# Results for: “Mathematics”

Title | Author | Publisher | Format | Buy | Remix | ||
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## Multiplication and Division: Decimals: SSAT-ISEE Arithmetic |
Ace Academics | Ace Academics | ePub | ||||

## Similar Polygons: GRE Geometry |
Ace Academics | Ace Academics | ePub | ||||

## 13 |
Parmananda Gupta | Laxmi Publications | |||||

148 DIFFERENTIAL GEOMETRY AND CALCULUS OF VARIATIONS Example 4. Find the extremal of the functional z a −a (λy + 1 2 µy″2) dx which satisfy the boundary conditions y(– a) = 0, y′(– a) = 0, y(a) = 0, y′(a) = 0. z FGH IJ K 1 µy″ 2 dx . −a 2 1 2 Let F(x, y, y′, y″) = λy + µy″ 2 d d2 Fy ′ + F =0 The Euler-Poisson equation is Fy − dx dx 2 y ″ Here Fy = λ, Fy′ = 0, Fy″ = µy″ Sol. Given functional is ∴ (1) λ− ⇒ a λy + d d2 (0) + (µy″ ) = 0 ⇒ dx dx 2 µ d4 y dx 4 +λ =0 ⇒ ...(1) d4 y λ =– 4 µ dx d3 y λ = − x + c1 3 µ dx ⇒ d2y ⇒ dx 2 =− λ x2 + c1x + c2 µ 2 dy λ x3 x2 =− + c1 + c2x + c3 2 dx µ 6 ⇒ λ x4 x3 x2 + c3x + c4 + c1 + c2 6 2 µ 24 This is the equation of the extremals. The boundary conditions are y(– a) = 0, y′(– a) = 0, y(a) = 0, y′(a) = 0 ⇒ y= − y(– a) = 0 ⇒ − λa4 c1a 3 c2 a2 – c3a + c4 = 0 − + 24µ 6 2 − ...(4) λa4 c1 a3 c2 a2 + + + c3a + c4 = 0 24µ 6 2 ...(5) λa3 c1a 2 + + c2a + c3 = 0 6µ 2 ...(6) − y′(a) = 0 ⇒ c1 a3 – 2c3a = 0 ⇒ c1a2 + 6c3 = 0 3 (4) + (6) ⇒ c1a2 + 2c3 = 0 Solving (7) and (8), we get c1 = 0, c3 = 0 (3) – (5) ⇒ (4) (3) ⇒ – λa 3 + 0 – c2a + 0 = 0 6µ ⇒ − λa 4 a2 −0+ 24µ 2 ...(3) λa 3 c1 a2 – c2a + c3 = 0 See All Chapters |
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## Medians, Altitudes, Bisectors: GED Geometry |
Ace Academics | Ace Academics | ePub | ||||

## Properties of Numbers: GED Algebra |
Ace Academics | Ace Academics | ePub | ||||

## Simultaneous Equations: SAT Math II Algebra I |
Ace Academics | Ace Academics | ePub | ||||

## Sets: GRE Algebra |
Ace Academics | Ace Academics | ePub | ||||

## 7 |
Parmananda Gupta | Laxmi Publications | |||||

73 CURVATURE AND TORSION The centre of the osculating circle at point P is called the centre of curvature of the curve C at the point P. By the definition of contact between curves, the osculating circle to the curve C at point P can be considered as the intersection of a sphere with at least 3-point contact with the curve C at point P and a plane with at least 3-point contact with C at P. If κ ≠ 0 at P, then the osculating plane at P is the unique plane having at least 3-point contact with the curve C at P. In particular, if τ ≠ 0 in addition to κ ≠ 0 at P, then the osculating plane is the unique plane having exactly 3-point contact with C at P. Therefore the osculating circle to a curve at a point always lies on the osculating plane to the curve at that point, provided κ ≠ 0 at the point under consideration. Thus, the osculating circle to a curve at a point can be considered as the intersection of a sphere with at least 3-point contact with the curve at that point and the osculating plane to the curve at the point under consideration, provided κ ≠ 0. See All Chapters |
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## Simplify Fractions: GED Arithmetic |
Ace Academics | Ace Academics | ePub | ||||

