Right triangle

Abstract: An integral equation technique is employed to obtain eigenvalues and eigenmodes of the homogeneous Holmholtz equation for a two‐ or three‐dimensional closed region of arbitrary shape with arbitrary first‐order homogeneous boundary conditions. The method is described for general (i.e., nonseparable) geometries, with a discussion of the simplification introduced by having a separable geometry given in an Appendix. A numerical example is given for a nonseparable geometry, i.e., a two‐dimensional right triangle of arbitrary enclosed angle with Neumann boundary conditions. Results for the special case of an isosceles right triangle agree very well with a known analytical solution.

Abstract: A model of velocity distribution among microchannels with triangle manifolds is proposed. According to the flow behaviors analyzed by Fluent, the manifolds are divided into several approximate rectangular channels, and then an equivalent simplified resistance network model is developed to establish the relationships between the velocities and pressure drops in microchannels and approximate rectangular channels. The velocity distributions are calculated under two situations, respectively, considering and ignoring singular losses. The outcomes of the present study are compared with Fluent's simulated results to analyze the effects of singular losses on the velocity distributions. It indicates that the proposed model is suitable for the calculation of velocity distribution among microchannels with obtuse angled or right triangle manifolds under low Reynolds numbers. The premise of ignoring singular losses is that the frictional pressure drops are three times larger than the singular pressure drops in each flow loop. The manifold optimization results indicate that the velocity distribution among microchannels with right triangle manifolds is much more uniform than that of the corresponding one with obtuse angled manifolds. © 2009 American Institute of Chemical Engineers AIChE J, 55: 1969–1982, 2009

Abstract: --PART I METHODS OF TEACHING SECONDARY MATHEMATICS Chapter 1 The Challenge of Teaching *Today's Students, Mathematics, and Society's Need Chapter 2 Planning for Instruction *Long-Range Planning of the Curriculum *Unit Plans *Short-Range Planning *Differentiated Instruction *Cooperative Learning *Mathematical Tasks *Final Thoughts on Lesson Planning Chapter 3 Teaching More Effective Lessons *Motivational Techniques *Classroom Questioning *Strategies for Teaching More Effective Lessons *Literacy in Mathematics *Writing Chapter 4 The Role of Problem-Solving *A Psychnological View of Problem Solving *Problem-Solving Preliminaries *An Introduction to Problem Solving *The Ten Problem-Solving Strategies *Creating Mathematical Problems *Creativity in Problem Solving Chapter 5 Using Technology to Enhance Mathematics Instruction *Calculators *Computers Chapter 6 Assessment *Assessment for Monitoring Student Progress *Assessment for Making Instructional Decisions *Evaluating Student Achievement Chapter 7 Enriching Mathematics Instruction *Enriching Mathematics Instruction with a Historical Approach *Enrichment Techniques for All Levels *The Gifted Student *Using Calculators to Enrich Instruction *Models and Manipulatives That Enrich Instruction Chapter 8 Extracurricular Activities in Mathematics *The Mathematics Club *Mathematics Teams *Mathematics Contests *Mathematics Projects *The Mathematics Fair *Cooperation with a University *The School Mathematics Magazine *The Mathematics Assembly Program *Guest Speakers Program *Class Trips of Mathematical Significance *Peer Teaching Program *The Computer *The Bulletin Board PART II ENRICHMENT UNITS FOR THE SECONDARY SCHOOL CLASSROOM Cross-Catalogue of Enrichment Units *Constructing Odd-Order Magic Squares *Constructing Even-Order Magic Squares *Introduction to Alphametics *A Checkerboard Calculator *The Game of Nim *The Tower of Hanoi *What Day of the Week Was It? *Palindromic Numbers *The Fascinating Number Nine *Unusual Number Properties *Enrichment with a Handheld Calculator *Symmetric Multiplication *Variations on a Theme--Multiplication *Ancient Egyptian Arithmetic *Napier's Rods *Unit Pricing *Successive Discounts and Increases *Prime and Composite Factors of a Whole Number *Prime Numeration System *Repeating Decimal Expansions *Peculiarities of Perfect Repeating Decimals *Patterns in Mathematics *Googol and Googolplex *Mathematics of Life Insurance *Geometric Dissections *The Klein Bottle *The Four-Color Map Problem *Mathematics on a Bicycle *Mathematics and Music *Mathematics in Nature *The Birthday Problem *The Structure of the Number System *Excursions in Number Bases *Raising Interest *Reflexive, Symmetric, and Transitive Relations *Bypassing an Inaccessible Region *The Inaccessible Angle *Triangle Constructions *The Criterion of Constructibility *Constructing Radical Lengths *Constructing a Pentagon *Investigating the Isosceles Triangle Fallacy *The Equiangular Point *The Minimum-Distance Point of a Triangle *The Isosceles Triangle Revisited *Reflective Properties of the Plane *Finding the Length of a Cevian of a Triangle *A Surprising Challenge *Making Discoveries in Mathematics *Tessellations *Introducing the Pythagorean Theorem *Trisection Revisited *Proving Lines Concurrent *Squares *Proving Points Collinear *Angle Measurement with a Circle *Trisecting a Circle *Ptolemy's Theorem *Constructing pi *The Arbelos *The Nine-Point Circle *The Euler Line *The Simson Line *The Butterfly Problem *Equicircles *The Inscribed Circle and the Right Triangle *The Golden Rectangle *The Golden Triangle *Geometric Fallacies *Regular Polyhedra *An Introduction to Topology *Angles on a Clock *Averaging Rates--The Harmonic Mean *Howlers *Digit Problems Revisited *Algebraic Identities *A Method for Factoring Trinomials of the Form: ax2 + bx + c *Solving Quadratic Equations *The Euclidean Algorithm *Prime Numbers *Algebraic Fallacies *Sum Derivations With Arrays *Pythagorean Triples *Divisibility *Fibonacci Sequence *Diophantine Equations *Continued Fractions and Diophantine Equations *Simplifying Expressions Involving Infinity *Continued Fraction Expansion of Irrational Numbers *The Farey Sequence *The Parabolic Envelope *Application of Congruence to Divisibility *Problem Solving--A Reverse Strategy *Decimals and Fractions in Other Bases *Polygonal Numbers *Networks *Angle Trisection--Possible or Impossible? *Comparing Means *Pascal's Pyramid *The Multinomial Theorem *Algebraic Solution of Cubic Equations *Solving Cubic Equations *Calculating Sums of Finite Series *A General Formula for the Sum of Series of the Form tr *A Parabolic Calculator *Constructing Ellipses *Constructing the Parabola *Using Higher Plane Curves to Trisect an Angle *Constructing Hypocycloid and Epicycloid Circular Envelopes *The Harmonic Sequence *Transformations and Matrices *The Method of Differences *Probability Applied to Baseball *Introduction to Geometric Transformations *The Circle and the Cardioid *Complex-Number Applications *Hindu Arithmetic *Proving Numbers Irrational *How to Use a Computer Spreadsheet to Generate Solutions to Certain Mathematics Problems *The Three Worlds of Geometry *piie Mix *Graphical Iteration *The Feigenbaum Plot *The Sierpinski Triangle *Fractals Appendix Additional Exercises Index About the Authors