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Computed Tomography in Multivariable Calculus (UMAP)

Yves Nievergelt

 Mathematics Topic:Calculus 3 Application Areas:Medicine, imaging Prerequisites:In general: the ability to ponder a problem that does not come with a canned recipe for its solution. In particular, from multivariable calculus: limits and partial derivatives of elementary (rational, trigonometric, or exponential) functions, integrals along lines in the plane and along lines or planes in space, changes and permutations of variables in multivariable integrals, dot products, and small linear systems.

| ©1997 by COMAP, Inc. | Tools for Teaching 1996 | 58 pages |

This module seeks to: 1) strengthen students' intuition in multidimensional geometry; 2) consolidate students' command of multivariable integrals; 3) provide exercises at a level between mechanical and theoretical; and 4) show applications of calculus that are important but rarely appear in calculus texts. This module provides exercises with line integrals and multivariable integrals for use in multivariable calculus courses, at a level intermediate between mechanical problems and abstract proofs. The exercises do not require any background on any applications.

INTRODUCTION

PHYSICAL MODEL OF CT SCANNERS

THE ALGEBRAIC RECONSTRUCTION TECHNIQUE (ART)
Reconstructions from Projections Are Already in Linear Algebra
Planar ART
Spatial ART

The Set of All Straight Lines in the Plane
Integrals of Functions over Straight Lines

APPROXIMATE IDENTITIES
Weight Functions with One Dimension
Approximate Identities with One Dimension
Approximate Identities with Several Dimensions

Filtered Back-Projection

FURTHER TOPICS
Fan-Beam Tomography
Nuclear Magnetic Resonance (NMR)
Historical Notes on Computed Tomography

SAMPLE PROBLEMS FOR EXAMINATIONS

SOLUTIONS TO THE ODD-NUMBERED EXERCISES

SOLUTIONS TO THE SAMPLE PROBLEMS FOR EXAMINATIONS

REFERENCES

ACKNOWLEDGMENTS