## app-1 |
N.P.Bali; P.N.Gupta; Dr C.P.Gandhi | Laxmi Publications | |||||

I APPENDIX Tables Table I : Inverse Quantities, Powers, Roots, Logarithms x 1/x 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 1.000 0.909 0.833 0.769 0.714 0.667 0.625 0.588 0.556 0.526 0.500 0.476 0.454 0.435 0.417 0.400 0.385 0.370 0.357 0.345 0.333 0.323 0.312 0.303 0.294 0.286 0.278 0.270 0.263 0.256 0.250 0.244 0.238 0.233 0.227 0.222 0.217 0.213 0.208 0.204 0.200 0.196 0.192 0.189 0.185 x2 1.000 1.210 1.440 1.690 1.960 2.250 2.560 2.890 3.240 3.610 4.000 4.410 4.840 5.290 5.760 6.250 6.760 7.290 7.840 8.410 9.000 9.610 10.24 10.89 11.56 12.25 12.96 13.69 14.44 15.21 16.00 16.81 17.64 18.49 19.36 20.25 21.16 22.09 23.04 24.01 25.00 26.01 27.04 28.09 29.16 x3 1.000 1.331 1.728 2.197 2.744 3.375 4.096 4.913 5.832 6.859 8.000 9.261 10.65 12.17 13.82 15.62 17.58 19.68 21.95 24.39 27.00 29.79 32.77 35.94 39.30 42.88 46.66 50.65 54.87 59.32 64.00 68.92 74.09 79.51 85.18 91.12 97.34 103.8 110.6 117.6 125.0 132.7 140.6 148.9 157.5 x 1.000 1.049 1.095 1.140 1.183 1.225 1.265 1.304 1.342 1.378 1.414 1.449 1.483 1.517 1.549 1.581 1.612 1.643 1.673 1.703 1.732 1.761 1.789 See All Chapters |
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## Chapter 2 Designing Conceptual Units in Mathematics |
Chris Weber | Solution Tree Press | ePub | ||||

This chapter is about When designing units of instruction, our goal must be mastery, not coverage—depth of understanding, not breadth of topics addressed—and we will describe just such a process for building units of instruction. First, how should units be organized and sequenced? We recommend that units be coherently organized, both horizontally within a grade level and vertically between grade levels. Since a sense of number is the basis for all mathematics, we recommend that standards and learning targets that build students’ sense of number be frontloaded within a grade level’s instructional year. Within grade levels, we recommend that topics such as addition and subtraction, graphing, and algebra be included within individual units and the units that are adjacent to them to allow for more continuity of teaching and learning and greater depth of study. Between grade levels, coherence of topics allows for collaboration between grade levels and Tier 2 intervention, which is particularly relevant in smaller schools where there are one or two teachers per grade level. See All Chapters |
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## Addition: Fractions: GED Arithmetic |
Ace Academics | Ace Academics | ePub | ||||

## Conic Sections: CLEP Precalculus |
Ace Academics | Ace Academics | ePub | ||||

## Variables: CLEP Algebra I |
Ace Academics | Ace Academics | ePub | ||||

## 9 |
Parmananda Gupta | Laxmi Publications | |||||

3 Surfaces in Space 1. INTRODUCTION A sphere, a portion of a cylinder are examples of surfaces in space. Till now we have been discussing the differential geometry of curves in space. In the present chapter , we shall learn few quantities regarding surfaces in space. 2. SURFACE IN SPACE We know that a curve in space is the locus of a point whose coordinates x, y, z are functions of a single parameter. On the same lines, we shall define a surface in space as the locus of a point whose coordinates are functions of two independent parameters. A surface in space is defined as the locus of a point whose position vector relative to a fixed origin may be expressed as a function of two independent parameters. P(x, y, z) Thus, a surface S in space may be represented by a S vector function r(u, v) = x(u, v)i + y(u, v)j + z(u, v)k, r(u, v) where u and v are independent parameters such that (u, v) ∈ R, where R is some region in the uv-plane. O(origin) Here r(u, v) is the position vector of the point P on the v surface S and (x(u, v), y(u, v), z(u, v)) are the cartesian coordinates of the point P. The above representation of a surface is called a parametric representation of a surface and is due to Gauss. Also, (u, v) are called the curvilinear coordinates of the point P. In particular cases, See All Chapters |