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## Multivariable calculus

### Unit 1: Lesson 3

Visualizing scalar-valued functions

# Interpreting graphs with slices

3d graphs can be a lot to take in, but it helps to imagine slicing them with planes parallel to the x-axis or y-axis and relate them with two-dimensional graphs.  Created by Grant Sanderson.

## Want to join the conversation?

• What software are you using to make 3 dimensional graphs.
• What would happens if you held z as a constant?
• f(x,y) and z are equivalent since the output of f(x,y) IS z. Therefore if you held z as a constant you're holding f(x,y) constant. It would simply be a constant function.
• At this video the graph f(x,y)=cos(x)sin(y) is shown. And by the way it looks like a surface of some pure material pictured by atomic force microscope.
So the question is: Do engineers use this kind of multivariable calculus to determine the semiconducting properties??
• Seems like they do as we learn this on the second semester on Engineering Mathematics of Mechatronics programme in Germany.
• I think the fact that the axes are not named (or indicated by color, somehow) in the 3D visualization hurts the explanation a little bit. Also, both the x and y slices were shown "upwards", there was a cut right before the y slice that rotated the graph. That was a little bit confusing also.
• Is it possible to have the axis named? I kept getting confused which axis was x or y. Or is there a better way of reading it?

In all of my textbooks it has been z straight up, x on the left and then y on the right.
• Okay, question and point.
What if we tried to take a slice that wasn't exactly very "nice"? That is, let's say we take some sort of diagonal slice. My intuition tells me that since in the 2D plane a line (which become sort of planes in 3D?) is y as a function of x, so let's say in our diagonal slice, we let z or y = 2x. But does that graph out the desired function?
(I'm having trouble visualizing this, so a picture may help.)
If my intuition is correct, and we simply let one variable equal something in terms of the other, what would it look like if we tried a different, nonlinear function? Like y = x^2 or y = 1/(x*z) ?
Thanks!
• If you want to intersect two functions f(x,y) and g(x,y) you would get a system of two equations with three variables. Solving this is not always straight forward, and may sometimes give you nothing, a line, a point, or some other set of points in R3. In your first example (y=2x) with the f(x,y) used in the video (f(x,y) = cos(x)sin(y)) you would have f(x)=cos(x)sin(2x). With y=x^2 you have f(x) = cos(x)sin(x^2). Your final example is more difficult. Given z = cos(x)sin(y) and y=1/(x*z) you can with a little manipulation find x*cos x=(csc y)/y however you cannot simplify any further.
(1 vote)
• Can you slice it in diagonal? Would that mean x and y has some sort of fixed relationship?
(1 vote)
• This might have already been asked many times, also it has nothing to do with the content, but what program is being used to show the graphs?
(1 vote)
• What types of questions would this info help us answer?
(1 vote)
• What software do you use for your graphing illustrations?
(1 vote)

## Video transcript

- [Voiceover] So in the last video I described how to interpret three dimensional graphs and I have another three dimensional graph here, it's a very bumpy guy. And this happens to be the graph with the function f of x, y is equal to cosine of x multiplied by the sine of y and you know I could also say like, that this graph represents I could also say that this graph represents z is equal to that whole value because we think about the output of the function as the z coordinate of each point and what I want to do here is describe how you can interpret the relationship between this graph and these functions that you know by taking slices of it. So for example let's say that we took a slice with this plane here and what this plane here is is it represents the value x equals zero and you can kinda see that because this is the x-axis. So when you're at zero on the x-axis you know, you pass through the origin and then the values of y and z can go freely so you end up with this plane. And let's say you want to just consider where this cuts through the graph, okay, so we'll limit our graph just down to the point where it cuts it and I'm going to draw a little red line over that spot. Now what you might notice here that red line looks like a sinusoidal wave in fact it looks exactly like the sine function itself you know, passes through the origin it starts by going up and this makes sense if we start to plug things into the original form here. Because if you take f and you plug in x equals zero but then we still let y range freely what it means is that you're looking at cosine of zero multiplied by sine of y And what is cosine of zero? cosine of zero evaluates to one so this whole function should look just like sine of y in that when we let y run freely the output, which is still represented by the z coordinate, will give us this graph that's just a normal two-dimensional graph that we're probably familiar with. And let's try this at a different point. Let's see what would happen if instead of plugging in x equals zero let's imagine that we plugged in y equals zero and this time before I graph it and before I show everything that goes on let's just try to figure out purely from the formula here what it's going to look like when we plug in y equals zero. So now I'm going to write over on the other side. We have f of x will still run freely y is going to be fixed as zero and what this means is we have cosine of x so maybe expect to see something that looks kinda like a cosine graph and then sine of zero. Except, what is sine of zero? Sine of zero cancels out and just becomes zero which multiplied by cosine of x means everything cancels out and becomes zero so what you'd expect is that this is going to look like a constant function that's constantly equal to zero. And let's see if that's what we get. So I'm going to slice it with y equals zero here and you look at the y axis we see when it's zero and x and z both run freely. I'm going to chop off my graph at that point, and indeed it chops it just at this straight line, the straight line that goes right along the x-axis. But let's say that we did a different constant value of y. Rather than y equals zero, and we'll erase all of this, let's say that I cut things at some other value. So in this case what I've chosen is y is equal to pi halves and it looks kinda like we've got a wave here and it looks like a cosine wave and you can probably see where this is going. This is when x is running freely and if we start to imagine plugging this in I'll just actually write it out. We've got cosine of x and then y is held at a constant sine of pi halves. Sine of pi halves is just, this just always equals one so we could replace this with one which means the function as a whole should look like cosine x. So again the multi-variable function we've frozen y and we're letting x range freely and it ends up looking like a cosine function and I think a really good way to understand a given three-dimensional graph when you see it, let's say you, you look back at the original graph when we don't have anything going on. Get rid of that little line. So you've got this graph and it looks wavy and bumpy and a little bit hard to understand at first but if you just think in terms of holding one variable constant it boils down always into a normal two-dimensional graph and you can even think about, as you're lighting planes, kinda slide back and forth what that means for the amplitude of the wave that you see and things like that. This become especially important, by the way, when we introduce a notion of partial derivatives.