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### Course: Multivariable calculus > Unit 1

Lesson 6: Visualizing multivariable functions (articles)# Multidimensional graphs

Examples and limitations of graphing multivariable functions.

## Background

## What we're building to

- Graphing a function with a two-dimensional input and a one-dimensional output requires plotting points in three-dimensional space.
- This ends up looking like a surface in three-dimensions, where the height of the surface above the
-plane indicates the value of the function at each point.$xy$

## Reviewing graphs of single-variable functions

Graphs are, by far, the most familiar way to visualize functions for most students. Before generalizing to multivariable functions, let's quickly review how graphs work for single-variable functions.

Suppose our function looks like this:

To plot a single input, like $x=1$ , we first compute $f(1)$ :

Then we mark the point $(1,f(1))$ on the $xy$ -plane. In this case, that means marking $(1,4)$ .

When we do this for all possible inputs $x$ , not just $1$ , we see what all the points of the form $(x,f(x))$ look like.

Unless $f(x)$ is some exotic or sporadic function that gives wildly different values as $x$ changes slightly, the result will be a smooth-looking curve.

## Adding one more dimension

So what can we do for functions with a two-dimensional input and a one-dimensional output? Perhaps something like this:

Associating inputs with outputs requires three numbers—two for the inputs and one for the output.

Inputs | Output | ||
---|---|---|---|

To represent these associations using a graph, we plot points in

*three dimensions*.- The association
is plotted with the point$(0,0)\to 10$ .$(0,0,10)$ - The association
is plotted with the point$(1,0)\to 7$ .$(1,0,7)$ - In general, the goal is to represent all points of the form
for some pair of numbers$(x,y,f(x,y))$ and$x$ .$y$

The resulting graph is shown below. The video shows this graph rotating, which hopefully will help you get a feel for the three-dimensional nature of it. You can also see the $xy$ -plane—which is now the input space—below the graph.

This means for any given point $(x,y)$ on the plane, the vertical distance between that point and the graph indicates the value of $f(x,y)$ . That vertical direction is usually referred to as the $xy$ -plane is called the

**, and the third axis which is perpendicular to the**$z$ -direction**.**$z$ -axisAs long as the value of $f(x,y)$ changes continuously as $x$ and $y$ change values, which is almost always the case in functions we deal with in practice, the graph ends up looking like some sort of surface.

## Example 1: The bell curve

Function: $f(x,y)={e}^{-({x}^{2}+{y}^{2})}$

Graph:

Let's analyze what's going on with this function. First, let's look inside the exponent of ${e}^{-({x}^{2}+{y}^{2})}$ and think about the value ${x}^{2}+{y}^{2}$ .

**Question**: How can you interpret the value

When the point $(x,y)$ is far from the origin, the function ${e}^{-({x}^{2}+{y}^{2})}$ will look like ${e}^{\text{(some big negative number)}}$ , which is nearly zero. This means the distance between the graph and the $xy$ -plane at those points will be tiny. When $x=0$ and $y=0$ , on the other hand, ${e}^{-({x}^{2}+{y}^{2})}={e}^{-0}=1$ , which is what gives us the bulge in the middle.

**Reflection Question**: The graph above has rotational symmetry, in the sense that it will look the same if we rotate it in any way about the

**Example 2**: Waves

Function: $f(x,y)=\mathrm{cos}(x)\cdot \mathrm{sin}(y)$

Graph:

One way to get an intuition for the function $f(x,y)=\mathrm{cos}(x)\mathrm{sin}(y)$ —and multivariable functions in general—is to see what happens when one of the inputs is held constant.

For example, what happens when we fix the value of $x$ to be $2$ ? Usually, we are plotting all the points that look like this:

By holding $x$ constant at $2$ , we are limiting our view to points that look like this:

There is a very nice way to interpret this geometrically:

The points in space where $x=2$ , which is to say all the points of the form $(2,y,z)$ , make up a plane. Why? Imagine slicing the graph with this plane. The points where the plane and the graph intersect—drawn in red above—are the points on our graph where $x=2$ .

So why would this be helpful for understanding the graph?

We have basically turned the multivariable function $f(x,y)=\mathrm{cos}(x)\mathrm{sin}(y)$ into a single variable function:

In fact, the curve we get from slicing the three-dimensional graph at $x=2$ has the same shape as the two-dimensional graph of $g(y)$ .

In this way, you can understand the three-dimensional graph of a multivariable function one slice at a time by holding one variable constant and looking at the resulting two-dimensional graph.

## Example 3: One input, two outputs

You can also graph a function with a one-dimensional input and a two-dimensional output—although, for whatever reason, this is not commonly done.

Function: $f(x)=({x}^{2},\mathrm{sin}(x))$

Points plotted: $(x,{x}^{2},\mathrm{sin}(x))$

Graph:

In this case, only $x$ runs freely, while the $y$ and $z$ values on the graph are both dependent on $x$ .

If we rotate the image so we can look squarely at the $xy$ -plane, the graph looks like $f(x)={x}^{2}$ . Another way to say this is that when we $xy$ -plane, it gives the graph of $f(x)={x}^{2}$ .

**project**the graph onto theSimilarly, rotating the image so that we're looking squarely at the $xz$ -plane makes the image look like the graph of $f(x)=\mathrm{sin}(x)$ .

