JMM Mini4- Sydney Opera House Part 2
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This document provides a summary of Part IIb of a minicourse on elementary mathematics and architecture focusing on the Sydney Opera House. It includes diagrams of the building's design with labeled segments. Additionally, it covers curvature and combinatorics of surfaces, defining Gaussian curvature and discussing properties of geodesic triangles on surfaces, including a formula for calculating triangle area using edge lengths and excess angle sum.
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The document contains 15 multiple choice questions about lines, planes, triangles, and other geometric concepts. Some key details are:
- Questions ask about finding equations of lines and planes, properties of triangles like circumscribed circles and distances between points, and determining coordinates of missing vertices.
- Geometry concepts tested include lines, planes, triangles, parabolas, progressions, and distances.
- The answer key is provided at the end listing the correct choice for each question.Ip entrance test paper 1
Ip entrance test paper 1APEX INSTITUTE
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At Apex Institute, we endeavour to amalgamate the ancient values of uttermost dedication and responsibility that gave teaching a very pious and sacred status, with latest teaching methodologies so as to make the act of learning, a delectable experience. The arduous task is accomplished by a team of diligent academicians and entrepreneurs, who with the years of experience have mastered the techniques to transform even the most abstruse topics easy to comprehend.
Great emphasis is laid on the concepts and fundamentals with a smooth and systematic transition to the advanced levels of teaching. The interaction between students and teachers is always encouraged so as to provide a very healthy environment for growth and learning. During the process of training the students for IIT-JEE/Engg./Medical entrance examinations, great emphasis is laid on the fundamentals so as to excel in the CBSE examination also.
Illustrious faculty and excellent teaching techniques provide you will an infallible approach to success only at Apex Institute.
Our Mission
Generating excellence both in academic and competitive aspects thereby training the youth for most gruelling entrance level exams viz. IIT-JEE/Engg.& Medical entrance.
Our Vision
Generating excellence in academic, exploring creative areas of the dormant self, nurturing talent & aptitude & including perseverance among the students to keep themselves abreast of the latest developments in the field of competitive examinations.
Our Objectives
1. Striving for the best possible education to ambitious students aspiring to join corporate world as medical professionals.
2. Creating a comprehensive understanding of the subjects & help developing strategies to meet arduous competitions among the students to match, excel & ameliorate the competition oriented education.
reas 2
reas 2KalculosOnline
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This document contains 20 multiple choice questions about calculating areas of geometric shapes such as triangles, circles, sectors, trapezoids, and polygons. The shapes are often comprised of or related to other geometric elements like diameters, radii, chords, tangents. Questions involve using properties of shapes, trigonometric functions, and formulas to determine the area based on given measurements or relationships between elements of the figures. An answer key is provided at the end listing the correct choice for each question.Isometric projections for engineering students
Isometric projections for engineering studentsAkshay Darji
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The document discusses isometric projections and isometric drawing. It begins by explaining the limitations of orthographic views and how isometric projections show all three dimensions of an object in a single view. It then defines the principles and types of projection, including orthographic, pictorial, axonometric, isometric, dimetric and trimetric. The remainder of the document focuses specifically on isometric projection, defining isometric axes, lines, planes and drawings. It provides examples of how to construct isometric views of various objects from their orthographic projections.IJCER (www.ijceronline.com) International Journal of computational Engineerin...
IJCER (www.ijceronline.com) International Journal of computational Engineerin...ijceronline
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The document summarizes research on the stability derivatives in pitch and roll of an oscillating supersonic delta wing with straight leading edges. A piston theory is used to obtain closed-form solutions for the stiffness and damping derivatives in pitch, as well as the roll damping derivative. Results show the derivatives vary with parameters like Mach number, incidence angle, and pivot position. Comparisons are made to other studies, showing good agreement for stiffness derivative but disagreement for damping derivative at high angles, possibly due to shock wave detachment. The roll damping derivative decreases with Mach number initially before becoming independent, and increases with aspect ratio.Line integral.ppt
Line integral.pptMichaelTegegn
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This document discusses line integrals and Green's theorem. It defines line integrals as integrals of scalar or vector fields along a curve, parameterized by arc length. Line integrals may depend on the path taken between two points, but are path-independent for conservative vector fields. Green's theorem relates line integrals around a closed curve to a double integral over the enclosed region, equating the line integral to the curl of the vector field integrated over the region. An example demonstrates using Green's theorem to evaluate a line integral as a double integral.Fundamentos
FundamentosKalculosOnline
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The document contains 15 multiple choice questions related to geometry. The questions cover topics like angles, triangles, circles, and quadrilaterals. Some questions provide diagrams to accompany the problem statements. The last part of the document lists the answers to each question in the form of a key or answer grid.All in one page trigo
All in one page trigozabidah awang
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1) Angles can be measured in degrees, minutes, or radians. Trigonometric functions relate to the sides of a right triangle and depend on the angle of rotation.
2) Positive angles are measured clockwise from the positive x-axis, negative angles counterclockwise.
3) The value of a trig function for any angle can be determined using a calculator, right triangles, or trig identities involving reference angles.犖犖迦牽犢犖犖ム犖犖迦犢仰犖狆材顕犖犖犖巌
犖犖迦牽犢犖犖ム犖犖迦犢仰犖狆材顕犖犖犖巌Ruangrit Ritruangdej
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The document discusses different geometric transformations including translation, reflection, and rotation. Translation moves every point of a figure by the same displacement vector. Reflection produces a mirror image of a figure across a line. Rotation turns a figure about a fixed point called the center of rotation through a given angle.JMM Mini4- Sydney Opera House Part 2
- 1. Elementary Mathematics and Architecture MAA Minicourse, Boston, January 4- 8, 2012 Alexander Hahn University of Notre Dame Part IIb: The Sydney Opera House (continued)
- 4. B C B C A A O O A1 A2 B B C C A A O O A3 A4
- 6. 0 100 200 300 feet
- 7. C C B B A (a) O A (b)
- 8. rib number 12 ridge beam segment top cut-off segment standard rib segment standard rib segment standard rib segment 246 foo t ra d iu s standard rib segment standard rib segment pedestal tie beam column (a) (b) standard rib segment number 7 (c)
- 11. Curvature and Combinatorics of Surfaces
- 15. M x S
- 16. Gauss map unit normals U(q) U(q) U(p) U(p) A q p S on the surface the unit sphere
- 17. M x S The Gaussian curvature of the surface at x is Area A K(x) = l i m S 0 Area S (Consider a flat piece of surface on M)
- 22. h r R
- 24. c(T) = V - E + F = 1 C g T a A b B
- 26. a T b g M A geodesic triangle T on M has area R 2 (a + b + g - p)