The Jacobian matrix

1. Formula
   Observation

   f([x; y]) = [$x + \sin (y)$; $y + \sin (x)$]

f

Vector-valued function from $R^2$ to $R^2$ representing a nonlinear transformation.

$R^2$

1. Formula
   Observation

   f([x; y]) = [$x + \sin (y)$; $y + \sin (x)$]

x

First input coordinate of the vector [x; y].

R

1. Formula
   Observation

   f([x; y]) = [$x + \sin (y)$; $y + \sin (x)$]

y

Second input coordinate of the vector [x; y].

R

1. Formula
   Observation

   (-2, 1)

(-2, 1)

Specific point in the domain around which the local linear behavior is analyzed.

$R^2$

1. Formula
   Observation

   ∂x

∂x

A tiny step in the x direction in the input space.

R

1. Formula
   Observation

   $f_1(x, y) = x + \sin (y)$

$f_1(x, y)$

First scalar-valued component function of f.

$R^2$

1. Formula
   Observation

   $f_2(x, y) = y + \sin (x)$

$f_2(x, y)$

Second scalar-valued component function of f.

$R^2$

1. Formula
   Observation

   Top-right yellow formula shows $f_1(x,y)$ in the first row of a column vector.

2. Audio
   Observation

   Speaker refers to $f_1$ as the x-component of the output when discussing ∂$f_1$/∂x.

$f_1(x,y)$

First component function of the transformation, equal to $x+\sin(y)$.

Real-valued function of variables x and y.

1. Formula
   Observation

   Top-right yellow formula shows $f_2(x,y)$ in the second row of a column vector.

2. Audio
   Observation

   Speaker refers to $f_2$ as the y-component of the output when discussing ∂$f_2$/∂x.

$f_2(x,y)$

Second component function of the transformation, equal to $y+\sin(x)$.

Real-valued function of variables x and y.

1. Formula
   Observation

   Formula uses (x,y) as inputs to $f_1$ and $f_2$.

2. Diagram
   Observation

   Left graph is a coordinate plane with grid lines and labeled axes.

(x,y)

Input coordinates in the domain plane being transformed.

Two-dimensional input space shown on the left graph.

1. Audio
   Observation

   Speaker says taking a tiny dx step in the input space corresponds to movement in the output space.

2. Animation
   Observation

   A green arrow appears in the inset around 00:00:01 and represents the result of an x-direction input step.

dx

Tiny step in the positive x direction in the input space.

Infinitesimal input displacement used for local linearization.

1. Formula
   Observation

   Green handwritten entry ∂$f_1$/∂x is written into the top-left position of the large bracketed matrix.

2. Audio
   Observation

   Speaker identifies this as the x-component of the movement caused by a tiny dx step, scaled by the size of that initial change.

$\frac{\partial f_1}{\partial x}$

Partial derivative of the first output component with respect to x; the horizontal component of the transformed x-basis direction.

Entry in the first row, first column of the Jacobian matrix.

1. Formula
   Observation

   f([x; y]) = [$x + \sin (y)$; $y + \sin (x)$]

2. Audio
   Observation

   In the last video we were looking at this particular function... taking every point x y in space to the point x plus sine y, y plus sine of x.

The video defines a specific nonlinear vector-valued function f that maps a 2D input vector [x; y] to an output vector [$x + \sin (y)$; $y + \sin (x)$].

1. Input is a 2D vector [x; y]

2. Output is a 2D vector

1. Audio
   Observation

   even though the function as a whole... looks rather complicated, around that one point, it looks like a linear function. It's locally linear.

2. Animation
   Observation

   Grid lines deform into curved lines globally, but within the zoomed-in square, they appear as straight lines.

A nonlinear transformation can appear linear when viewed in a sufficiently small neighborhood around a specific point. This property is called local linearity.

1. Considered around a specific point in the domain

2. Requires zooming in on a neighborhood

1. Vector-valued function definition

1. Audio
   Observation

   what matrix is going to tell you the linear function that this looks like. And this is going to be some kind of two by two matrix.

2. Formula
   Observation

   Empty 2x2 matrix brackets drawn on screen.

1. The exact entries of the matrix are not yet filled in during this clip.

The Jacobian matrix is a 2x2 matrix that represents the local linear approximation of a vector-valued function around a specific point.

1. Applies to vector-valued functions with 2D inputs and outputs

1. Local linearity of nonlinear transformations

1. Audio
   Observation

   instead of writing the entire function as something with a vector valued output, I'm going to go ahead and represent this as two separate scalar valued functions.

2. Formula
   Observation

   $f_1(x, y) = x + \sin (y)$, $f_2(x, y) = y + \sin (x)$

To analyze the transformation more easily, the vector-valued function f is decomposed into two scalar-valued component functions, $f_1$ and $f_2$.

1. Original function is vector-valued

1. Vector-valued function definition

1. Formula
   Observation

   Visible yellow formula defines [$f_1(x,y)$; $f_2(x,y)$] = [$x+\sin (y)$; $y+\sin (x)$].

