Envisioning Tensors and Tensor Equation Invariance ©
Ronald D. Kriz, Assoicate Professor
Department of Engineering Science and Mechanics, and
Director of the University Visualization and Animation Group
Virginia Polytechnic Institute and State University
Mecit Yaman, Ph.D Graduate Candidate
(Advisor: Professor M. Harting)
Department of Physics
University of Cape Town, South Africa

Tensors and their invariant equations have been used in the applied sciences to mathematically model complex multivariate relations in a concise and simple format. Often applied scientists envision these multivariate and complex relationships as "visual mental models" in a glyph format. These visual mental models are more or less clear psychical images that convey a cognitive understanding of the physical model being studied, e.g. gradients in space and time of tensoral properties convey our fundamental idea of a comoving derivative of that property where these gradients are often imagined in the "minds eye" as visual objects (glyphs) whose shapes and colors represent tensoral components that change with space and time. Because many of these ideas are common to each individuals' visual creative cognitive process, these images can be shared with others as images because of recent advances in computer graphics. Here we explore how recent advances in computer graphics can capture this visual cognitive process and communicate some of the more fundamental ideas in mechanics and the applied sciences. With regard to the fundamental idea of tensor equation invariance, it is demonstrated that the idea of quantitative mathematical invariance associated with tensor equations can be used to qualitatively envision the same invariance associated with physical laws in a glyph format. Consequently envisioning invariance enables scientists to see and understand the qualitative content of equations associated with physical laws, e.g. equilibrium in Cauchy's equation was envisioned and understood graphically as a glyph at a point or 3D gradient of glyphs surrounding that point. These concepts are developed and envisioned by creating eigenvalue-eigenvector glyphs, that is one of three visual methods used when envisioning scientific information.

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Zeroth Order Tensors and
Tensor Equation Invariance
Envisioning Scalar Gradients ©:

Extracting Relationships
From Scalar Functions ©

Second Order Tensors and
Tensor Equation Invariance
Sij Definitions and Refs.

Fourth Order Tensors and
Tensor Equation Invariance
NSF Visualization Contest

ICCES03 Pub & Pres

Creating σij Glyphs / README
Example-1: Four σij Glyphs: PNS, HWY, Reynolds, and Quadric
Example-2: Residual Stress Gradients
Creating Cijkl Glyphs / README
Example: Calcium Formate

Send comments to: rkriz@vt.edu
Ronald D. Kriz, Short Bio
Engineering Science and Mechanics
College of Engineering
Virginia Tech
Blacksburg, Virginia 24061

Content Chronology:
Created April 2005
Moved to ESM Dept Web-server September 2008
Moved to Off-Campus Web-server November 2014

DVD: ESM Content