Tilted Planes and Curvature in Three-Dimensional Space: Explorations of Partial Derivatives

Andrew Grossfield

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Conference: 2018 ASEE Mid-Atlantic Section Spring Conference
Location: Washington, District of Columbia
Publication Date: April 6, 2018
Start Date: April 6, 2018
End Date: April 7, 2018
Page Count: 25
DOI: 10.18260/1-2--29499
Permanent URL: https://peer.asee.org/29499
Download Count: 1457

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Andrew Grossfield P. E. Vaughn College of Aeronautics & Technology

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Throughout his career Dr. Grossfield has combined an interest in engineering design and mathematics. He studied Electrical Engineering at the City College of New York, graduating with a BSEE. He attended the NYU Courant Institute at night, obtaining an M.S. degree in mathematics, while designing circuitry full time during the day for aerospace/avionics companies. He earned his doctorate studying Continuum Mechanics under the direction of L. M. Milne Thomson, CBE at The University of Arizona. He is a member of ASEE, IEEE, and MAA. Grossfield@IEEE.org is his email address.

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Abstract

Tilted Planes and Curvature in Three-Dimensional Space: Explorations of Partial Derivatives

By

Andrew Grossfield, Ph. D., P. E. Abstract

Many engineering students encounter and algebraically manipulate partial derivatives in their fluids, thermodynamics or electromagnetic wave theory courses. However it is possible that unless these students were properly introduced to these symbols, they may lack the insight that could be obtained from a geometric or visual approach to the equations that contain these symbols.

We accept the approach that just as the direction of a curve at a point in two-dimensional space is described by the slope of the straight line tangent to the curve at that point, the orientation of a surface at a point in 3-dimensional space is determined by the orientation of the plane tangent to the surface at that point.

A straight line has only one direction described by the same slope everywhere along its length; a tilted plane has the same orientation everywhere but has many slopes at each point. In fact the slope in almost every direction leading away from a point is different. How do we conceive of these differing slopes and how can they be evaluated?

We are led to conclude that the derivative of a multivariable function is the gradient vector, and that it is wrong and misleading to define the gradient in terms of some kind of limiting process. This approach circumvents the need for all the unnecessary delta-epsilon arguments about obvious results, while providing visual insight into properties of multi-variable derivatives and differentials.

This paper provides the visual connection displaying the remarkably simple and beautiful relationships between the gradient, the directional derivatives and the partial derivatives. We find that the altitude above the horizontal coordinate plane varies sinusoidally with direction. The properties of multivariable derivatives can be easily grasped in terms of the properties of the orientation or tilt of planes in a 3-dimensional Cartesian coordinate system.

The spotlight is turned on the curvature or deviation from the tangent plane in terms of the classic second degree surfaces that prevails almost everywhere on well-behaved, that is, continuous and smooth (differentiable), warped surfaces. Here too the curvature is found to vary sinusoidally, only at twice the frequency and raised or lowered vertically. We see the significance of that wonderful intrinsic point property of surfaces, the Gaussian curvature and what it reveals about the differences between the curvature at the mountain passes and the curvature of the mountaintops and valleys.

This visual treatment of fundamental mathematical theory should serve as an introduction to precollege students of what lies ahead in their continuing study of mathematics.

Citation
Format

Grossfield, A. (2018, April), Tilted Planes and Curvature in Three-Dimensional Space: Explorations of Partial Derivatives Paper presented at 2018 ASEE Mid-Atlantic Section Spring Conference, Washington, District of Columbia. 10.18260/1-2--29499

TY - CPAPER
AB - Tilted Planes and Curvature in Three-Dimensional Space:
Explorations of Partial Derivatives

Andrew Grossfield, Ph. D., P. E.
Abstract

Many engineering students encounter and algebraically manipulate partial derivatives in their fluids, thermodynamics or electromagnetic wave theory courses. However it is possible that unless these students were properly introduced to these symbols, they may lack the insight that could be obtained from a geometric or visual approach to the equations that contain these symbols.

We accept the approach that just as the direction of a curve at a point in two-dimensional space is described by the slope of the straight line tangent to the curve at that point, the orientation of a surface at a point in 3-dimensional space is determined by the orientation of the plane tangent to the surface at that point.

A straight line has only one direction described by the same slope everywhere along its length; a tilted plane has the same orientation everywhere but has many slopes at each point. In fact the slope in almost every direction leading away from a point is different. How do we conceive of these differing slopes and how can they be evaluated?

We are led to conclude that the derivative of a multivariable function is the gradient vector, and that it is wrong and misleading to define the gradient in terms of some kind of limiting process. This approach circumvents the need for all the unnecessary delta-epsilon arguments about obvious results, while providing visual insight into properties of multi-variable derivatives and differentials.

This paper provides the visual connection displaying the remarkably simple and beautiful relationships between the gradient, the directional derivatives and the partial derivatives. We find that the altitude above the horizontal coordinate plane varies sinusoidally with direction. The properties of multivariable derivatives can be easily grasped in terms of the properties of the orientation or tilt of planes in a 3-dimensional Cartesian coordinate system.

The spotlight is turned on the curvature or deviation from the tangent plane in terms of the classic second degree surfaces that prevails almost everywhere on well-behaved, that is, continuous and smooth (differentiable), warped surfaces. Here too the curvature is found to vary sinusoidally, only at twice the frequency and raised or lowered vertically. We see the significance of that wonderful intrinsic point property of surfaces, the Gaussian curvature and what it reveals about the differences between the curvature at the mountain passes and the curvature of the mountaintops and valleys.

This visual treatment of fundamental mathematical theory should serve as an introduction to precollege students of what lies ahead in their continuing study of mathematics.

AU - Andrew Grossfield P. E.
CY - Washington, District of Columbia
DA - 2018/04/06
PB - ASEE Conferences
TI - Tilted Planes and Curvature in Three-Dimensional Space: Explorations of Partial Derivatives
UR - https://peer.asee.org/29499
DO - 10.18260/1-2--29499
ER -