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## Multivariable calculus

### Course: 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.
• I think it's called Grapher, a built-in program for Macs. Take a look: https://en.wikipedia.org/wiki/Grapher.

Credits to Tom, from the previous section.
• 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.
• What software do you use for your graphing illustrations?
• 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.
• Can you slice it in diagonal? Would that mean x and y has some sort of fixed relationship?
• 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!