Multivariable functions | Multivariable calculus | Khan Academy

1. Formula
   Observation

   The board shows $f(x)=x^2$ in yellow.

2. Audio
   Observation

   The speaker says ordinary-calculus functions have a single input and output a single number, calling this a single-variable function.

$f(x)$

A single-variable function with one input variable $x$; in the displayed example it is defined by $f(x)=x^2$.

$x$ is treated as a single number; no explicit domain or codomain is stated.

1. Formula
   Observation

   The variable $x$ appears inside $f(x)=x^2$.

2. Audio
   Observation

   The speaker refers to the single variable in the ordinary-calculus example.

$x$

The single input variable of the ordinary single-variable function example.

A number on a number line; no specific numerical range is given.

1. Formula
   Observation

   The board shows $f(x,y)=x^2+y$ in purple.

2. Audio
   Observation

   The speaker says a multivariable function handles multiple variables and may output a number depending on both inputs, giving $x^2+y$ as an example.

$f(x,y)$

A multivariable function with two input variables $x$ and $y$; in the displayed scalar-output example it is defined by $f(x,y)=x^2+y$.

$(x,y)$ is treated as a pair of numbers, later described as a point in the $xy$-plane; no explicit domain or codomain is stated.

1. Formula
   Observation

   The board writes $f(x,y)$ and later draws separate number lines labeled $x$ and $y$, then an $xy$-plane with a point.

2. Audio
   Observation

   The speaker says common notation uses $x,y$, but could also use $x,y,z$ or $x_1,x_2,x_3$, and that two such quantities are better thought of as a single point in the $xy$-plane.

$x,y$

Two input variables of a multivariable function; they can be viewed separately as numbers or together as coordinates of one point in the $xy$-plane.

Each is a number; together they form a point in the plane. No explicit bounds are given.

1. Formula
   Observation

   The board shows $f(x,y)=\begin{bmatrix}3x\\2y\end{bmatrix}$ in purple.

2. Audio
   Observation

   The speaker says a multivariable function could also output a vector and gives an invented example with entries $3x$ and $2y$.

$f(x,y)=\begin{bmatrix}3x\\2y\end{bmatrix}$

A vector-valued multivariable function taking two input variables and returning a column vector whose components depend on $x$ and $y$.

Input $(x,y)$ is a pair of numbers; output is a two-component vector. The exact ambient space is not named.

1. Formula
   Observation

   The output is written vertically as $\begin{bmatrix}3x\\2y\end{bmatrix}$.

2. Audio
   Observation

   The speaker says that when multiple numbers go into the output, the convention is to think of them as a vector.

$\begin{bmatrix}3x\\2y\end{bmatrix}$

A column vector used to represent multiple output numbers of a function.

A two-entry vector with first component $3x$ and second component $2y$.

1. Diagram
   Observation

   Two horizontal number lines are drawn, one marked $x$ and one marked $y$; they are then replaced by perpendicular axes forming an $xy$-plane with a single point plotted.

2. Audio
   Observation

   The speaker says instead of thinking of $x$ and $y$ as separate entities, one should think about the $xy$-plane and a single point, so the function takes a point to a number or a point to a vector.

$(x,y)$ as a point

The ordered pair of input values interpreted geometrically as one point in the $xy$-plane rather than two independent number-line objects.

The plane is shown qualitatively; no coordinate values are assigned to the plotted point.

1. Formula
   Observation

   Handwritten formula appears as $f(x,y) = x^2$.

2. Audio
   Observation

   Speaker says the function has a two-dimensional input and would be f of x y.

$f(x,y)$

A scalar-valued multivariable function with two-dimensional input (x,y).

Two-dimensional input space; output is a single number.

1. Audio
   Observation

   Speaker refers to inputs living on this xy plane and later says what we are looking at is the xy plane, all of the input space.

2. Diagram
   Observation

   A gray horizontal plane with yellow axes is shown beneath the 3D graph.

xy plane

The two-dimensional input space for functions visualized by height graphs or contour plots.

Input space for functions with two-dimensional input.

1. Audio
   Observation

   Speaker says a single number is their output, and the height of the graph is going to correspond with that output.

2. Animation
   Observation

   A vertical axis rises from the xy plane while a teal checkered surface sits above it.

height of the graph

The vertical coordinate representing the scalar output value of a function $f(x,y)$.

Used for scalar-valued functions with two-dimensional input.

1. Audio
   Observation

   Speaker says the lines here called contour lines tell you which inputs all share a constant output value.

2. Diagram
   Observation

   Concentric closed curves are visible over a color-coded 2D field.

contour lines

Curves in the input plane joining points where the function has the same output value.

