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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.
Table of Contents:
INTRODUCTION
PHYSICAL MODEL OF CT SCANNERS
THE ALGEBRAIC RECONSTRUCTION TECHNIQUE (ART)
Reconstructions from Projections Are Already in Linear Algebra
Planar ART
Spatial ART
RADON'S TRANSFORM
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
INVERSION OF THE RADON TRANSFORM
The Adjoint Radon Transform
Filtered BackProjection
FURTHER TOPICS
FanBeam Tomography
Nuclear Magnetic Resonance (NMR)
Historical Notes on Computed Tomography
SAMPLE PROBLEMS FOR EXAMINATIONS
SOLUTIONS TO THE ODDNUMBERED EXERCISES
SOLUTIONS TO THE SAMPLE PROBLEMS FOR EXAMINATIONS
REFERENCES
ACKNOWLEDGMENTS
ABOUT THE AUTHOR