2. Diagram
   Observation

   Left side shows a warped grid and a highlighted inset region, indicating a visualized transformation of the plane.

The clip studies the specific map sending (x,y) to ($x+\sin(y)$,\ $y+\sin(x)$). The speaker interprets this as a transformation of the plane and uses its local effect on tiny input steps to build the Jacobian matrix.

1. The map takes two real input variables x and y.

2. The output is a two-dimensional vector with components $f_1$ and $f_2$.

1. Audio
   Observation

   Speaker explains that a tiny dx step in the input space produces a 2D movement in the output space, whose x-component is a little partial change in $f_1$ and whose y-component is a little partial change in $f_2$.

2. Formula
   Observation

   The entries ∂$f_1$/∂x and ∂$f_2$/∂x are written into the first column of the matrix.

For this transformation, the local effect of moving only in the x direction is recorded by two numbers: how much the first output component changes and how much the second output component changes. Dividing those tiny output changes by the size of the initial input nudge rescales them into finite rates that remain meaningful under zooming.

1. The input perturbation is taken in the x direction.

2. The derivatives describe the corresponding output components after scaling by the size of the input step.

1. Example nonlinear transformation from $R^2$ to $R^2$

1. Formula
   Observation

   The large green bracketed matrix has ∂$f_1$/∂x in the top-left slot and ∂$f_2$/∂x in the bottom-left slot by the end of the clip.

2. Audio
   Observation

   Speaker explicitly associates these entries with the consequences of a tiny step in the x direction.

The first column of the Jacobian records where the infinitesimal x-direction input step is sent in the output plane. Its top entry is the horizontal output component and its bottom entry is the vertical output component.

1. Applies to a map from $R^2$ to $R^2$.

2. The column corresponds to the transformed x-basis direction.

1. Example nonlinear transformation from $R^2$ to $R^2$
2. Partial derivatives as components of the response to an infinitesimal input step

1. Audio
   Observation

   Speaker states that when representing a linear transformation with a matrix, the first column tells where the first basis vector goes and the second column shows where the second basis vector goes.

2. Diagram
   Observation

   The right column of the matrix is left open while the speaker prepares to fill it from the upward-step vector.

1. The second-column entries themselves are not written before the clip ends.

The Jacobian is assembled column by column: each column is the image of one input basis direction under the local linear approximation. In this clip, the x-basis direction gives the first column, and the y-basis direction is about to give the second column.

1. Used for a linear transformation represented by a matrix.

2. Each column corresponds to the image of a basis vector in the domain.

1. First column of the Jacobian from the x-direction basis vector

1. Formula
   Observation

   The screen shows [$f_1(x,y)$; $f_2(x,y)$] = [$x + \sin (y)$; $y + \sin (x)$].

2. Audio
   Observation

   The speaker refers to the output components as $f_1$ and $f_2$.

The clip works with a concrete two-variable, two-output function. Its first component is $f_1(x,y)=x+\sin (y)$, and its second component is $f_2(x,y)=y+\sin (x)$. This example supplies the rows of the later matrix: row 1 comes from $f_1$, row 2 comes from $f_2$.

1. The function has two input variables x and y.

2. The output has two components $f_1$ and $f_2$.

3. The discussion focuses on behavior near the point (-2,1).

1. Formula
   Observation

   A 2x2 bracketed array is built on screen with entries ∂$f_1$/∂x, ∂$f_1$/∂y, ∂$f_2$/∂x, ∂$f_2$/∂y.

2. Audio
   Observation

   The speaker explains each entry as an output change caused by a tiny input step in one coordinate direction.

3. Formula
   Observation

   The label 'Jacobian Matrix' is written beneath the array.

For this two-output, two-input example, the Jacobian matrix is organized so that each row corresponds to one output component and each column corresponds to one input direction. The top row records how $f_1$ changes with respect to x and y; the bottom row records how $f_2$ changes with respect to x and y.

1. Applies to a function with two outputs and two inputs in this clip.

2. Each entry is a partial derivative of one output component with respect to one input variable.

3. The matrix is meant to be evaluated at a specified point, here (-2,1).

1. Example vector-valued function used to motivate the Jacobian

1. Audio
   Observation

   The speaker says evaluating the partials at the point gives a concrete 2x2 matrix representing the linear transformation that the function looks like once you have zoomed in.

2. Diagram
   Observation

   The left side shows a grid and a highlighted small region around the point, visually matching the idea of zooming into local behavior.

The clip presents the Jacobian not merely as a table of symbols but as the matrix of the linear map that approximates the nonlinear function near a chosen point. After evaluating the partial derivatives at (-2,1), the resulting numerical 2x2 matrix describes the local stretching, shearing, and rotation-like behavior seen in a sufficiently small neighborhood.

1. The interpretation is local, tied to a specific base point.

2. The function must have the relevant partial derivatives available at that point.

3. The matrix represents a linear approximation, not the full nonlinear map.

1. Structure of the 2x2 Jacobian matrix

1. Audio
   Observation

   The speaker repeatedly describes entries as the change in an output component caused by a tiny step in one input direction, divided by the size of that step.