Contour plot of a scalar-valued function of two variables.

1. Audio
   Observation

   Speaker says these are called parametric surfaces.

2. Animation
   Observation

   A flat checkered sheet bends into a cylinder and then into a torus in 3D space.

parametric surfaces

Surfaces obtained by moving a two-dimensional input into three dimensions and looking at the resulting output shape.

Mappings from two dimensions to three dimensions.

1. Audio
   Observation

   The speaker welcomes viewers to multivariable calculus and says the first thing to get straight is what the word multivariable means.

2. Diagram
   Observation

   The title "Multivariable calculus" is visible at the top left.

The clip introduces multivariable calculus as the branch being studied and immediately frames the central preliminary question as the meaning of "multivariable" before doing any calculus operations.

1. This is an introductory framing statement rather than a formal mathematical definition.

1. Audio
   Observation

   The speaker says ordinary-calculus functions have a single input, some kind of number, and output just a single number, and calls this a single-variable function.

2. Formula
   Observation

   The example $f(x)=x^2$ is written on the board.

A single-variable function is presented as the familiar calculus case: it takes one numerical input and returns one numerical output. The example $f(x)=x^2$ illustrates this pattern.

1. One input variable.

2. Output is a single number in the displayed example.

1. $f(x)$
2. $x$

1. Audio
   Observation

   The speaker says a multivariable function handles multiple variables, commonly written with inputs like $x,y$, and may output something that depends on both of those.

2. Formula
   Observation

   The board shows $f(x,y)=x^2+y$.

A multivariable function is defined here as a function that handles multiple variables at once. The example $f(x,y)=x^2+y$ shows a function of two inputs whose output depends on both.

1. More than one input variable.

2. In the example, two inputs $x$ and $y$.

3. The output depends on the inputs.

1. $f(x,y)$
2. $x,y$

1. Audio
   Observation

   The speaker says the function could also output a vector, then states the convention: if there's multiple numbers that go into the output, think of it as a vector; if there's multiple numbers that go into the input, write them sideways and think of them as a point in space.

2. Formula
   Observation

   The board shows $f(x,y)=\begin{bmatrix}3x\\2y\end{bmatrix}$.

The clip distinguishes two notational conventions: multiple output numbers are usually represented as a vector, while multiple input numbers are often written horizontally and interpreted as a point in space.

1. Applies when a function has several output numbers or several input numbers.

2. The speaker explicitly says this convention is not set in stone.

1. $f(x,y)=\begin{bmatrix}3x\\2y\end{bmatrix}$
2. $\begin{bmatrix}3x\\2y\end{bmatrix}$
3. $x,y$

1. Audio
   Observation

   The speaker says that although $x$ and $y$ could be thought of as two separate numbers on number lines, it would probably be more accurate to call the subject multidimensional calculus, because one should think about the $xy$-plane and a single point.

2. Diagram
   Observation

   Separate number lines for $x$ and $y$ are drawn, then erased and replaced by perpendicular axes with a point plotted in the plane.

The input pair of a two-variable function is reinterpreted geometrically: rather than two isolated numbers, $(x,y)$ is a single point in the $xy$-plane. This motivates describing the function as taking a point to a number or a point to a vector.

1. Discussed for two-variable functions.

2. The plane is introduced qualitatively, without assigning coordinates to the drawn point.

1. $(x,y)$ as a point
2. Multivariable function

1. Audio
   Observation

   Speaker says when you have multivariable functions, graphs become three dimensional, but these only really apply to functions that have some kind of two dimensional input and a single number as their output.

2. Animation
   Observation

   A teal checkered bowl-like surface is shown above a gray xy plane with a vertical axis.

For functions with a two-dimensional input and a single numerical output, the graph is drawn in three dimensions: the input point lies in the xy plane and the output value is represented by height above that plane.

1. The function has two-dimensional input.

2. The function has a single-number output.

1. xy plane
2. height of the graph

1. Audio
   Observation

   Speaker says these functions can also be visualized just in two dimensions, flattening things out, where we visualize the entire input space and associate a color with each point.

2. Audio
   Observation

   Speaker says the color tells you roughly the size of that output, and the lines here called contour lines tell you which inputs all share a constant output value.

3. Formula
   Observation

   The handwritten example is $f(x,y) = x^2$.

4. Diagram
   Observation

   A 2D color map with concentric curves is shown.

A scalar multivariable function can be flattened into a two-dimensional picture of the whole input space. Color encodes approximate output magnitude, and contour lines mark sets of inputs with equal output values.

