This blog post explains the calculus of inverse functions through a geometric lens, focusing on the Legendre transform. It illustrates how the derivative of a function relates to the derivative of its inverse by visualizing the tangent lines to both curves. Because the graph of an inverse function is simply the original function reflected across the line y=x, their tangent lines at corresponding points are also reflections. This reflection swaps the roles of rise and run, demonstrating why the derivative of the inverse is the reciprocal of the original function's derivative at corresponding points. The post then introduces the Legendre transform as a way to characterize a function by its tangent lines, connecting it to the concept of duality and setting the stage for future exploration of its applications in physics and optimization.
This blog post, titled "Geometrically understanding calculus of inverse functions (2023)," delves into the relationship between a function and its inverse, specifically focusing on how their derivatives interact and the geometric implications of this relationship. The author begins by establishing the foundational concept of an inverse function, reminding the reader that it essentially reverses the input-output mapping of the original function. This reversal is visually represented by reflecting the function's graph across the line y=x. This reflection is crucial for understanding the subsequent exploration of derivatives.
The core of the post lies in demonstrating the reciprocal relationship between the derivatives of a function and its inverse. It meticulously explains that the derivative of the inverse function at a particular point is the reciprocal of the derivative of the original function evaluated at the corresponding "swapped" point (i.e., where the x and y values are interchanged). This reciprocal relationship isn't simply stated; rather, it is meticulously derived using both algebraic manipulation of the defining equation of an inverse function and a compelling geometrical argument. The geometric argument leverages the aforementioned reflection across the line y=x. By considering the tangent lines to both the function and its inverse at corresponding points, the author shows that the reflection process effectively swaps the rise and run of these tangent lines, resulting in reciprocal slopes, and therefore reciprocal derivatives.
The author then elaborates on the visual interpretation of this derivative relationship, explaining how the slope of the tangent line to the original function translates to the slope of the tangent line to the inverse function after reflection. This visualization reinforces the reciprocal nature of the derivatives and provides a deeper, more intuitive understanding than a purely algebraic approach.
Further extending this geometric perspective, the post touches upon the concept of the Legendre transform, highlighting its connection to the duality between points and lines. It hints at how the Legendre transform can be understood through the lens of supporting lines to a convex function and how this relates to the geometry of inverse functions. However, this discussion is primarily introductory, serving more as a teaser for future exploration than a comprehensive explanation. The post concludes by suggesting that this geometric understanding of inverse functions and their derivatives forms a foundational basis for appreciating the Legendre transform and its applications in areas like physics and optimization, promising a more detailed exploration of this connection in subsequent posts.
Summary of Comments ( 4 )
https://news.ycombinator.com/item?id=43895852
HN users generally praised the clarity and visual approach of the blog post explaining inverse functions and the Legendre transform. Several appreciated the geometric intuition provided, contrasting it with more abstract or algebraic explanations they'd encountered previously. One commenter suggested the post could be improved by clarifying the relationship between the Legendre transform and convex conjugate functions. Another highlighted the connection to supporting hyperplanes, offering additional geometric insight. Some users mentioned the practical applications of the Legendre transform in fields like physics and machine learning, further emphasizing the value of the explanation. A few commenters engaged in a brief discussion about the notation used in the post and alternative conventions.
The Hacker News post discussing the blog post "Geometrically understanding calculus of inverse functions" has a modest number of comments, offering a few different perspectives.
One commenter appreciates the clear visual explanation provided by the blog post, specifically mentioning the effective use of color-coded diagrams to illustrate the relationship between the derivative of a function and its inverse. They highlight how this visual approach aids in understanding the concept beyond rote memorization of the formula.
Another commenter builds upon this appreciation for the visual approach by suggesting an interactive element. They propose that allowing users to manipulate the graph and observe the changes in the derivative and inverse function's derivative in real-time would further enhance the learning experience. This interactivity, they argue, would provide a more intuitive grasp of the underlying principles.
A separate thread of discussion arises concerning the Legendre transform, which is touched upon in the blog post as a future topic. One commenter points out the significance of the Legendre transform in various fields like physics and thermodynamics, expressing anticipation for the author's subsequent post on the subject. Another commenter contributes to this thread by providing a link to a resource that offers a different perspective on the Legendre transform. This resource, they suggest, might complement the author's geometric approach and offer a broader understanding of the concept.
Finally, there's a brief exchange about the notation used for the inverse function. While acknowledging the common usage of f⁻¹(x), one commenter expresses a preference for g(x) to avoid potential confusion with 1/f(x). This preference, they explain, stems from encountering contexts where such confusion could arise. Another commenter responds by pointing out the widespread acceptance of f⁻¹(x) as the standard notation for the inverse function, implying that while alternative notations exist, the established convention is unlikely to change.
In summary, the comments on the Hacker News post primarily commend the blog post's visual clarity in explaining the calculus of inverse functions. They also express interest in the forthcoming discussion of the Legendre transform and briefly touch upon notational conventions for inverse functions. While not extensive, the comments provide valuable feedback and demonstrate engagement with the concepts presented in the blog post.