If we consider a function from 3-dimensional space to 1-d real line (like a scalar-valued f(x,y,z) kind of function) and look at it as a transformation, we will see a 3d space being squeezed to a line. Similarly in a 1-input 3-output vector valued function we will see the real line to take a curvy shape in 3-space. But, is it possible to devise a function from 1D space to 3D space that will convert a line to a surface ? Intuitively it seems unlikely to have such a vector valued function, but is it really mathematically impossible?

Nice question! :-) After thinking about it for a while I'm come to the conclusion that it is NOT possible. Anytime you only have one _argument_ in your function/transformation then as you change that argument we will draw a curve in the N-dimensional space depending on the dimensions of our output vector. Now I suppose we could write another mathematical function that takes this output and create a surface using its outputs, but I can't think of a way to create a surface by itself. So as far as I can tell we would need (at least) a a function with a 2-dimensional domain (2 operands). If anyone else has additional thoughts feel free to share them with us, but this makes sense to me at least. Hope this helps! - Convenient Colleague

What about transformations from space to plane, from line to space, and from space to line?

Yes, they are possible. But they might've been left out of this article for sake of brevity.

Can anyone explain a little bit more about the challenge question? I guessed it correctly, but couldn`t understand it even with the answer.

The 𝑥-coordinate of the output, 𝑥² + 𝑦², is always greater than or equal to 0, which means that the output will always be on or to the right of the 𝑦-axis. The absolute value of the 𝑦-coordinate of the output, |𝑥² − 𝑦²|, is always less than or equal to the 𝑥-coordinate of the output, which means that the output's distance to the 𝑥-axis is never greater than its distance to the 𝑦-axis. Thereby the output must be below the line 𝑦 = 𝑥 and above the line 𝑦 = −𝑥.

Is there a way of following the space to space transformation?

I suppose just replaying the animation over and over and following each individual point! :)

Beautiful! Which tool did you use to animate this? What is the math behind the animation? Is it some convex multiplication?

Some of these visualizations were definitely done with _Manim_ (which is an python library made by the instructor, Grant Sanderson). I can't tell if all of them were made with _Manim_ though

Is there a special name for function of n variable to n+1 dimensions, where the n+1th coordinate is zero? For example, f(x,y)=(x,y,0). I think this is related to "embedding", but I'm not sure.

I think you're talking about fixing the function to a specific limitation or parameter, right?

Main content

Course: Multivariable calculus > Unit 1

Lesson 6: Visualizing multivariable functions (articles)

Transformations

Google Classroom

Here we see how to think about multivariable functions through movement and animation.

Background

Multivariable functions

The idea of transformations

In all of our methods for visualizing multivariable functions, the goal is to somehow see the connection between the input and the output of a function.

With graphs, this means plotting points whose coordinates include both input and output information.
With contour maps this means marking which input values will go to certain output values.
With parametric functions, you mark where the input lands in the output space.
With vector fields you plot the output as a vector whose tail sits at the input.

The thought behind transformations is to simply watch (or imagine) each input point moving to its corresponding output point.

It can be a bit of a mind-warp to view functions as transformations if you never have before, so if it feels confusing at first, that's okay.

To whet your appetite for what this might look like, here's a video from the parametric surface article which shows how a certain function transforms a square into a torus (doughnut shape):

Khan Academy video wrapper

See video transcript

Concept over precision

Thinking about functions as transformations can be very powerful for a few reasons:

We are not constrained as much by dimension. Both the input and the output can have either one, two or three dimensions, and there will be a way to concretely think about what the function is doing.
Even when the dimensions are too big to look at, thinking in terms of a transformation at least allows for a vague idea of what's happening in principle. For example, we can know that a function from $100$ ‍-dimensional space to $20$ ‍-dimensional space is "flattening" down $80$ ‍ dimensions, perhaps analogous to squishing three-dimensional space onto the line.
This idea generalizes more easily to functions with different types of inputs and outputs, such as functions of the complex numbers, or functions that map points of the sphere onto the $x y$ ‍-plane.
Understanding functions in this capacity will make it easier to see the connections between multivariable calculus and linear algebra.

However, with all that said, it should be stressed that transformations are most powerful as an understanding of what functions do, not as a precise description. It would be rare to learn the properties of a given function by observing what it looks like as a transformation.

Example 1: From line to line

Let's start simple, with a single-variable function.