2. Formula
   Observation

   The matrix entries shown are ∂$f_1$/∂x, ∂$f_1$/∂y, ∂$f_2$/∂x, ∂$f_2$/∂y.

Each entry answers a separate question: if the input moves a tiny amount in one coordinate direction, how much does one particular output component change per unit of that input step? In the displayed matrix, the first row tracks responses of $f_1$, while the second row tracks responses of $f_2$; the left column tracks x-direction perturbations, and the right column tracks y-direction perturbations.

1. The perturbation is taken along one coordinate axis at a time.

2. The ratio is understood in the infinitesimal or local sense used for partial derivatives.

3. In this clip the inputs are x and y and the outputs are $f_1$ and $f_2$.

1. Example vector-valued function used to motivate the Jacobian

1. Audio
   Observation

   Speaker says dividing the tiny output change by the size of the initial tiny change scales it up to a normal-sized vector that does not shrink as we zoom in further and further.

For a tiny input displacement, the corresponding tiny output displacement divided by the size of the input displacement gives a finite quantity that remains meaningful under further zooming.

1. There is a differentiable local response to a tiny input step.

2. The output change is compared against the size of the initial input change.

For the infinitesimal x-step considered in the clip, the resulting scaled components define stable partial-derivative quantities.

1. Audio
   Observation

   Speaker says the first column tells where the first basis vector goes and the second column shows where the second basis vector goes.

When a linear transformation is represented by a matrix, its columns are the images of the domain basis vectors.

1. A linear transformation is being represented by a matrix with respect to chosen bases.

For the standard basis-vector interpretation used in the explanation of the Jacobian.

1. Audio
   Observation

   The speaker says all of this is specific to the point we started at, (-2,1), and that evaluating each partial derivative there gives numbers forming a concrete 2x2 matrix.

2. Formula
   Observation

   The point (-2,1) is written next to the function definition.

1. The audio contains one inconsistent restatement as 'two negative one,' but the visible coordinate and earlier speech support (-2,1).

The matrix of partial derivatives shown on screen represents a local linear transformation only after each entry is evaluated at the chosen input point (-2,1).

1. A specific base point has been selected, here (-2,1).

2. The relevant partial derivatives exist at that point.

3. The function is being approximated locally rather than globally.

For the example function and the chosen point (-2,1), each partial derivative entry is evaluated at that point to produce a concrete 2x2 matrix.

1. Audio
   Observation

   The speaker says, 'this matrix here that's full of all of the different partial derivatives has a very special name ... the Jacobian ... or more fully you'd call it the Jacobian matrix.'

2. Formula
   Observation

   The words 'Jacobian Matrix' are written under the matrix.

The matrix whose entries are the partial derivatives of the output components with respect to the input variables is called the Jacobian, or more fully the Jacobian matrix.

1. The function has multiple output components and multiple input variables.

2. The entries are arranged by output component and input direction as shown.

In this two-by-two example, the displayed array of partial derivatives is named the Jacobian matrix.

1. Audio
   Observation

   The speaker says it is much more than just a way of recording what all the partial derivatives are, and that there is a reason for organizing it like this in particular.

1. The sentence is cut off before the speaker completes the explanation.

The Jacobian is presented as more than a mere list of partial derivatives; its arrangement is intended to encode the local linear behavior of the function.

1. The partial derivatives are organized into the matrix layout shown in the clip.

2. The context is local analysis near a chosen point.

For the example under discussion, the speaker asserts a nontrivial reason for this specific organization, but the full justification is not completed within the clip.

1. Audio
   Observation

   The speaker states that the Jacobian matrix is fundamentally supposed to represent what a transformation looks like when you zoom in near a specific point.

2. Formula
   Observation

   The matrix of partial derivatives is displayed while this statement is made.

The Jacobian matrix represents what a transformation looks like when you zoom in near a specific point.

1. There is a transformation under consideration.

2. A specific point in its domain or displayed plane is chosen for local analysis.

For the transformation and point discussed in the clip.

1. Audio
   Observation

   think of just a tiny step to the right... a little partial x... what that turns into after the transformation is going to be some tiny step in the output space... It's no longer purely in the x direction, it has some rightward component, but now also some downward component.

2. Animation
   Observation

   A green arrow representing ∂x is drawn in the zoomed-in box, pointing diagonally down and to the right.

1. Expression

   $$\partial x$$
   Explanation

   Start with a tiny step in the positive x-direction in the input space.

   Justification

   Definition of partial derivative direction.

   Shown in the video
2. Expression

   $$\text{Transformed } \partial x$$
   Explanation

   Apply the nonlinear transformation f to this tiny step.

   Justification

   Action of the function f.

   Shown in the video
3. Expression

   $$\text{Diagonal vector in output space}$$
   Explanation

   Observe the resulting vector in the output space, which has both horizontal and vertical components.

   Justification

   Visual representation of the transformed step.

   Shown in the video

The first column of the Jacobian matrix will represent the x and y components of this transformed tiny step ∂x.