1. The function is scalar-valued.

2. The input space is two-dimensional.

1. Three-dimensional graphs of scalar multivariable functions
2. contour lines

1. Audio
   Observation

   Speaker says surfaces in three dimensional space look like graphs, but they actually deal with a much different animal that you could think of as mapping two dimensions and ... moving into three dimensions.

2. Audio
   Observation

   Speaker says these are called parametric surfaces.

3. Animation
   Observation

   A flat checkered sheet is bent into a cylinder and then into a torus.

Parametric surfaces are produced by taking a two-dimensional input and moving it into three-dimensional space; the focus is on the shape of the output surface rather than on how each point gets there.

1. The mapping takes two-dimensional input.

2. The output is viewed as a surface in three dimensions.

1. Three-dimensional graphs of scalar multivariable functions
2. parametric surfaces

1. Audio
   Observation

   Speaker says another fun one is a vector field, where every input point is associated with some kind of vector, which is the output of the function there.

2. Audio
   Observation

   Speaker says this would be a function with a two dimensional input and then two dimensional output because each of these are two dimensional vectors.

3. Diagram
   Observation

   Yellow arrows are placed throughout a 2D grid.

4. Animation
   Observation

   Small droplets move along the directions indicated by the arrows.

A vector field assigns a vector to each input point. In the example shown, the function has two-dimensional input and two-dimensional output, and the arrows indicate the vector attached to each point.

1. Each input point is assigned a vector.

2. In the displayed example, both input and output are two-dimensional.

1. vector field

1. Audio
   Observation

   Speaker says one of my all-time favorite ways to think about multivariable functions is to just take the input space ... and watch them move to their output.

2. Audio
   Observation

   Speaker says this is going to be a function that also outputs in two dimensions, and I'm just going to watch every single point move over to where it's supposed to go.

3. Animation
   Observation

   A regular grid deforms into a skewed grid.

Another way to understand a multivariable function is to view it as a transformation: start with the input space, then track where each point moves in the output space. The video uses a 2D-to-2D example and shows the grid being distorted.

1. The function maps points from an input space to an output space.

2. The example shown is two-dimensional input to two-dimensional output.

1. function as transformation

1. Diagram
   Observation

   The screen displays a Cartesian coordinate system with x-axis ranging from -7 to 6 and y-axis from -3 to 3. Multiple straight lines are drawn on the grid, forming a pattern of intersecting diagonal lines.

The visual shows level curves (or contour lines) for a multivariable function. These are sets of points $(x,y)$ in the domain where the function takes a constant value $c$, i.e., $f(x,y) = c$. For linear functions, these level curves are straight lines.

1. The function is defined on a 2D domain.

2. The level curves correspond to constant values of the function.

1. Audio
   Observation

   The speaker says it would probably be more accurate to call it multidimensional calculus because instead of thinking of $x$ and $y$ as separate entities, one thinks about the $xy$-plane and a single point.

2. Diagram
   Observation

   The visual transition from two number lines to an $xy$-plane supports this claim.

1. This is presented as the speaker's interpretive claim about terminology, not as a formally proved theorem.

For functions of several variables, it is more accurate conceptually to think in terms of multidimensional calculus, because the inputs are points in a plane or space rather than separate one-dimensional entities.

1. The function has multiple input variables.

2. Those inputs are being interpreted geometrically.

Informal universal claim about how multivariable functions should be conceptualized.

1. Audio
   Observation

   The speaker says, "this isn't set in stone, but the convention is to usually think if there's multiple numbers that go into the output, think of it as a vector. If there's multiple numbers that go into the input, just kind of write them more sideways like this and think of them as a point in space."

1. The speaker explicitly marks this as a convention rather than a necessary rule.

By common convention, multiple output numbers are treated as a vector, while multiple input numbers are written horizontally and treated as a point in space.

1. The function has multiple output numbers or multiple input numbers.

General convention statement, explicitly qualified as not absolute.

1. Audio
   Observation

   Speaker says these only really apply to functions that have some kind of two dimensional input ... and a single number as their output.

Three-dimensional graph visualization applies to functions with two-dimensional input and a single-number output.

1. The function has two-dimensional input.

2. The function has scalar output.

For the class of functions described in the video.

1. Audio
   Observation

   Speaker says the lines here called contour lines tell you which inputs all share a constant output value.

Contour lines join input points that share a constant output value.

1. The picture is a contour plot of a scalar-valued function.

For points lying on the same contour line.

1. Audio
   Observation

   Speaker says surfaces in three dimensional space ... look like graphs, but they actually deal with a much different animal that you could think of as mapping two dimensions ... into three dimensions.