$f (x) = x^{2} - 3$ ‍

Consider all the input-output pairs.

$x$ ‍ (input)	$x^{2} - 3$ ‍ (output)
$- 2$ ‍	$1$ ‍
$- 1$ ‍	$- 2$ ‍
$0$ ‍	$- 3$ ‍
$1$ ‍	$- 2$ ‍
$2$ ‍	$1$ ‍
$⋮$ ‍	$⋮$ ‍

What would it look like for all the inputs on the number line to slide over onto their corresponding output? If we pictured the input space as one number line, and the output space as another number line, we might get a motion like this:

Khan Academy video wrapper

See video transcript

Alternatively, since in this case the input space and output space are really the same thing, a number line, we could think of the line transforming onto itself, dragging each point

x

to where the point

x^{2} - 3

started off, like this:

Khan Academy video wrapper

See video transcript

Example 2: From line to plane

Now let's take a function with a one-dimensional input and a two-dimensional output, like

\begin{array}{r} f (x) = (\cos (x), \frac{x}{2} \sin (x)) \end{array}

Again we consider all input-output pairs.

Inputs $x$ ‍	Outputs $(\cos (x), \frac{x}{2} \sin (x))$ ‍
$0$ ‍	$(1, 0)$ ‍
$\frac{π}{2}$ ‍	$(0, \frac{π}{4})$ ‍
$π$ ‍	$(- 1, 0)$ ‍
$⋮$ ‍	$⋮$ ‍

Imagine all possible inputs on the number line sliding onto their corresponding outputs. This time, since the outputs have two coordinates, they live in the

x y

-plane.

Khan Academy video wrapper

See video transcript

Notice, the final image of the warped and twirled number line inside the

x y

-plane is what we would have drawn if we interpreted

f

as a parametric function, but this time, we can actually see which input points end up where on the final curve.

Let's take a moment to watch it again and follow some specific inputs as they move to their outputs.

\begin{aligned} 0 & \to f (0) = (\cos (0), 0 \sin (0)) = (1, 0) \\ \frac{π}{2} & \to f (\frac{π}{2}) = (\cos (\frac{π}{2}), \frac{π}{4} \sin (\frac{π}{2})) = (0, π / 4) \\ π & \to f (π) = (\cos (π), \frac{π}{2} \sin (π)) = (- 1, 0) \end{aligned}

Khan Academy video wrapper

See video transcript

Example 3: Simple plane to plane transformation

Consider a

90^{\circ}

rotation of the plane (arrows are pictured just to help follow the transformation):

Khan Academy video wrapper

See video transcript

This could be considered a way to visualize a certain function with a two-dimensional input and a two-dimensional output. Why?

This transformation moves points in two-dimensional space to other points in two-dimensional space. For example, the point that starts at

(1, 0)

ends at

(0, 1)

. The point that starts at

(1, 2)

ends at

(- 2, 1)

, etc. The function describing this transformation is

$f (x, y) = (- y, x)$ ‍

For any given point, like

(3, 4)

, this function

f

tells you where that point lands after you rotate the plane

90^{\circ}

counterclockwise, (in this case

(- 4, 3)

Example 4: More complicated plane to plane transformation

Now let's look at a more complicated function with a two-dimensional input and a two-dimensional output:

$f (x, y) = (x^{2} + y^{2}, x^{2} - y^{2})$ ‍.

Each input is a point on the plane, such as

(1, 2)

, and it moves to another point on the plane, such as

(1^{2} + 2^{2}, 1^{2} - 2^{2}) = (5, - 3)

. When we watch every point on the plane slide over to its corresponding output point, it looks as if a copy of the plane is morphing:

Khan Academy video wrapper

See video transcript

Notice, all the points end up on the right side of the plane. This is because the first coordinate of the output is

x^{2} + y^{2}

, which must always be positive.

Challenge question: In the transformation above, representing the function

f (x, y) = (x^{2} + y^{2}, x^{2} - y^{2})

, notice that all points end up in the sideways-

V

-shaped region between the lines

x = y

and

x = - y

. Which of the following numerical facts explains this?

