Transformations, part 1 | Multivariable calculus | Khan Academy

Khan Academy
5 May 201605:16
EducationalLearning
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TLDRThe video script explores the concept of visualizing multivariable functions through the lens of transformations. It starts with a simple one-dimensional function, illustrating how inputs move to outputs, and then delves into multivariable functions with a one-dimensional input and a two-dimensional output. The script uses the example of a function mapping to a point in a plane defined by cosine and sine of the input, emphasizing the visual movement from input to output space. This approach offers insights into whether the space is stretched or compressed, and it parallels parametric plots, but with the advantage of retaining input information.

Takeaways
  • πŸ“Š The script introduces various ways to visualize multivariable functions, including 3D graphs, contour maps, vector fields, and parametric functions.
  • πŸ”„ The concept of functions as transformations is highlighted, emphasizing the movement of points from an input space to an output space.
  • πŸ“ The input space is depicted as a blob, which could represent any multidimensional space, including real number lines or 3D space.
  • πŸ“ Similarly, the output space is also represented as a blob, and could be a real number line, the x-y plane, or a 3D space.
  • πŸ”‘ The function is defined as a mapping from inputs to outputs, associating each input with a corresponding output.
  • πŸ“ˆ An example of a one-dimensional function, f(x) = x^2 - 3, is used to illustrate how inputs move to their respective outputs.
  • πŸŽ› A simple animation is suggested to visualize the movement of numbers from the input to the output in the function f(x) = x^2 - 3.
  • πŸ“ The script then moves to multivariable functions, considering a function with a one-dimensional input and a two-dimensional output.
  • 🌐 An example function f(x) = (cos(x), x*sin(x)) is given to demonstrate the transformation of a single input variable into a two-dimensional output.
  • πŸ“Š The visualization of the function f(x) = (cos(x), x*sin(x)) as a transformation helps in understanding whether the space is stretched or squished.
  • πŸ”„ The script concludes by comparing the transformation view with parametric plots, noting that the former retains the input information.
Q & A

  • What are the different ways to visualize multivariable functions mentioned in the script?

    -The script mentions three-dimensional graphs, contour maps, vector fields, and parametric functions as ways to visualize multivariable functions.

  • What is the abstract concept of input and output space as described in the script?

    -The script describes input and output spaces as abstract 'blobs' that could represent any dimensional space, such as the real number line or three-dimensional space, where functions map inputs to outputs.

  • How does the script suggest visualizing a function without using a graph?

    -The script suggests visualizing a function as a transformation where you watch the actual points of the input space move to the output space.

  • What is the example of a one-dimensional function given in the script?

    -The example given is the function f(x) = x squared minus three, which is visualized by watching how the input values move to their corresponding outputs.

  • How does the script describe the transformation of the number zero in the given function?

    -The script describes the transformation of zero as moving to negative three, since zero squared minus three equals negative three.

  • What is the significance of the animation in the script when visualizing transformations?

    -The animation helps to visually demonstrate how each input number moves to its corresponding output, providing a clearer understanding of the function's transformation.

  • What is the multivariable function example provided in the script?

    -The multivariable function example is f(x) = (cosine of x, x sine of x), which has a one-dimensional input and a two-dimensional output.

  • How does the script explain the transformation of the number zero in the multivariable function?

    -The script explains that when zero is plugged into the multivariable function, it moves to the point one-zero, as cosine of zero is one and zero times sine of x is zero.

  • What is the transformation of the number pi in the multivariable function as described in the script?

    -The transformation of pi results in the point negative one-zero, as cosine of pi is negative one and pi times sine of pi is zero.

  • How does the script relate the idea of transformations to parametric plots?

    -The script relates transformations to parametric plots by stating that watching the input space move to the output space is similar to what you would see in a parametric plot, but with the added benefit of retaining input information.

  • What is the script's intention for the next video?

    -The script indicates that the next video will discuss how to interpret functions with two-dimensional input and output as transformations.

Outlines
00:00
πŸ“ˆ Visualizing Functions as Transformations

This paragraph introduces the concept of visualizing multivariable functions as transformations, rather than traditional methods like three-dimensional graphs or contour maps. The speaker prefers to think abstractly about functions, considering an input space and an output space, both represented as 'blobs' that could be any dimensional space. The function's role is to map inputs to outputs, and visualization techniques aim to show these associations. The paragraph provides a simple example of a one-variable function, f(x) = x^2 - 3, and discusses the idea of watching how inputs move to their corresponding outputs, suggesting an animation to visualize this movement.

05:02
πŸ”„ Exploring Multivariable Functions with Animation

The second paragraph expands on the concept of transformations with multivariable functions, specifically focusing on a function with a one-dimensional input and a two-dimensional output. The function given is f(x) = (cos(x), x*sin(x)), and the speaker illustrates how different inputs like zero and pi would be transformed to their respective outputs in the output space. The paragraph emphasizes the visual appeal of watching each element of the input space move to the output space, providing a clearer understanding of the function's effect. It also touches on the concept of space transformation, such as stretching or squishing, and relates this visualization method to parametric plots, highlighting the advantage of retaining input information in this approach.