In other words, this function $f(x)=({x}^{2},\mathrm{sin}(x))$ is a way to combine the two functions $f(x)={x}^{2}$ and $f(x)=\mathrm{sin}(x)$ into one, and its graph captures the information of both in one image.

## Limitations

As soon as you try to apply this process to functions with higher-dimensional inputs or outputs, you'll run out of dimensions which you can comfortably visualize.

For instance, consider the function $f(x,y)=({x}^{2},{y}^{2})$ with a two-dimensional input and a two-dimensional output. Graphing it would require four dimensions of space! This is because we would need to plot all points of the form $(x,y,{x}^{2},{y}^{2})$ .

In practice, when people think about graphs of higher dimensional functions, like $f(x,y,z)={x}^{2}+{y}^{2}+{z}^{2}$ , they usually start by considering graphs of some simpler function with a two-dimensional input and one-dimensional output like $f(x,y)={x}^{2}+{y}^{2}$ . This is a sort of conceptual prototype.

Such a prototype can help make sense out of certain operations, and can give you a feel for what starts to happen as your input space becomes multidimensional. At the end of the day, the actual computations are performed purely symbolically for the higher dimensional function.

## Want to join the conversation?

- can anyone show me how can i plot these 3d graphs on matlab or mathematica or maxima or scilab or anyother app? :((10 votes)
- Found the link below really helpful

http://www.mathworks.com/company/newsletters/articles/using-matlabs-meshgrid-command-and-array-operators-to-implement-one-and-two-variable-functions.html(9 votes)

- can you make everything a little clearer? I am having trouble understanding(4 votes)
- In order to discriminate what each value (or set of values), when pumped into the function, will return, you must have one dimension per variable.

Take this analogy: you have a bag of apples, and a bag of oranges. You want to express how many you gain per second, at any given second, and make a model of it. So you section the floor into "blocks" and label each block with a time. Then you take however many apples, and however many bananas that you gained and place a apple or banana in that section, respectively.

We can't display how many apples AND bananas we gained by only placing apples in the marked off section: we need both apples AND bananas so people can discriminate between them, right? Same thing with a graph. We can't display two values with only one dimension: we need (at least) one dimension per value.

All is great if we only have 2 or 3 values in total, but what happens when we get to 4? Or 5? Or 100? Can you imagine a 100 dimension grid? I can't.

I hope I helped. Please tell me if I completely missed what you were struggling with.(7 votes)

- In the second paragraph under the header 'Limitations', f(x,y) = (x^2,y^2) is said to be a function of two inputs and one output, while in the third paragraph, f(x,y,) = (x^2 + y^2) is said to be a function of two inputs and one output.

I don't understand the how f(x,y,) can represent either one output, or two. Also, I don't understand why different notation is used between (x^2,y^2) and (x^2 + y^2) if they are both two inputs. Sorry if the question is unclear or doesn't make sense!(2 votes)- Hi there, do not get confused, output is given in the form of f(x,y) this means that we are getting two outputs by putting the value of x and y in to the function, so this function produces two values one is x and y for each given or certain value of x and y. Eg: F(x,y) = sin(x) , tg (y) here I have independent values x and y this is so clear, to my function I am producing one sin(x) output for x values, and one tg (y) output for any y. Pay attention; If I write F(x,y) = sin(x).tg(y) in this case my function returns only one value for each given x and y. ( dot and semicolon)(4 votes)

- Shouldn’t the function with two outputs and one input, when looked at so only the x and z plane are visible, look like a sinusoidal function whose period decreases as distance from the origin increases?(3 votes)
- How do I learn benchmark angles? I can't seem to understand it. Please help me Khan Academy Guirdians(3 votes)
- In example 3, what would the image look like if we looked at it from the yz plain?(2 votes)
- In the 𝑦𝑧-plane, the points plotted are (𝑥², sin(𝑥)),

which means that the curve would look like the functions

𝑓₁(𝑥) = sin(√𝑥) and 𝑓₂(𝑥) = −sin(√𝑥).(3 votes)

- In example 3, why would f(x) = sin(x) outputs in xz plane ? Why not in xy, like f(x) = x² ?(2 votes)
- because in xy plane only parabola is getting formed do a little math on this you will get it(1 vote)

- What is the difference between a surface and curve? Thanks(1 vote)
- The curver is a curved line in the multi-dimentional space, while the surface typically has more dimensions. You've already seen 2-dimentional surfaces above. Also for example there exists an opinion that our universe is a 3-dimentional surface wrapping 4-dimentional sphere of space-time. I hope I gave you a clue :)(2 votes)

- "Such a prototype can help make sense out of certain operations, and can give you a feel for what starts to happen as your input space becomes multidimensional."

Please elaborate. What operations are we talking about here?(1 vote) - In Example 3 one input three outputs how is the height of the spiral growing in height (seems to be growing) when it is a sin function moving in waves(1 vote)
- I also first thought it looked that way, but it is just an "optical illusion". The square term for y = f(x) = x^2 grows very large fast and this can sort of give the impression, that z is also growing, when in reality, it is only y. It is just the viewing angle combined with having a single constant thickness line.(1 vote)