Choose 1 answer:
(Choice A)
$x^{2} + y^{2} = x^{2} - y^{2}$ ‍ and $x^{2} + y^{2} = - (x^{2} - y^{2})$ ‍
(Choice B)
$| x^{2} + y^{2} | \geq 0$ ‍
(Choice C)
$x^{2} + y^{2} \geq 0$ ‍ and $| x^{2} - y^{2} | \leq | x^{2} + y^{2} |$ ‍

You could describe this "V" shaped region as follows: It is all the points $(a, b)$ ‍ such that
$a \geq 0$ ‍,
$b \leq a$ ‍,
$b \geq - a$ ‍
To get a feel for these last two properties, you can imagine fixing a value of $a$ ‍, like $a = 2$ ‍, and letting $b$ ‍ vary between $- 2$ ‍ and $2$ ‍. What do the points $(2, b)$ ‍ look like?
You could write these properties even more compactly as
$a \geq 0$ ‍
$| b | \leq | a |$ ‍
If we look at the function above, the outputs are all of the form $(x^{2} + y^{2}, x^{2} - y^{2})$ ‍ for some $x$ ‍ and $y$ ‍. Therefore checking whether a given output is in the $V$ ‍-shaped region comes down to whether or not
$x^{2} + y^{2} \geq 0$ ‍
$| x^{2} - y^{2} | \leq | x^{2} + y^{2} |$ ‍

Example 5: From plane to line

Next think of a function with a two-dimensional input and a one-dimensional output.

$f (x, y) = x^{2} + y^{2}$ ‍,

The corresponding transformation will squish the

x y

-plane onto the number line.

Khan Academy video wrapper

See video transcript

Such squishification can make it hard to follow everything that's going on, so for the sake of a precise and clear description, you would be better off using a graph or a contour map. Nevertheless, it can be a helpful concept to keep in the back of your mind that what function from two dimensions to one dimension does is squish the plane onto the line in a certain way.

For instance, this gives a new way to interpret the level sets in a contour map: they are all the points of the plane which scrunch together into a common point on the line.

Example 6: From plane to space

Functions with a two-dimensional input and three-dimensional output map the plane into three-dimensional space. For instance, such a transformation might look like this (the red and blue lines are just to help keep track of what happens to the

x

and

y

directions):

Khan Academy video wrapper

See video transcript

Analogous to the one-to-two dimensions example above, our final image reflects the surface we would get by interpreting the function as a parametric function.

Example 7: From space to space

Functions from three dimensions to three dimensions can be seen as mapping all three-dimensional space onto itself. With this many variables, actually looking at the transformation can be a combination of horrifying, beautiful, and confusing. For instance, consider this function:

$f (x, y, z) = (y z, x z, x y)$ ‍

Here's what it looks like as a transformation.

Khan Academy video wrapper

See video transcript

It might be pretty, but it's a serious spaghettified mess to actually try to follow.

Final thoughts

Transformations can provide wonderful ways to interpret properties of a function once you learn them. For instance, constant functions squish their input space to a point, and discontinuous functions must tear apart the input space during the movement.

These physical interpretations can become particularly helpful as we venture into the topics of multivariable calculus, in which one runs the risk of learning concepts and operations symbolically without an underlying understanding of what's happening.

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Min ChanHong
Posted 8 years ago. Direct link to Min ChanHong's post “I would like to suggest i...”
I would like to suggest in this article to implement the idea of Jacobian determinant to introduce the unit area or volume of a transformed object.
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(20 votes)
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Asmundur Gudjonsson
Posted 8 years ago. Direct link to Asmundur Gudjonsson's post “Can someone explain to me...”
Can someone explain to me better what exactly the reflection question is? I don't understand it, although I feel like I understand the material.
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(5 votes)
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- dguynot
  Posted 11 days ago. Direct link to dguynot's post “I think it's when one thi...”
  I think it's when one thing goes to another thing.
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  (1 vote)
pittkor
Posted 4 years ago. Direct link to pittkor's post “Can anyone explain a litt...”
Can anyone explain a little bit more about the challenge question? I guessed it correctly, but couldn`t understand it even with the answer.
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(2 votes)
Answer
- Jerry Nilsson
  Posted 4 years ago. Direct link to Jerry Nilsson's post “The 𝑥-coordinate of the ...”
  The 𝑥-coordinate of the output, 𝑥² + 𝑦²,
  is always greater than or equal to 0,
  which means that the output will always be
  on or to the right of the 𝑦-axis.
  
