The Junyeong Diagram: A Visual Flow-Based Approach to the Multivariable Chain Rule
jun yeong ha · 2025
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)_