Parametric surfaces resemble graphs visually but correspond to mappings from two dimensions into three dimensions rather than scalar height over a plane.

1. The object is described as a surface in three-dimensional space generated from two-dimensional input.

For the surfaces discussed in this segment.

1. Audio
   Observation

   Speaker says every input point is associated with some kind of vector, which is the output of the function there.

2. Audio
   Observation

   Speaker says this would be a function with a two dimensional input and then two dimensional output because each of these are two dimensional vectors.

In a vector field, the output at each input point is a vector; in the example shown, the function has two-dimensional input and two-dimensional output.

1. The object is a vector field.

2. The displayed example uses planar vectors.

For each input point in the field.

1. Audio
   Observation

   Speaker says understanding functions as transformations is going to be a great way to connect those two.

Viewing multivariable functions as transformations provides a connection between multivariable calculus and linear algebra.

1. The function is interpreted by moving input points to output points.

For the transformation-based interpretation discussed in the video.

1. Animation
   Observation

   First two horizontal number lines are drawn, one labeled $x$ and one labeled $y$; then they are erased and replaced by perpendicular axes forming an $xy$-plane with a point.

2. Audio
   Observation

   The speaker moves from saying $x$ and $y$ could be separate numbers to saying one should think about the $xy$-plane and a single point.

1. Expression

   $$x \text{ and } y \text{ as separate numbers}$$
   Explanation

   The speaker first allows the naive view that the two inputs are separate quantities, each living on its own number line.

   Justification

   Directly stated in the audio and shown by two separate horizontal lines.

   Shown in the video
2. Expression

   $$(x,y) \text{ as a point in the } xy\text{-plane}$$
   Explanation

   The picture then replaces the two lines with perpendicular axes and a single plotted point, showing that the pair is better understood as one location in the plane.

   Justification

   Visual transformation on the board plus the speaker's explanation that one should think about the $xy$-plane and a single point.

   Shown in the video
3. Expression

   $$f:\text{point}\to\text{number} \quad \text{or} \quad f:\text{point}\to\text{vector}$$
   Explanation

   After the reinterpretation, the function is described as taking a point to a number or a point to a vector.

   Justification

   Explicitly stated by the speaker after the plane diagram appears.

   Shown in the video

The geometric meaning of a two-variable input is a single point in the $xy$-plane, so a multivariable function is naturally viewed as mapping points to numbers or vectors.

1. Audio
   Observation

   Speaker says you can imagine a fluid flowing, so here's a bunch of droplets like water, and they kind of flow along that.

2. Audio
   Observation

   Speaker says that actually turns out to give insight about the underlying function.

3. Animation
   Observation

   Small droplets move through the arrow field.

1. Expression

   $$$$
   Explanation

   Start with a vector field in which each input point carries a vector.

   Justification

   Directly stated by the speaker and shown as arrows on the plane.

   Shown in the video
2. Expression

   $$$$
   Explanation

   Imagine droplets placed in the field and let them move along the local vector directions.

   Justification

   The speaker explicitly suggests imagining a fluid flowing and the animation shows droplets moving with the arrows.

   Shown in the video
3. Expression

   $$$$
   Explanation

   Observe the resulting motion pattern to gain insight about the underlying function.

   Justification

   The speaker says this flow picture gives insight about the underlying function.

   Shown in the video

The flowing-droplet picture is an intuitive way to read qualitative information from a vector field.

1. Audio
   Observation

   Speaker says take the input space ... and watch them move to their output.

2. Audio
   Observation

   Speaker says I'm just going to watch every single point move over to where it's supposed to go.

3. Animation
   Observation

   A regular grid becomes a skewed grid.

1. Expression

   $$$$
   Explanation

   Begin with the input space drawn as a regular coordinate grid.

   Justification

   The animation first shows an undistorted Cartesian grid.

   Shown in the video
2. Expression

   $$$$
   Explanation

   Apply the function by sending each input point to its corresponding output point.

   Justification

   The speaker describes watching every point move to where it is supposed to go.

   Shown in the video
3. Expression

   $$$$
   Explanation

   Track the images of the grid lines after all points move.

   Justification

   The animation shows the formerly straight grid lines becoming slanted and curved relative to the original axes.

   Shown in the video

The deformed grid visualizes the function as a transformation of the whole input space.

1. Formula
   Observation

   The board writes $f(x)=x^2$.

2. Audio
   Observation

   The speaker uses this as the ordinary-calculus example of a function with a single input and single output.

Illustrate the familiar calculus setting with a function of one variable.

