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Freedom, Constraint And Control In Multivariable Calculus

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2001 Annual Conference


Albuquerque, New Mexico

Publication Date

June 24, 2001

Start Date

June 24, 2001

End Date

June 27, 2001



Page Count


Page Numbers

6.508.1 - 6.508.14

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Paper Authors

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Andrew Grossfield

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NOTE: The first page of text has been automatically extracted and included below in lieu of an abstract

Session 1665

Freedom, Constraint and Control in Multivariable Calculus

Andrew Grossfield College of Aeronautics


Certainly, everyone interested in technology should possess an understanding of the models of deterministic, continuous multivariable control. The study of multivariable calculus can be viewed as a natural extension of the unfortunately named "single variable" calculus. Ordinary "single variable" calculus is the study of equations in two variables, F(x, y) = 0. Equations in two variables provide mechanisms for studying: a) continuous curves in the two dimensional plane, b) continuous models of control (where one variable controls a second variable), c) continuous models of time variation (signals, trends or evolution) or d) situations where two variables track together continuously. These functions and curves of ordinary calculus are real entities having observable and predictable properties.

What happens when a single variable depends on or is controlled by several other variables? How do we visualize and treat the situation when several variables depend on or are controlled by a single variable? What if several variables control several other variables? How will small changes in the controlling variables affect the controlled variables?

This paper begins with a discussion of the importance of counting variables at the beginning of multivariable problems, proceeds to a classification of the entities of multivariable calculus and continues to describe the concept of the derivative as it pertains to each of the entities of multivariable calculus.

Similar to the functions of calculus, the objects of multivariable calculus, including space curves, warped surfaces and models of multivariable control are also real and also have observable and predictable properties. Every additional variable provides an additional degree of freedom by adding a dimension to the space spanned by the collection of variables. Every additional equation provides a constraint, which removes a dimension from the space spanned by the variables.

Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition Copyright ! 2001, American Society for Engineering Education

Grossfield, A. (2001, June), Freedom, Constraint And Control In Multivariable Calculus Paper presented at 2001 Annual Conference, Albuquerque, New Mexico.

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