1. Audio
   Observation

   Speaker starts from a tiny dx step, identifies the x-component of the output movement as ∂$f_1$/∂x and the y-component as ∂$f_2$/∂x, then writes both into the first column.

2. Formula
   Observation

   The matrix entries ∂$f_1$/∂x and $f_2$/∂x appear sequentially in the left column.

1. Expression

   $$$$
   Explanation

   Take a tiny step dx in the input space.

   Justification

   This is the starting perturbation named by the speaker.

   Shown in the video
2. Expression

   $$$$
   Explanation

   Observe the resulting 2D movement in the output space.

   Justification

   The transformation sends the input nudge to some output displacement.

   Shown in the video
3. Expression

   $$\frac{\partial f_1}{\partial x}$$
   Explanation

   Identify the horizontal component of that output movement as the partial change in $f_1$ per unit x-step.

   Justification

   Speaker explicitly calls this the x-component of the movement caused by dx.

   Shown in the video
4. Expression

   $$\frac{\partial f_2}{\partial x}$$
   Explanation

   Identify the vertical component of the same output movement as the partial change in $f_2$ per unit x-step.

   Justification

   Speaker explicitly calls this the y-component caused by the initial step to the right.

   Shown in the video
5. Expression

   $$\begin{bmatrix} \frac{\partial f_1}{\partial x} \\ \frac{\partial f_2}{\partial x} \end{bmatrix}$$
   Explanation

   Place these two scaled components into the first column of the matrix.

   Justification

   The visual matrix is filled top-to-bottom with exactly these entries.

   Shown in the video

The first column of the Jacobian is the vector of partial derivatives with respect to x.

1. Audio
   Observation

   Speaker introduces a tiny step in the y direction, asks what it becomes after the transformation, points to a vector with upward and rightward components, and says he will write its components as the second column.

2. Animation
   Observation

   A red upward arrow appears in the inset around 00:01:40, followed by a blue diagonal arrow around 00:01:41.

1. The actual symbolic entries for the second column are not completed within this clip.

1. Expression

   $$$$
   Explanation

   Consider instead a tiny step upward in the input space, i.e. in the y direction.

   Justification

   Speaker explicitly contrasts this with the earlier x-direction step.

   Shown in the video
2. Expression

   $$$$
   Explanation

   Look at what that upward input step becomes after applying the transformation.

   Justification

   This is the question posed aloud by the speaker.

   Shown in the video
3. Expression

   $$$$
   Explanation

   The resulting output vector still has an upward component but also a rightward component.

   Justification

   Speaker describes the transformed vector in exactly these terms while pointing to the inset.

   Shown in the video
4. Expression

   $$$$
   Explanation

   Write those two components as the second column of the matrix.

   Justification

   Speaker states this directly and the right column remains reserved for that purpose.

   Shown in the video

The second column will record the output components produced by an infinitesimal y-direction input step.

1. Audio
   Observation

   The speaker constructs the matrix entry by entry, first discussing the x-component of the output change and then the y-component, each caused by a tiny step in the y direction.

2. Formula
   Observation

   The on-screen matrix is filled progressively with ∂$f_1$/∂x, $f_1/y$, ∂$f_2$/∂x, ∂$f_2$/∂y.

1. Expression

   $$\begin{bmatrix} f_1(x,y) \\ f_2(x,y) \end{bmatrix} = \begin{bmatrix} x + \sin(y) \\ y + \sin(x) \end{bmatrix}$$
   Explanation

   Start from the concrete vector-valued function whose two output components will be differentiated.

   Justification

   This is the function written at the top of the board in the clip.

   Shown in the video
2. Expression

   $$\frac{\partial f_1}{\partial x}$$
   Explanation

   The first entry records how the first output component $f_1$ changes when the input moves a tiny amount in the x direction.

   Justification

   The speaker explicitly ties the top-left entry to the x-component of the output change caused by a tiny x-step.

   Shown in the video
3. Expression

   $$\frac{\partial f_1}{\partial y}$$
   Explanation

   The second entry in the first row records how $f_1$ changes when the input moves a tiny amount in the y direction.

   Justification

   The speaker describes this as the change in $f_1$ caused by some tiny step in the y direction divided by the size of that step.

   Shown in the video
4. Expression

   $$\frac{\partial f_2}{\partial x}$$
   Explanation

   The first entry in the second row records how the second output component $f_2$ changes under a tiny x-direction input step.

   Justification

   The speaker shifts to the y-component of the output and attributes it to the initial tiny step in the input space.

   Shown in the video
5. Expression

   $$\frac{\partial f_2}{\partial y}$$
   Explanation

   The final entry records how $f_2$ changes under a tiny y-direction input step.

   Justification

   The speaker identifies this as the change of $f_2$ caused by dy, or by that little partial y.

   Shown in the video
6. Expression

   $$\begin{bmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\ \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y} \end{bmatrix}$$
   Explanation

   Putting the four entries together yields the Jacobian matrix shown on screen.

   Justification

   This is exactly the completed bracketed array written beneath the function definition.

   Shown in the video

The matrix assembled in the clip is the Jacobian matrix for the example function, with rows indexed by output components and columns indexed by input directions.