1. One input variable $x$.

2. The displayed rule is $f(x)=x^2$.

Show what a single-variable function looks like.

1. Expression

   $$f(x)=x^2$$
   Explanation

   The function takes one number $x$ and returns one number, namely the square of $x$.

   Justification

   Written directly on the board and described verbally as the ordinary-calculus case.

   Shown in the video

$f(x)=x^2$ is a single-variable function.

Matches the speaker's description of one input and one output.

1. Formula
   Observation

   The board writes $f(x,y)=x^2+y$.

2. Audio
   Observation

   The speaker says one might imagine a number that depends on $x$ and $y$ in some way like $x^2+y$.

Give an example of a function of two variables whose output is a number.

1. Two input variables $x$ and $y$.

2. The displayed rule is $f(x,y)=x^2+y$.

Show a multivariable function with numerical output.

1. Expression

   $$f(x,y)=x^2+y$$
   Explanation

   The output depends on both inputs: it squares $x$ and adds $y$.

   Justification

   Directly written on the board and introduced verbally as a possible dependence on $x$ and $y$.

   Shown in the video

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

Consistent with the speaker's statement that the output can be a number depending on both variables.

1. Formula
   Observation

   The board writes $f(x,y)=\begin{bmatrix}3x\\2y\end{bmatrix}$.

2. Audio
   Observation

   The speaker says the function could also output a vector and invents an example with entries $3x$ and $2y$.

1. The speaker explicitly says he is just making stuff up here, so the example is illustrative rather than motivated by a prior problem.

Give an example of a function of two variables whose output has multiple components.

1. Two input variables $x$ and $y$.

2. The displayed rule is $f(x,y)=\begin{bmatrix}3x\\2y\end{bmatrix}$.

Show a multivariable function with vector output.

1. Expression

   $$f(x,y)=\begin{bmatrix}3x\\2y\end{bmatrix}$$
   Explanation

   The function takes the pair $(x,y)$ and returns a column vector whose first component is $3x$ and second component is $2y$.

   Justification

   Written directly on the board and described verbally as a vector-output example.

   Shown in the video

$f(x,y)=\begin{bmatrix}3x\\2y\end{bmatrix}$.

Matches the speaker's convention that multiple output numbers are treated as a vector.

1. Formula
   Observation

   Handwritten text reads $f(x,y) = x^2$.

2. Audio
   Observation

   Speaker says this particular one is an x squared.

3. Diagram
   Observation

   A colored contour plot fills the screen.

1. The speaker also mentions maybe some complicated thing, but no second explicit formula is written on screen in this clip.

Visualize a scalar function of two variables in the plane using color and contour lines.

1. The function is $f(x,y)=x^2$.

2. The input space is the xy plane.

Show how output size and equal-output sets appear in a two-dimensional visualization.

1. Expression

   $$f(x,y)=x^2$$
   Explanation

   Write the example function with two-dimensional input and scalar output.

   Justification

   The formula is visibly handwritten on screen and named in the audio.

   Shown in the video
2. Expression

   $$$$
   Explanation

   Represent the whole input plane and assign each point a color according to the size of its output.

   Justification

   The speaker explains that color tells roughly the size of the output.

   Shown in the video
3. Expression

   $$$$
   Explanation

   Draw contour lines to mark inputs sharing a constant output value.

   Justification

   The speaker defines contour lines this way while the plot shows concentric curves.

   Shown in the video

The example is displayed as a color-coded contour plot of $f(x,y)=x^2$, with contour lines marking equal output values.

The visual matches the spoken description: a 2D input plane, color encoding output magnitude, and contour lines for constant output.

1. Diagram
   Observation

   Yellow arrows fill a 2D grid.

2. Animation
   Observation

   Droplets move along the arrow directions.

3. Audio
   Observation

   Speaker says this would be a function with a two dimensional input and then two dimensional output because each of these are two dimensional vectors.

1. The exact algebraic formula for the displayed vector field is not given in this clip.

Visualize a function whose output at each point is a vector.

1. Each input point is assigned a vector.

2. The example shown has two-dimensional input and two-dimensional output.

Show how the vector field can be understood geometrically.

1. Expression

   $$$$
   Explanation

   Place a vector arrow at each input point in the plane.

   Justification

   The diagram shows arrows distributed over a grid.

   Shown in the video
2. Expression

   $$$$
   Explanation

   Interpret the arrows as the function outputs at those points.

   Justification

   The speaker states that each vector is the output of the function there.

   Shown in the video
3. Expression

   $$$$
   Explanation

   Add moving droplets to suggest flow along the field.

