428 Chapters
  Title Author Publisher Format Buy Remix
Medium 9781576336984

Multiplication and Division: Decimals: SSAT-ISEE Arithmetic

Ace Academics Ace Academics ePub
Medium 9781576336588

Similar Polygons: GRE Geometry

Ace Academics Ace Academics ePub
Medium 9789380298818

13

Parmananda Gupta Laxmi Publications PDF

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

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

λa 4 a2

−0+

24µ

2

...(3)

λa 3 c1 a2

– c2a + c3 = 0

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Medium 9781576336403

Medians, Altitudes, Bisectors: GED Geometry

Ace Academics Ace Academics ePub
Medium 9781576336397

Properties of Numbers: GED Algebra

Ace Academics Ace Academics ePub
Medium 9781576337783

Simultaneous Equations: SAT Math II Algebra I

Ace Academics Ace Academics ePub
Medium 9781576336571

Sets: GRE Algebra

Ace Academics Ace Academics ePub
Medium 9789380298818

7

Parmananda Gupta Laxmi Publications PDF

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.

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Medium 9781576336380

Simplify Fractions: GED Arithmetic

Ace Academics Ace Academics ePub
Medium 9788131807781

app-1

N.P.Bali; P.N.Gupta; Dr C.P.Gandhi Laxmi Publications PDF

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

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Medium 9781936763313

Chapter 2 Designing Conceptual Units in Mathematics

Chris Weber Solution Tree Press ePub

This chapter is about what we want students to learn and how we will know if they learn it (DuFour et al., 2010). We build the case for common assessments as a vital part of the unit-design process and describe how to craft them. As the key lever of RTI, common assessments establish the target that students and staff are working toward and provide the evidence that can be used to extend learning for some and intervene with others. In the second half of this chapter, we provide an example unit for grade 3.

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.

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Medium 9781576336380

Addition: Fractions: GED Arithmetic

Ace Academics Ace Academics ePub
Medium 9781576337387

Conic Sections: CLEP Precalculus

Ace Academics Ace Academics ePub
Medium 9781576337363

Variables: CLEP Algebra I

Ace Academics Ace Academics ePub
Medium 9789380298818

9

Parmananda Gupta Laxmi Publications PDF

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,

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