Mindmap
Keywords
πŸ’‘Multivariable Functions
Multivariable functions are mathematical functions that take multiple inputs to produce a single output. In the context of the video, these functions are central to understanding how different inputs can affect an output in a multidimensional space. The script discusses various ways to visualize these functions, such as three-dimensional graphs and contour maps, emphasizing the importance of visualizing the transformation process from input to output.
πŸ’‘Visualization
Visualization in the video refers to the graphical representation of functions to better understand their behavior. It is a key concept because it helps in comprehending how multivariable functions operate by mapping inputs to outputs in a visual format. Examples include three-dimensional graphs and contour maps, which are used to represent the function's values across different input variables.
πŸ’‘Transformation
A transformation, as described in the video, is the process of mapping inputs to outputs through a function. It's a conceptual framework for understanding how data points move from the input space to the output space. The video uses this concept to illustrate the movement of points, providing a dynamic way to visualize the function's effect.
πŸ’‘Input Space
The input space in the video is the set of all possible inputs to a function. It is depicted as a 'blob' in the script to represent the abstract concept of where inputs come from, which could be a line for one-dimensional inputs or a three-dimensional space for more complex inputs. The input space is fundamental to understanding transformations as it defines the domain from which the function takes values.
πŸ’‘Output Space
The output space is the set of all possible results that a function can produce. In the video, it is also represented as a 'blob' to symbolize the abstract concept of where outputs go. It is crucial for understanding the range of a function and how the function's behavior affects the output values.
πŸ’‘Graph
A graph is a visual representation of the relationship between inputs and outputs of a function. In the script, the graph is mentioned as a common way to visualize single-variable functions, such as the example of f(x) = x squared minus three, which would be visualized as a parabolic shape.
πŸ’‘Contour Map
A contour map is a type of visualization used for multivariable functions, where lines (contours) connect points of equal output value. The video mentions contour maps as one of the ways to visualize multivariable functions, providing a way to see how the function's output changes across different regions of the input space.
πŸ’‘Vector Fields
Vector fields are a visualization technique used to represent multivariable functions where each point in the input space is associated with a vector that indicates the direction and magnitude of the function's output. Although not explicitly detailed in the script, the concept is implied as part of the discussion on visualizing multivariable functions.
πŸ’‘Parametric Functions
Parametric functions are a way to represent functions where the output is defined in terms of a parameter. In the video, the script mentions parametric functions as another method to visualize functions, particularly highlighting how they can be used to understand transformations from input to output.
πŸ’‘Animation
Animation in the context of the video refers to the dynamic visualization of the transformation process. It is used to demonstrate how input values move to their corresponding output values, providing a more intuitive understanding of the function's behavior. The script describes an animation where numbers move to their respective outputs, illustrating the concept of transformation.
πŸ’‘Cosine and Sine
Cosine and sine are trigonometric functions that are part of the script's example of a multivariable function with a one-dimensional input and a two-dimensional output. The function f(x) = (cos(x), x*sin(x)) is used to illustrate how inputs are transformed into outputs in a two-dimensional space, with cosine and sine determining the position of the output point.
Highlights

The video discusses different ways to visualize multivariable functions, including 3D graphs, contour maps, vector fields, and parametric functions.

The presenter's favorite way to think about functions is as a transformation, mapping inputs to outputs.

Functions can be visualized abstractly with an input space and output space represented as 'blobs'.

Visualization methods like graphs and contour maps associate input-output pairs.

The concept of transformation involves watching points in the input space move to the output space.

A simple example is given with a one-dimensional function f(x) = x^2 - 3.

Instead of a graph, the function is visualized by showing how inputs like 0, 1, and 3 move to their corresponding outputs.

An animation demonstrates the movement of numbers from the input to the output space.

For single-variable functions, this method provides a clear sense of how inputs transition to outputs.

The video then explores multivariable functions with a one-dimensional input and a two-dimensional output.

A specific function f(x) = (cos(x), x*sin(x)) is introduced to illustrate multivariable transformations.

Examples are given for inputs like 0 and Ο€, showing their movement to the points (1,0) and (-1,0) respectively.

The transformation process is visualized for the multivariable function, showing how each element of the input space moves.

The presenter notes that this method helps to understand if the space gets stretched or squished during the transformation.

The transformation visualization is compared to a parametric plot, highlighting the retention of input information.

The video concludes with a teaser for the next video, which will discuss transformations for functions with two-dimensional inputs and outputs.

Transcripts

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Multivariable FunctionsVisualizationTransformationGraphsContour MapsVector FieldsParametric FunctionsInput-OutputAnimationMathematicsEducational