ES.1802 September 24, 2026
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Overview
The class develops a way to understand functions of two variables by turning their 3D surfaces into level curves, or constant-height slices, in the xy-plane. Examples including z = x² + y², z = x² - y², and z = sin(x)sin(y), alongside laser-cut models, VR plots, and Desmos or Mac Grapher, lead to directional derivatives and partial derivatives.
Key takeaways
- For z = x² + y², fixing z = c gives circles of radius √c, so the surface is a paraboloid rather than a cone.
- The surface z = x² - y² rises along the x-axis and falls along the y-axis; its zero level set is the pair of lines y = ±x.
- A level curve records points in the xy-plane with the same function value, making it a 2D map of a 3D surface.
- The directional derivative measures change per unit distance in a chosen direction; it is zero along a level curve and greatest in the perpendicular uphill direction.
- Desmos and Mac Grapher can display both 3D surfaces and 2D heat maps, where color represents the function’s z-value.
- The shortest route to Fortnite’s safe-zone boundary is perpendicular to it, providing a geometric analogy for moving across contours in the direction of greatest increase.
Chapters
- The class shifts from functions of x to functions of x and y, written as z = f(x,y).
- Students preview surfaces through examples including x² + y², x² - y², and sin(x)sin(y).
- The guiding idea is to understand 3D functions through familiar 2D representations.
- Students arrange laser-cut pieces labeled with z-values to reconstruct a 3D mountain-like surface.
- Viewing the surface from above projects its horizontal slices onto the xy-plane.
- These constant-height slices are called level curves or contours, similar to elevation lines on a map.
- Setting z to a constant c gives x² + y² = c, a circle centered at the origin.
- At z = 0 the level curve is a point; at z = 1 it has radius 1, and at z = 2 its radius is √2.
- Because radius grows as √z rather than linearly with height, the surface is a paraboloid, not a cone.
- For z = 0, the level set x² - y² = 0 consists of the lines y = x and y = -x.
- Positive and negative level values produce hyperbolas opening along different axes.
- The surface rises along the x-axis and falls along the y-axis, creating a saddle shape, also compared to Pringles chips.
- A VR headset lets a student enter and inspect the graph of sin(x)sin(y), which forms repeating peaks and troughs like an egg carton.
- The class also views a saddle, a bell-shaped surface, a function with asymptotic behavior, and a plane using a 3D plotting interface.
- The instructor invites students to try the headsets and notes that the plotting software can display many functions.
- Desmos and the built-in Mac Grapher are presented as tools for visualizing functions of two variables.
- A 3D plot of x² + y² can also be represented as a heat map in the xy-plane, with color encoding the z-value.
- The plotted color map makes the relationship between a surface and its level curves visible.
- At a point on a contour, moving along the level curve leaves the height unchanged, so the rate of change in that direction is zero.
- Moving perpendicular to the contour in the uphill direction produces the greatest increase in height.
- Moving perpendicular in the opposite direction decreases height, while other directions produce intermediate rates of change.
- When Fortnite’s safe zone shrinks, the shortest route from a player’s position to the zone boundary is perpendicular to that boundary.
- The shortest route mirrors the direction of greatest increase across level curves on a surface.
- The class names the change in a function per unit distance along a specified direction the directional derivative.
- The class introduces the partial derivative with respect to x, written ∂f/∂x, as a derivative in a specific coordinate direction.
- On a level curve, ∂f/∂x is zero where the curve has a horizontal tangent, because a small step along x does not change the function value there.
- The closing discussion previews further work with partial derivatives and directional derivatives in the next class.
Transcript excerpt
0:00 Yes. >> All right, everyone. Welcome to Thursday. Today's topic isn't on the uh >> waiting for >> what do I want to say? Okay, >> this is the part of the class where we get to kick up the dimensions of uh functions, surfaces, and more of what you thought you would do in multiple calculus before I came at you with this stuff. >> This is on the test. >> This is not on the test. Yes, >> this is on a test. >> And so begin the snores. >> Yes, >> it's on a test. >> But it's on a test. Yeah. Who's Who said that? I I like this person. Go Barbara. You tell. All right. Shout out to >> If it wasn't on the test.
0:49 >> Yeah, if it wasn't on the test. >> Well, this is like people get the question. Wait, is this gonna be on the test? >> No. So much >> single variable so I can remember. >> It's crazy. >> And you've got some some f ofx. If anything, today's kind of chill because we're going to maybe even have VR headsets, but we'll learn a lot. >> Okay. >> Or and we'll learn a lot. Why should the R be in the opposite direction of what we expect for learning? We are going to move from writing functions in terms of x to writing functions of multiple variables. It might look something like this is some cos of x and y. Okay.
1:37 I made a promise when this class started which was that looking at stuff in three dimensions was going to be able to boil down to looking at stuff in two dimensions. Let's fulfill the promise. And to do that, I would like to ask you to describe the following functions
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Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, ChristianAltoSax.