   Justification

   The animation shows droplets traveling in the directions indicated by the arrows.

   Shown in the video

The vector field is represented by arrows at each point, with droplet motion providing an intuitive flow picture of the underlying function.

The spoken explanation and the animation agree: arrows encode vector outputs and droplets trace their direction field.

1. Animation
   Observation

   A regular grid deforms into a skewed grid.

2. Audio
   Observation

   Speaker says this is going to be a function that also outputs in two dimensions, and I'm just going to watch every single point move over to where it's supposed to go.

1. The exact formula of the transforming function is not stated in this clip.

Understand a 2D-to-2D multivariable function by watching the input space move to the output space.

1. The function takes points in two-dimensional space.

2. The function outputs in two dimensions.

Visualize the effect of the function on the entire input plane.

1. Expression

   $$$$
   Explanation

   Start with the standard input grid.

   Justification

   The animation begins with evenly spaced horizontal and vertical lines.

   Shown in the video
2. Expression

   $$$$
   Explanation

   Move every input point to its output location.

   Justification

   The speaker explicitly describes watching every single point move to where it is supposed to go.

   Shown in the video
3. Expression

   $$$$
   Explanation

   Observe the resulting distorted grid as the image of the input space.

   Justification

   The final frame shows slanted grid lines replacing the original orthogonal grid.

   Shown in the video

The function is visualized as a transformation that sends the regular grid to a skewed grid.

The before-and-after animation directly supports the spoken transformation interpretation.

1. Diagram
   Observation

   The text "Multivariable calculus" is visible at top left, and "Grant" is handwritten at top right.

2. Audio
   Observation

   The speaker introduces the topic and says his name is Grant.

1. Title text "Multivariable calculus"

2. Handwritten name "Grant"

1. The name "Grant" is written after the title is already present.

1. The topic title remains at the top left.

The clip opens by identifying the course topic and the instructor before any mathematical notation appears.

1. Diagram
   Observation

   The blue subheading "Multivariable functions" appears below the main title.

1. Blue text "Multivariable functions"

1. A new subheading appears under the main title.

1. The main title remains visible.

The lesson narrows from the broad course topic to the specific subject of multivariable functions.

1. Animation
   Observation

   The formula $f(x)=x^2$ is written in yellow, and an arrow is drawn toward the $x$ inside $f(x)$.

2. Audio
   Observation

   The speaker explains that ordinary-calculus functions have a single input and calls $x$ the single variable.

1. Formula $f(x)=x^2$

2. Arrow pointing to $x$

1. The formula is built stroke by stroke.

2. An arrow is added to emphasize the input variable.

1. The formula remains on screen after being written.

The visual emphasis on $x$ supports the definition of a single-variable function.

1. Animation
   Observation

   The formula $f(x,y)=x^2+y$ is written in purple below the earlier example.

2. Audio
   Observation

   The speaker introduces multivariable functions and gives this as an example of a number depending on $x$ and $y$.

1. Formula $f(x,y)=x^2+y$

1. A second function rule appears beneath the first.

1. The earlier single-variable example stays visible above.

The stacked formulas contrast one-variable and two-variable functions.

1. Animation
   Observation

   Two horizontal yellow number lines are drawn, one marked with $x$ and one marked with $y$.

2. Audio
   Observation

   The speaker says $x$ and $y$ could be thought of as two separate numbers on separate number lines.

1. Upper number line labeled $x$

2. Lower number line labeled $y$

1. Two distinct horizontal lines are drawn in sequence.

1. Both lines are presented as one-dimensional number lines.

This visualizes the less accurate viewpoint of treating the inputs as separate scalar entities.

1. Animation
   Observation

   The two number lines are erased and replaced by perpendicular axes forming an $xy$-plane, with a single point plotted.

2. Audio
   Observation

   The speaker says it would probably be more accurate to think about the $xy$-plane and a single point.

1. No coordinate values are attached to the plotted point.

1. Perpendicular axes

2. Plotted point in the plane

1. The separate number lines disappear.

2. A plane diagram appears in their place.

3. A single point is marked in the plane.

1. The discussion remains about the same two input variables $x$ and $y$.

The animation demonstrates the conceptual shift from two independent numbers to one ordered-pair location.

1. Animation
   Observation

   The formula $f(x,y)=\begin{bmatrix}3x\\2y\end{bmatrix}$ is written in purple below the scalar-output example.

2. Audio
   Observation

   The speaker says the function could also output a vector and gives this invented example.

