Abstract
The paper presents analyses of multivariable calculus learning-teaching phenomena through the lenses of DNR-based instruction, focusing on several foundational calculus concepts, including cross product, linearization, total derivative, Chain Rule, and implicit differentiation. The analyses delineate the nature of current multivariable calculus instruction and theorize the potential consequences of the two types of instruction to student learning. The following are among the consequences revealed: understanding the concept of function as a covariational processes, and the concept of derivative as a linear approximation; understanding the relation between composition of functions and their Jacobian matrices; understanding the idea underlying the concept of parametrization and the rationale underpinning the process of implicit differentiation; thinking in terms of structure, appending external inputs into a coherent mental representation, and making logical inferences.
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Notes
While this reasoning trajectory holds the gist of the IMF theorem, instructors may choose a more precise treatment; namely, there exist unique neighborhoods \({N}_{u}\) of \(u\) and \({N}_{v}\) of \(v\) and a function \(\phi\): \({N}_{u}\to {N}_{v}\) such that \(F\left(x,y,z\right)=0\), \((x,y)\epsilon {N}_{u}\), and \(v\epsilon {N}_{v}\) if and only if \(z=\phi (x,y)\).
The relationships are expressed in the following theorem:
Let \(F:{R}^{n}\to R\) be a differentiable function for which \(\frac{\partial F}{\partial {x}_{n}}\left({\varvec{a}}\right)\ne 0\), then by the IFT \(F=0\) defines \({x}_{n}\) as a differentiable function \(\phi\). The partial derivatives of \(\phi\) are: \(\frac{\partial \phi }{\partial {x}_{i}}\left({x}_{1}, \dots , {x}_{n-1}\right)=-\frac{\frac{\partial F}{\partial {x}_{i}}\left({\varvec{a}}\right)}{\frac{\partial F}{\partial {x}_{n}}\left({\varvec{a}}\right)}\) for \(i=1, 2, \dots , n-1\).
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Harel, G. The learning and teaching of multivariable calculus: a DNR perspective. ZDM Mathematics Education 53, 709–721 (2021). https://doi.org/10.1007/s11858-021-01223-8
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DOI: https://doi.org/10.1007/s11858-021-01223-8