INTRODUCTION The chain rule in multivariable calculus is often taught in a symbolic and memorization-based manner. However, for early learners and students, this can obscure the intuitive flow of variable dependencies. In this paper, I propose a new visual method, the _Junyeong Ha Diagram_, which structures composite functions from right to left and interprets differentiation as a stepwise movement of fraction-like paths through variable layers. CLASSICAL CHAIN RULE In a standard multivariable case, given: \[ z = f(x, y), \quad x = g(u, v), \quad y = h(u, v) \] the chain rule for partial derivatives is: \[ {\partial u} = {\partial x} \cdot {\partial u} + {\partial y} \cdot {\partial u} \] This formulation, while mathematically correct, provides little structural insight into the dependency paths. THE JUNYEONG HA DIAGRAM Instead of symbolic abstraction, if z=f(x,y), x=(u,v) y=(u,v), the Junyeong Ha Diagram visualizes the functional dependencies as: \[ z \rightarrow (x, y) \rightarrow (u, v) \] Here, the variable on the far right (e.g., u) is the DENOMINATOR, initiating the differentiation. Each path from u to z via intermediate variables (e.g., x or y) represents a product of partial derivatives. The rule is: - Fix the input variable (e.g., u) as the denominator. - Trace all paths leading from u to z. - Multiply each partial derivative along each path. - Sum the products over all valid paths. \[ {\partial u} = \left({\partial x} \cdot {\partial u} \right) + \left( {\partial y} \cdot {\partial u} \right) \] EXAMPLE Given: \[ x = u^2 + v, \quad y = uv, \quad z = x^2 + y^3 \] Compute: \[ {\partial u} = \left(2x \cdot 2u\right) + \left(3y^2 \cdot v\right) \] EDUCATIONAL IMPLICATIONS Stewart’s calculus textbook draws a lot of tree diagrams to explain the chain rule, but I think they’re too complicated. My method is way simpler and more intuitive. You just look at the variables from left to right, fix the one on the far right as the denominator, and trace the paths back. That’s it. It makes partial derivatives much easy to understand and calculate without memorizing formulas. CONCLUSION The Junyeong Ha Diagram presents an intuitive, visual approach to understanding the multivariable calculus chain rule. It encourages flow based reasoning and simplifies the analysis of composite function derivatives. This structure has the potential to assist in both education and algorithmic differentiation systems. REFERENCES - Stewart, J. (2020). _Calculus, 9/E (metric version)_
MOTIVATION When I read a book on general relativity, I learned that when a star becomes a black hole, it collapses and its radius shrinks, forming an event horizon. But I had a question: _Where does that collapsed radius go?_ After thinking deeply, I came up with an idea — maybe the collapsed radius doesn’t just disappear, but forms new space inside the black hole. If this idea is true, it could be related to an interesting hypothesis: what if our universe exists inside a black hole? INTRODUCTION In classical general relativity, the gravitational collapse of a massive star leads to the formation of a black hole. As the star shrinks under its own gravity, its radius decreases toward the Schwarzschild radius. At this stage, an event horizon forms, and according to classical understanding, all matter collapses toward a singularity. However, what happens to the “space” that once existed inside the star? Does it disappear? Or could it, under the right description of spacetime, be transformed into a different structure? CONCEPTUAL HYPOTHESIS We propose that as the star collapses and its radius shrinks, the spatial region that the star once occupied is not annihilated. Instead, that spatial information may be encoded or “re-created” inside the event horizon as a new internal space. This is not to say that a volume is preserved in a Euclidean sense, but rather that spacetime curvature and topology might rearrange in such a way that the interior structure becomes larger than suggested by the external Schwarzschild radius. This idea resonates with some speculative models in theoretical physics, such as: - The hypothesis that our universe could lie inside a black hole - Bounce cosmologies where singularities are avoided - The concept of emergent geometry in quantum gravity QUALITATIVE FRAMEWORK Assume a star of mass M collapses into a Schwarzschild black hole with radius $r_s = {c^2}$. The spatial volume occupied by the original star (V₀) becomes smaller as the radius contracts. However, we hypothesize that the collapse reorganizes the topology of spacetime such that a new volume Vin forms inside the black hole: \[ V_{} \sim f(V_0, r_s) \] where f is an unknown mapping potentially defined by a future quantum gravity theory. CONCEPTUAL VISUALIZATION OF INTERNAL SPACE GENERATION When a massive star collapses under its own gravity and passes within its Schwarzschild radius, it forms a black hole. The standard interpretation of general relativity suggests that anything inside the event horizon becomes causally disconnected from the outside universe. However, the fate of the “lost” volume — the space the star originally occupied — is not directly addressed. In this study, we hypothesize that the collapsing matter does not destroy the original volume, but instead reorganizes the topology of spacetime, forming an internal region within the black hole that preserves or transforms the spatial volume into a new structure. Assume a star of initial volume \( V_0 \) and mass \( M \) collapses to a Schwarzschild black hole with radius: \[ r_s = {c^2} \] We propose that an internal volume \( V_{} \) is created according to an unknown mapping: \[ V_{} \sim f(V_0, r_s) \] where \( f \) is a function to be determined by a future theory of quantum gravity or topological extension of spacetime. This idea resonates with certain cosmological models that posit our universe as residing inside a black hole. The core hypothesis encourages new ways to think about spacetime structure under extreme gravitational collapse. DISCUSSION Although this idea is speculative and lacks a concrete metric description, it suggests a new way of interpreting black hole interiors. Instead of a singular point, the core of the black hole might contain extended structure—possibly even large enough to host an entire universe, consistent with some cosmological models. CONCLUSION This work presents a conceptual perspective on gravitational collapse, where the shrinking of stellar radius gives rise to an emergent inner space. While the idea is not formalized mathematically, it provides an intuitive motivation for deeper studies in quantum gravity and black hole topology. REFERENCES - S. Carroll, _Spacetime and Geometry_, Addison-Wesley (2004) - L. Susskind, _The Black Hole War_, Little, Brown (2008) - C. Rovelli, _Planck Stars_, Int. J. Mod. Phys. D 23, 1442026 (2014) - M. Boas, _Mathematical Methods in the Physical Sciences_, Wiley (2006)