use the gradient to find the flat plane that just touches a curved surface at one point.

Key ideas

The gradient of a function points straight out from its level surface, so it is perpendicular to that surface and its tangent plane. That makes it a normal vector, a direction that sticks straight out from a surface. 20:02
To find a tangent plane for a surface written as an equation, first treat the surface as a level set of a function w. Then take the gradient of w. It gives the normal vector of the tangent plane. 23:33
Plug the given point into the gradient to get the exact normal vector at that spot. In the lecture's example, the gradient at (2, 1, 1) is (4, 2, -2). 24:22
The tangent plane's equation is the normal vector's numbers times the distance from the point, set equal to zero. In the example, that gives 4(x - 2) + 2(y - 1) - 2(z - 1) = 0. 29:08
A second way uses small changes. Setting the change in w to zero keeps you on the level surface, and the same plane equation comes out. The lecture says the gradient method is quicker. 27:07

Words to know

gradient: A list of numbers that shows how fast a function changes in each direction. For a function of x, y, and z, it is the three partial rates of change written together as a vector.
tangent plane: A flat plane that touches a curved surface at one point and matches its slope there. Near that point, the plane is the best flat stand-in for the surface.
normal vector: An arrow that points straight out from a surface, at a right angle to it.
level set: All the points where a function has the same fixed value. For example, all points where w equals 4 form one level set.

Example

Think of a smooth hill and a flat board you lay on top of it so the board just touches the hill at one spot. The board sits at the same tilt as the hill there, and a stick poking straight out of the board is the normal vector. The gradient gives you that stick's direction, which is how you find the board's tilt.

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

Part 4 · 30:00–40:00

A directional derivative tells you how fast a function changes when you move in any chosen direction, not just along the x or y axis.

Key ideas

Partial derivatives only tell you how a function changes when you move along the x axis or the y axis. A directional derivative asks the same question for any direction. 30:52
You pick a unit vector u to give the direction. You then move in a straight line in that direction, at unit speed, so the parameter s is the distance you have traveled. 32:01
The directional derivative is found by writing x and y as functions of s along the line, plugging them into w, and taking the derivative. This is a special case of the chain rule. 34:27
Picture slicing the graph with a vertical plane that points in the direction u. The slope of that slice is the directional derivative. The x and y partial derivatives are just two special slices. 37:28

Words to know

directional derivative: How fast a function changes when you move in one chosen direction, written dw/ds with a subscript for the direction u.
unit vector: An arrow with length exactly 1. It is used to show a direction without adding any size.
partial derivative: How a function changes when you move only along one axis, with the other variable held still.
slice: The cut made through a 3D graph by a flat vertical plane. The slice is a curve, and its slope tells you the rate of change.

Example

Imagine standing on a hill and walking in a straight line toward the northeast. The directional derivative tells you how steeply the ground goes up or down as you walk that exact way, even if you are not walking due north or due east. Each direction you could walk has its own slope.

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

Part 5 · 40:00–50:08

This part explains how to find how fast a function changes in any chosen direction, and how the gradient points to the direction of fastest increase.

Key ideas

The directional derivative is the slope of the graph when you walk in a chosen direction. To compute it, you take the dot product of the gradient with the unit vector (a direction arrow of length one) for that direction. 40:49
For the x direction, this gives the partial derivative with respect to x, so the directional derivative formula matches the familiar partial derivatives. 42:19
The dot product equals the length of the gradient times the cosine of the angle between the gradient and your direction. So the slope depends on how closely your direction lines up with the gradient. 43:51
The gradient points in the direction of fastest increase of the function. Its length is the slope in that fastest direction. Walking the opposite way gives the fastest decrease. 45:25
Moving perpendicular to the gradient means the function does not change, because you are moving along a level curve. The gradient is always perpendicular to level curves. 47:43

Words to know

Gradient: An arrow that points in the direction where a function grows fastest, with its length showing how steep that growth is.
Directional derivative: How steep the function is when you move in one chosen direction, like the slope you would feel walking that way.
Unit vector: A direction arrow with length exactly one. It tells you which way to go without adding any extra size.
Level curve: A line on a map (or contour plot) where the function has the same value everywhere along it, like a line of equal height on a hill map.

Example

Imagine standing on a hill in fog. The steepest way up is the direction the gradient points, and walking straight across the side of the hill keeps you at the same height, like walking along a contour line on a map. Walking exactly the opposite way from the uphill direction takes you down the fastest.

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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