1. Audio
   Observation

   The speaker says the construction is specific to the starting point (-2,1), and that evaluating each partial derivative there gives numbers forming a concrete 2x2 matrix representing the local linear transformation.

1. No numerical values for the four partial derivatives are computed in this clip.

1. Expression

   $$(-2,1)$$
   Explanation

   Fix the base point at which the local behavior is being examined.

   Justification

   The speaker states that everything discussed is specific to the point where they started, (-2,1).

   Shown in the video
2. Expression

   $$\left.\frac{\partial f_1}{\partial x}\right|_{(-2,1)},\ \left.\frac{\partial f_1}{\partial y}\right|_{(-2,1)},\ \left.\frac{\partial f_2}{\partial x}\right|_{(-2,1)},\ \left.\frac{\partial f_2}{\partial y}\right|_{(-2,1)}$$
   Explanation

   Evaluate each symbolic partial derivative at the chosen point to turn it into a number.

   Justification

   The speaker says each partial derivative should be evaluated at the point to get some number.

   Shown in the video
3. Expression

   $$\begin{bmatrix} \left.\frac{\partial f_1}{\partial x}\right|_{(-2,1)} & \left.\frac{\partial f_1}{\partial y}\right|_{(-2,1)} \\ \left.\frac{\partial f_2}{\partial x}\right|_{(-2,1)} & \left.\frac{\partial f_2}{\partial y}\right|_{(-2,1)} \end{bmatrix}$$
   Explanation

   Assemble the evaluated numbers into the same 2x2 layout as before.

   Justification

   The speaker says this produces a very concrete 2x2 matrix.

   Shown in the video
4. Expression

   $$\text{local linear transformation}$$
   Explanation

   Interpret the resulting numerical matrix as the linear map that the function resembles after zooming in near the base point.

   Justification

   The speaker explicitly describes the matrix as representing the linear transformation that the function looks like once you have zoomed in.

   Shown in the video

Evaluating the Jacobian entries at (-2,1) converts the symbolic derivative array into a concrete matrix describing the local linear approximation of the nonlinear map.

1. Audio
   Observation

   The speaker connects local linearity to the Jacobian matrix and explains that understanding this representation makes the rest fall into place.

2. Diagram
   Observation

   A highlighted square region around (-2,1) is shown on the transformed grid.

3. Formula
   Observation

   The $2\times 2$ Jacobian matrix of partial derivatives is displayed beside the diagram.

1. Expression

   $$$$
   Explanation

   Start with a nonlinear transformation represented by a distorted grid on the left.

   Justification

   Observed directly from the diagram.

   Shown in the video
2. Expression

   $$$$
   Explanation

   Choose a specific point, here marked (-2,1), and focus on a small square neighborhood around it.

   Justification

   Observed directly from the highlighted region and point label.

   Shown in the video
3. Expression

   $$\begin{bmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\ \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y} \end{bmatrix}$$
   Explanation

   Associate that local picture with the matrix of first-order partial derivatives of the component functions.

   Justification

   Observed from the displayed formula labeled "Jacobian Matrix" and the speaker’s explanation.

   Shown in the video
4. Expression

   $$$$
   Explanation

   Interpret the resulting matrix as the linear description of how the transformation behaves near the chosen point.

   Justification

   Stated in the audio as the meaning of local linearity.

   Shown in the video

The Jacobian matrix is presented as the local linear representation of the transformation near the highlighted point.

1. Audio
   Observation

   we zoomed in on a specific point... negative two one... added a couple extra grid lines around it... this square shows the zoomed in version of that neighborhood.

2. Formula
   Observation

   (-2, 1)

3. Diagram
   Observation

   Left side shows a global grid with a highlighted point at (-2, 1). Right side shows a zoomed-in square grid.

Analyze the local behavior of the function f around the point (-2, 1).

1. Function f([x; y]) = [$x + \sin (y)$; $y + \sin (x)$]

2. Point of interest: (-2, 1)

Understand how the transformation affects points in a small neighborhood around (-2, 1).

1. Expression

   $$\text{Identify point } (-2, 1)$$
   Explanation

   Locate the specific point on the global grid.

   Justification

   Given in the problem setup.

   Shown in the video
2. Expression

   $$\text{Zoom in on neighborhood}$$
   Explanation

   Display a magnified view of the grid lines surrounding the point.

   Justification

   Visual aid for analyzing local behavior.

   Shown in the video

The zoomed-in view allows for detailed observation of how the grid lines deform locally.

Visual confirmation of the zoomed-in square matching the location of (-2, 1) on the global grid.

1. Formula
   Observation

   The displayed map is [$f_1(x,y)$; $f_2(x,y)$] = [$x+\sin (y)$; $y+\sin (x)$].

2. Audio
   Observation

   The speaker builds the Jacobian concept using this specific example rather than a generic abstract function only.

1. The clip explains the construction visually and verbally, but it does not show the final completed numeric/symbolic Jacobian for this specific map before ending.

Given the transformation $f_1(x,y)=x+\sin(y)$ and $f_2(x,y)=y+\sin(x)$, interpret the Jacobian matrix by examining tiny input steps in the x and y directions.