  The absolute value of the 𝑦-coordinate of the output, |𝑥² − 𝑦²|,
  is always less than or equal to the 𝑥-coordinate of the output,
  which means that the output's distance to the 𝑥-axis is
  never greater than its distance to the 𝑦-axis.
  Thereby the output must be below the line 𝑦 = 𝑥
  and above the line 𝑦 = −𝑥.
  Button navigates to signup page
  (10 votes)
Connor Albert
Posted a year ago. Direct link to Connor Albert's post “Math is really amazing!”
Math is really amazing!
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(6 votes)
Answer
ph20b005
Posted 3 years ago. Direct link to ph20b005's post “If we consider a function...”
If we consider a function from 3-dimensional space to 1-d real line (like a scalar-valued f(x,y,z) kind of function) and look at it as a transformation, we will see a 3d space being squeezed to a line. Similarly in a 1-input 3-output vector valued function we will see the real line to take a curvy shape in 3-space. But, is it possible to devise a function from 1D space to 3D space that will convert a line to a surface ? Intuitively it seems unlikely to have such a vector valued function, but is it really mathematically impossible?
Button navigates to signup pageComment on ph20b005's post “If we consider a function...”
(4 votes)
Answer
- Iron Programming
  Posted 3 years ago. Direct link to Iron Programming's post “Nice question! :-) After...”
  Nice question! :-)
  
  After thinking about it for a while I'm come to the conclusion that it is NOT possible. Anytime you only have one argument in your function/transformation then as you change that argument we will draw a curve in the N-dimensional space depending on the dimensions of our output vector.
  
  Now I suppose we could write another mathematical function that takes this output and create a surface using its outputs, but I can't think of a way to create a surface by itself.
  
  So as far as I can tell we would need (at least) a a function with a 2-dimensional domain (2 operands).
  
  If anyone else has additional thoughts feel free to share them with us, but this makes sense to me at least.
  
  Hope this helps!
  - Convenient Colleague
  Comment on Iron Programming's post “Nice question! :-) After...”
  (4 votes)
Arunabh Bhattacharya
Posted 7 years ago. Direct link to Arunabh Bhattacharya's post “What about transformation...”
What about transformations from space to plane, from line to space, and from space to line?
Button navigates to signup pageComment on Arunabh Bhattacharya's post “What about transformation...”
(4 votes)
Answer
- Potugadu
  Posted 6 years ago. Direct link to Potugadu's post “Yes, they are possible. B...”
  Yes, they are possible. But they might've been left out of this article for sake of brevity.
  Button navigates to signup page
  (3 votes)
raymondz
Posted 7 years ago. Direct link to raymondz's post “Is there a way of followi...”
Is there a way of following the space to space transformation?
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(2 votes)
Answer
- mand4796
  Posted 6 years ago. Direct link to mand4796's post “I suppose just replaying ...”
  I suppose just replaying the animation over and over and following each individual point! :)
  Button navigates to signup page
  (2 votes)
Ephraim Raj
Posted 6 years ago. Direct link to Ephraim Raj's post “Beautiful! Which tool did...”
Beautiful! Which tool did you use to animate this? What is the math behind the animation? Is it some convex multiplication?
Button navigates to signup pageComment on Ephraim Raj's post “Beautiful! Which tool did...”
(2 votes)
Answer
- Joaquin Badillo
  Posted 2 years ago. Direct link to Joaquin Badillo's post “Some of these visualizati...”
  Some of these visualizations were definitely done with Manim (which is an python library made by the instructor, Grant Sanderson). I can't tell if all of them were made with Manim though
  Button navigates to signup page
  (2 votes)
YehonathanShatz
Posted 6 years ago. Direct link to YehonathanShatz's post “Is there a special name f...”
Is there a special name for function of n variable to n+1 dimensions, where the n+1th coordinate is zero? For example, f(x,y)=(x,y,0).
I think this is related to "embedding", but I'm not sure.
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- mand4796
  Posted 6 years ago. Direct link to mand4796's post “I think you're talking ab...”
  I think you're talking about fixing the function to a specific limitation or parameter, right?
  Button navigates to signup page
  (2 votes)
PV PV
Posted 10 months ago. Direct link to PV PV's post “I took the opposite appro...”
I took the opposite approach to this, I am more comfortable learning how transformations work from one dimension to another rather than the actual math itself, afterall I'm only here for a broad general understanding
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(1 vote)
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