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Breaking Multivariable Calculus: Interactive Models Showing Where Our Favorite Formulas Fail.

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Title: Breaking Multivariable Calculus: Interactive Models Showing Where Our Favorite Formulas Fail.
Authors: Paul, Stepan (AUTHOR)
Source: Mathematics Magazine. Jun2026, Vol. 99 Issue 3, p277-283. 7p.
Subject Terms: Multivariable calculus, Three-dimensional printing, Mathematical singularities, Real variables
Abstract: Summary: We present interactive 3D printed models of the graphs of three different functions of two variables that each exhibit some kind of unexpected behavior with respect to a concept learned in multivariable calculus. The interactivity of each model is meant to help a learner make geometric sense of the pathological properties of each function and illustrate the need for the technical conditions for the relevant theorems. [ABSTRACT FROM AUTHOR]
Copyright of Mathematics Magazine is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Education Research Complete
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DbLabel: Education Research Complete
An: 194611117
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PubType: Academic Journal
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  Data: Breaking Multivariable Calculus: Interactive Models Showing Where Our Favorite Formulas Fail.
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  Data: Summary: We present interactive 3D printed models of the graphs of three different functions of two variables that each exhibit some kind of unexpected behavior with respect to a concept learned in multivariable calculus. The interactivity of each model is meant to help a learner make geometric sense of the pathological properties of each function and illustrate the need for the technical conditions for the relevant theorems. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Mathematics Magazine is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1080/0025570X.2025.2509467
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        Text: English
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        Type: general
      – SubjectFull: Three-dimensional printing
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      – SubjectFull: Mathematical singularities
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      – TitleFull: Breaking Multivariable Calculus: Interactive Models Showing Where Our Favorite Formulas Fail.
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              Text: Jun2026
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