1. Column-vector formula $\begin{bmatrix}3x\\2y\end{bmatrix}$

1. A third function rule appears, now with a vertical bracketed output.

1. The input remains $f(x,y)$.

The visual form of the output reinforces the convention that multiple output numbers are treated as a vector.

1. Animation
   Observation

   A teal checkered bowl-like surface hovers above a gray plane with yellow axes and a vertical axis.

2. Audio
   Observation

   Speaker discusses graphs becoming three dimensional and height corresponding to output.

1. teal checkered surface

2. gray xy plane

3. yellow coordinate axes

4. vertical axis

1. The viewpoint rotates around the surface.

2. The relation between input plane and height is emphasized.

1. The surface remains above the plane.

2. The vertical direction represents output height.

This visual demonstrates the graph of a scalar function of two variables, with height encoding the output value.

1. Diagram
   Observation

   A full-screen 2D color field with concentric curves appears.

2. Formula
   Observation

   Handwritten $f(x,y)=x^2$ is added near the top.

3. Audio
   Observation

   Speaker explains color and contour lines.

1. colored 2D field

2. contour curves

3. axes labels

4. handwritten formula

1. Color varies across the plane.

2. Concentric curves mark equal-value sets.

1. The picture remains in the input plane rather than lifting into 3D.

The function is represented entirely in two dimensions by encoding output magnitude as color and equal outputs as contour lines.

1. Animation
   Observation

   A flat checkered sheet bends into a cylinder and then into a torus in 3D space.

2. Audio
   Observation

   Speaker introduces surfaces in three dimensional space and calls them parametric surfaces.

1. flat checkered sheet

2. cylinder

3. torus

4. 3D axes

1. The sheet bends upward into a cylindrical form.

2. The cylindrical form closes into a torus.

1. The checker pattern stays attached to the surface.

2. The setting remains three-dimensional.

The animation illustrates a two-dimensional parameter domain being moved into three-dimensional space to form a surface.

1. Diagram
   Observation

   Yellow arrows are drawn at many points of a 2D grid.

2. Audio
   Observation

   Speaker defines a vector field as assigning a vector to each input point.

1. 2D grid

2. yellow arrows

1. Arrows appear at multiple input locations.

1. Each arrow is anchored to a point in the plane.

The arrows represent the vector output attached to each input point of the function.

1. Animation
   Observation

   Small droplets move through the arrow field.

2. Audio
   Observation

   Speaker says imagine a fluid flowing and that droplets flow along that.

1. droplets

2. arrow field

1. Droplets begin moving.

2. Their trajectories follow local arrow directions.

1. The underlying arrow field remains visible.

The motion of droplets gives an intuitive fluid-flow reading of the vector field.

1. Audio
   Observation

   The speaker first says one could think of $x$ and $y$ as two separate numbers on number lines, then says it would probably be more accurate to think about the $xy$-plane and a single point.

2. Animation
   Observation

   The board literally replaces two separate number lines with one plane containing a point.

Thinking of the inputs $x$ and $y$ only as two separate numbers on separate number lines.

For a function of two variables, the pair $(x,y)$ is better understood as a single point in the $xy$-plane, so the function maps a point to a number or to a vector.

1. Audio
   Observation

   The speaker says many people jump into partial derivatives and gradients, but he wants to spend videos talking about different ways to visualize multivariable functions first.

Believing one should immediately begin with computational tools such as partial derivatives and gradients.

The clip argues for first building visual intuition for the types of multivariable functions before moving into those calculus operations.

1. Audio
   Observation

   Speaker says surfaces in three dimensional space ... look like graphs, but they actually deal with a much different animal.

Because parametric surfaces are drawn in 3D, they may be mistaken for the same kind of object as graphs of scalar functions.

The video distinguishes them: graphs here mean height over a 2D input plane for scalar output, whereas parametric surfaces come from mapping two dimensions into three dimensions.

1. Audio
   Observation

   Speaker says don't worry if this doesn't make sense immediately.

A viewer might expect every previewed visualization to be fully understandable on first exposure.

The speaker explicitly frames this segment as a quick introduction and says more detailed videos will follow.

1. Audio
   Observation

   The speaker contrasts ordinary-calculus functions with a single input against multivariable functions handling multiple variables.

2. Formula
   Observation

   $f(x)=x^2$ is followed by $f(x,y)=x^2+y$ on the board.

The multivariable-function idea extends the single-variable case by allowing more than one input variable.

1. Formula
   Observation

   The board shows both $f(x,y)=x^2+y$ and $f(x,y)=\begin{bmatrix}3x\\2y\end{bmatrix}$.

2. Audio
   Observation

   The speaker says the output can commonly be a number, but it could also be a vector.