1. $f_1(x,y)=x+\sin(y)$

2. $f_2(x,y)=y+\sin(x)$

3. The plane is shown before and after transformation, with an inset zoom region.

4. The point (-2,1) is displayed near the formula.

Explain how the columns of the Jacobian arise from the local effects of basis-direction steps.

1. Expression

   $$\begin{bmatrix} f_1(x,y) \\ f_2(x,y) \end{bmatrix} = \begin{bmatrix} x+\sin(y) \\ y+\sin(x) \end{bmatrix}$$
   Explanation

   Start from the explicit nonlinear map being visualized.

   Justification

   This formula is visible on screen throughout the clip.

   Shown in the video
2. Expression

   $$\begin{bmatrix} \frac{\partial f_1}{\partial x} \\ \frac{\partial f_2}{\partial x} \end{bmatrix}$$
   Explanation

   Use a tiny x-step to determine the first column.

   Justification

   The speaker derives these entries from the output movement caused by dx and writes them into the left column.

   Shown in the video
3. Expression

   $$\begin{bmatrix} \frac{\partial f_1}{\partial y} \\ \frac{\partial f_2}{\partial y} \end{bmatrix}$$
   Explanation

   Use a tiny y-step to determine the second column.

   Justification

   The speaker introduces the upward-step case and says its components will occupy the second column; the full symbolic writing is not completed onscreen in this excerpt.

   Derived from the video

The Jacobian is built columnwise: the first column comes from the x-direction infinitesimal step, and the second column comes from the y-direction infinitesimal step.

The visible matrix structure and the spoken rule about basis-vector images confirm the columnwise construction.

1. Formula
   Observation

   The board shows the specific function [$f_1(x,y)$; $f_2(x,y)$] = [$x + \sin (y)$; $y + \sin (x)$] and the associated 2x2 matrix of partial derivatives.

2. Audio
   Observation

   The speaker uses this exact function to explain what each matrix entry means and names the resulting matrix the Jacobian matrix.

1. The clip does not compute the numerical values of the partial derivatives at (-2,1).

2. The clip does not state the final simplified symbolic entries such as 1, $\cos (y)$, $\cos (x)$, 1.

Given the vector-valued function [$f_1(x,y)$; $f_2(x,y)$] = [$x + \sin (y)$; $y + \sin (x)$], organize the local partial-derivative information into the matrix discussed in the clip.

1. $f_1(x,y)=x+\sin(y)$

2. $f_2(x,y)=y+\sin(x)$

3. The discussion is centered at the input point (-2,1).

Identify the matrix formed from the partial derivatives of the two output components with respect to the two input variables, and understand its local meaning.

1. Expression

   $$\begin{bmatrix} f_1(x,y) \\ f_2(x,y) \end{bmatrix} = \begin{bmatrix} x + \sin(y) \\ y + \sin(x) \end{bmatrix}$$
   Explanation

   Write down the two output components of the example function.

   Justification

   These formulas are displayed directly on the board.

   Shown in the video
2. Expression

   $$\begin{bmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\ \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y} \end{bmatrix}$$
   Explanation

   Arrange the four partial derivatives into a 2x2 matrix, with rows corresponding to $f_1$ and $f_2$ and columns corresponding to x and y.

   Justification

   This is the matrix built entry by entry in the clip and labeled Jacobian Matrix.

   Shown in the video
3. Expression

   $$\text{Evaluate at } (-2,1)$$
   Explanation

   To obtain the concrete local linear map, substitute the base point into each partial derivative.

   Justification

   The speaker says the construction is specific to the starting point (-2,1) and that evaluating there yields a concrete 2x2 matrix.

   Shown in the video

The example leads to the Jacobian matrix $\begin{bmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\ \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y} \end{bmatrix}$, intended for evaluation at (-2,1).

Verification would require computing the four partial derivatives and substituting (-2,1), but those numerical evaluations are not performed within this clip.

1. Animation
   Observation

   Global grid lines curve and distort, while the lines within the zoomed-in yellow square remain relatively straight.

1. Global coordinate grid

2. Zoomed-in square grid

3. Curved grid lines

1. Straight grid lines transform into curved lines globally.

2. Grid lines inside the zoomed-in square appear straight.

1. The point (-2, 1) remains the center of the zoomed-in view.

Demonstrates that while the overall transformation is highly nonlinear, it appears linear when restricted to a small enough neighborhood.

1. Animation
   Observation

   A green arrow labeled ∂x is drawn starting from the center of the zoomed-in square, pointing diagonally downwards and to the right.

1. Green arrow

2. Zoomed-in square grid

1. A new vector (green arrow) is introduced to represent the transformed tiny step in the x-direction.

1. The starting point of the arrow is the center of the zoomed-in square.

Visually represents the first column of the Jacobian matrix, showing how a pure x-step becomes a combination of x and y steps after transformation.

1. Diagram
   Observation

   Left half shows a distorted grid with a highlighted square inset and coordinate labels; the inset contains a small vector visualization.

2. Animation
   Observation

   A green arrow appears inside the inset early in the clip, then later a red upward arrow and a blue diagonal arrow appear.

1. Exact coordinates of the inset center are not fully legible from the sampled frames.

1. Warped 2D grid

2. Highlighted square inset

3. Green arrow

4. Red upward arrow

5. Blue diagonal arrow

6. Large green bracketed matrix placeholder

1. The clip first focuses on the result of an x-direction input step, represented by a green arrow in the inset.

2. Later attention shifts to a y-direction input step, represented by a red upward arrow and then a blue diagonal output arrow.

3. The matrix placeholder is progressively filled in the first column while the second column remains empty.

1. The transformation formula stays visible at the top right.

2. The overall layout of domain/output visualization plus matrix construction remains constant.

3. The point label (-2,1) remains visible near the formula.

The visuals connect infinitesimal input directions to their transformed output directions and then encode those output directions as columns of the Jacobian.

1. Animation
   Observation

   The top-left matrix entry ∂$f_1$/∂x is written first, then the bottom-left entry ∂$f_2$/∂x is added.

2. Formula
   Observation

   By around 00:01:00 the left column is complete while the right column is still blank.

1. Large green bracketed matrix

2. Entry ∂$f_1$/∂x

3. Entry ∂$f_2$/∂x

1. The matrix begins as an empty bracketed template.

2. The upper-left entry is filled with ∂$f_1$/∂x.

3. The lower-left entry is filled with ∂$f_2$/∂x.

4. The right column remains unfilled through the end of the clip.

1. The matrix remains a 2-by-2 template.

2. The transformation formula above stays unchanged.

The animation makes explicit that the x-basis direction contributes the entire first column before the y-basis direction is considered.

1. Diagram
   Observation

   The left half of the screen shows a coordinate grid with curved grid lines and a highlighted square region containing a red arrow.

2. Audio
   Observation

   The speaker talks about zooming in and the local linear transformation the function looks like.

1. The exact animated motion inside the highlighted region is not fully described in the sampled frames.

1. Cartesian-style input grid

2. Curved transformed grid lines

3. Highlighted square region

4. Red arrow inside the highlighted region

5. Function formula at top right

6. 2x2 matrix below the formula

1. The board progressively fills the matrix entries from left to right and top to bottom.

2. The label 'Jacobian Matrix' is added below the matrix.

3. The visual emphasis remains on the relationship between the local boxed region and the algebraic matrix.

1. The example function stays displayed at the top right throughout the clip.

2. The base point (-2,1) remains written beside the function.

3. The matrix layout remains 2x2 with rows for $f_1$, $f_2$ and columns for x, y.

The picture links a local neighborhood in the input-output geometry to the algebraic object built from partial derivatives. The highlighted box suggests the region where the nonlinear map is being approximated by a linear transformation, while the matrix records the coefficients of that approximation.

1. Diagram
   Observation

   White arrows are drawn from the point (-2,1) toward the four entries of the matrix.

2. Audio
   Observation

   The speaker says each partial derivative should be evaluated at the point to get a number.

1. The precise order in which all four arrows are drawn is approximate from the sampled frames.

1. Point label (-2,1)

2. Four matrix entries ∂$f_1$/∂x, $f_1/y$, ∂$f_2$/∂x, ∂$f_2$/∂y

3. White connecting arrows

1. Arrows are added from the point label to each matrix entry.

2. The visual connection reinforces that the symbolic entries are to be evaluated at the same base point.

1. The matrix structure itself does not change while the arrows are added.

2. The function definition remains unchanged above.

The arrows encode the logical step from a symbolic Jacobian to a numerical local linear map: every entry is evaluated at the same input point (-2,1).

1. Diagram
   Observation

   The frame is split between a transformed coordinate grid on the left and a handwritten Jacobian matrix on the right.

2. Animation
   Observation

   White arrows connect the point (-2,1) to the four entries of the matrix.

1. Distorted coordinate grid

2. Highlighted square region

3. Green point labeled (-2,1)

4. $2\times 2$ Jacobian matrix

5. Four white arrows

6. Label "Jacobian Matrix"

1. The overall layout remains fixed throughout the clip.

2. The arrows continue to indicate correspondence between the highlighted point and each matrix entry.

1. The matrix entries remain visible as ∂f₁/∂x, ∂f₁/∂y, f₂/∂x, ∂f₂/∂y.

2. The point (-2,1) remains highlighted.

3. The square neighborhood remains outlined on the transformed grid.

The visual organizes the lesson around one point in a transformed plane and the four partial derivatives that together form its Jacobian matrix.

1. Diagram
   Observation

   A yellow square is drawn over the warped grid around the marked point (-2,1).

2. Audio
   Observation

   The speaker describes zooming in near a specific point.

1. The exact numerical size of the square is not stated.

1. Yellow square

2. Warped grid lines

3. Point (-2,1)

1. The square isolates a small region of the transformed grid for local inspection.

1. The square stays centered around the same highlighted point during the clip.

2. The surrounding grid remains globally distorted.

The square functions as a visual stand-in for a small neighborhood, supporting the claim that the Jacobian describes local behavior rather than global shape.