Within the broad class of multivariable functions, the clip distinguishes scalar-output and vector-output cases.

1. Animation
   Observation

   Separate number lines for $x$ and $y$ are replaced by an $xy$-plane with a point.

2. Audio
   Observation

   The speaker says one should think about the $xy$-plane and a single point.

The geometric interpretation applies to multivariable functions by recasting their multiple inputs as one point in a plane.

1. Audio
   Observation

   The speaker links the plane-point viewpoint to the claim that "multidimensional calculus" would be more accurate.

The terminological claim depends on the prior geometric reinterpretation of inputs as points in a plane rather than separate scalars.

1. Audio
   Observation

   The speaker mentions partial derivatives and gradients as things one could jump into, but says he will first discuss ways to visualize multivariable functions.

1. Partial derivatives and gradients are named but not defined in this clip.

The clip contrasts immediate entry into computational calculus topics with first developing visual understanding of multivariable functions.

1. Audio
   Observation

   Speaker moves from 3D graphs to saying these functions also can be visualized just in two dimensions, flattening things out.

Contour plots are presented as a two-dimensional alternative to three-dimensional graphs for the same class of scalar multivariable functions.

1. Audio
   Observation

   Speaker defines contour lines as inputs sharing a constant output value.

Contour lines are the mechanism by which the contour-plot visualization encodes constant output values.

1. Audio
   Observation

   Speaker says surfaces look like graphs but deal with a much different animal, mapping two dimensions into three dimensions.

Both are drawn in three-dimensional space, but graphs represent scalar height over a plane while parametric surfaces represent outputs of a 2D-to-3D mapping.

1. Audio
   Observation

   Speaker says each input point is associated with a vector, which is the output of the function there.

The symbol 'vector field' is defined by the rule that the function output at each point is a vector.

1. Audio
   Observation

   Speaker says understanding functions as transformations is going to be a great way to connect those two.

The transformation viewpoint is presented as a bridge between multivariable calculus and linear algebra.

1. Audio
   Observation

   The speaker asks what the word multivariable means and answers by contrasting it with ordinary single-variable functions.

1. Topic framing: multivariable calculus
2. Single-variable function
3. Multivariable function

1. Formula
   Observation

   The board shows $f(x)=x^2$ and then $f(x,y)=x^2+y$.

1. Single-variable function
2. Multivariable function

1. Formula
   Observation

   The board shows $f(x,y)=\begin{bmatrix}3x\\2y\end{bmatrix}$.

2. Audio
   Observation

   The speaker says the output could also be a vector.

1. Convention for inputs and outputs with several numbers
2. Example of a vector-output multivariable function

1. Animation
   Observation

   Two number lines are replaced by an $xy$-plane with a point.

2. Audio
   Observation

   The speaker says it is more accurate to think about the $xy$-plane and a single point.

1. Viewing $(x,y)$ as a point in the $xy$-plane
2. Reinterpreting two input numbers as one plane point
3. Treating multivariable inputs as unrelated one-dimensional objects

1. Audio
   Observation

   The speaker says it would probably be more accurate to call it multidimensional calculus.

1. Terminological claim: multivariable vs. multidimensional
2. Viewing $(x,y)$ as a point in the $xy$-plane

1. Audio
   Observation

   The speaker says many people jump into partial derivatives and gradients, but he wants to discuss visualization first.

1. Starting multivariable calculus with operations before visualization
2. Topic framing: multivariable calculus

1. Audio
   Observation

   Speaker says graphs become three dimensional but only really apply to functions with two dimensional input and a single number output.

1. Three-dimensional graphs of scalar multivariable functions
2. Scope of 3D graph visualization

1. Audio
   Observation

   Speaker says contour lines tell you which inputs all share a constant output value.

1. Contour plot visualization
2. Meaning of contour lines

1. Audio
   Observation

   Speaker contrasts surfaces that look like graphs with mappings from two dimensions into three dimensions.

1. Parametric surfaces
2. Parametric surfaces versus graphs
3. Do not confuse parametric surfaces with ordinary graphs

1. Audio
   Observation

   Speaker defines a vector field as assigning a vector to each input point.

1. Vector fields
2. Output type of a vector field

1. Audio
   Observation

   Speaker describes taking the input space and watching points move to their output, then connects this to linear algebra.

1. Functions as transformations of space
2. Transformation viewpoint links to linear algebra

1. Formula
   Observation

   The handwritten example is $f(x,y)=x^2$.

2. Diagram
   Observation

   A contour plot is shown for that example.

1. Contour plot example for $f(x,y)=x^2$
2. Contour plot visualization