1. Audio
   Observation

   even though the function as a whole... looks rather complicated, around that one point, it looks like a linear function.

Because a function is globally nonlinear, its behavior cannot be approximated by a linear function anywhere.

Nonlinear functions exhibit local linearity; in a sufficiently small neighborhood around any point, they look like linear transformations.

1. Audio
   Observation

   Speaker stresses that the raw partial change is a tiny nudge, and dividing by the size of the initial tiny change gives something that does not shrink under further zooming.

One might think the tiny output displacement caused by dx is already the desired derivative quantity.

The derivative is obtained after scaling the tiny output change by the size of the initial input change, producing a finite rate that remains stable under zooming.

1. Audio
   Observation

   Speaker explicitly notes that the vertical component of the output step is still caused by the initial step to the right.

Because the input step is in the x direction, one may expect only $f_1$ to change.

For this transformation, an x-direction input step can produce both horizontal and vertical output changes, giving both ∂$f_1$/∂x and $f_2$/∂x.

1. Audio
   Observation

   Speaker says the transformed upward-step vector still has some upward component but also has a rightward component.

One may expect a purely upward input perturbation to remain purely vertical after transformation.

The local output vector for the y-step has both vertical and horizontal components, so the second Jacobian column generally contains two nonzero entries.

1. Audio
   Observation

   The speaker says it is much more than just a way of recording what all the partial derivatives are, and that there is a reason for organizing it like this in particular.

1. The full positive explanation is cut off before completion.

One might think the Jacobian is only a convenient table for storing partial derivatives.

The clip explicitly argues that the arrangement matters because the resulting matrix represents the local linear transformation of the function near the chosen point.

1. Audio
   Observation

   The speaker stresses that all of this is very specific to the point that we started at, (-2,1).

One might read the matrix as describing the function everywhere rather than near a particular input.

In this clip, the matrix is tied to the base point (-2,1); its meaning as a local linear transformation comes from evaluating the partial derivatives at that point.

1. Audio
   Observation

   The speaker stresses that the Jacobian matrix represents what the transformation looks like when you zoom in near a specific point.

2. Diagram
   Observation

   The left image shows a globally curved grid but singles out one small square region.

One may think the displayed Jacobian matrix describes the entire nonlinear transformation everywhere.

The clip emphasizes local linearity: the Jacobian is tied to the behavior near a specific point, here (-2,1), not to the full global distortion of the grid.

1. Audio
   Observation

   what matrix is going to tell you the linear function that this looks like.

The Jacobian matrix is the mathematical tool used to describe the local linear approximation of a nonlinear function.

1. Audio
   Observation

   instead of writing the entire function as something with a vector valued output, I'm going to go ahead and represent this as two separate scalar valued functions.

A vector-valued function contains multiple scalar-valued component functions, which are used to build the Jacobian matrix.

1. Audio
   Observation

   Speaker moves from the explicit map $f_1$,$f_2$ to the partial derivatives generated by a tiny dx step.

2. Formula
   Observation

   The formula for the map is visible while the matrix entries ∂$f_1$/∂x and ∂$f_2$/∂x are introduced.

The concrete nonlinear map is the object to which the partial-derivative interpretation is applied.

1. Formula
   Observation

   The two partial derivatives with respect to x are placed into the first column of the matrix.

2. Audio
   Observation

   Speaker describes these as the components of the movement caused by dx.

The first Jacobian column consists exactly of the two partial-derivative quantities extracted from the x-direction local response.

1. Audio
   Observation

   After constructing the first column, the speaker generalizes to the rule that matrix columns are images of basis vectors.

The completed first column is a special instance of the broader rule that each matrix column records where a basis vector goes.

1. Audio
   Observation

   Speaker uses the basis-vector-column rule to explain why the upward-step vector should become the second column.

The justification for assembling the Jacobian columnwise depends on the linear-algebra fact that matrix columns are images of basis vectors.

1. Formula
   Observation

   The function components $f_1$ and $f_2$ appear above the matrix whose entries are their partial derivatives.

2. Audio
   Observation

   The speaker builds the matrix directly from changes in $f_1$ and $f_2$.

The specific vector-valued function supplies the output components whose partial derivatives become the rows of the Jacobian matrix.

1. Audio
   Observation

   Each matrix entry is explained as an output change caused by a tiny input step in one coordinate direction.

2. Formula
   Observation

   The completed matrix displays those entries in a fixed row-column arrangement.

The general interpretation of each partial-derivative entry is embodied in the specific 2x2 layout shown on the board.

1. Audio
   Observation

   The speaker says evaluating the matrix at the point gives the linear transformation the function looks like once you have zoomed in.

Once the symbolic matrix is evaluated at a point, it is applied as the matrix of the local linear approximation.

1. Formula
   Observation

   The label 'Jacobian Matrix' is written directly under the 2x2 array of partial derivatives.

2. Audio
   Observation

   The speaker names that matrix the Jacobian, or more fully the Jacobian matrix.