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Multiplication and Division: Fractions: GED Arithmetic

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Sets: PSAT Algebra

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Medium 9781576336588

Circles - Regular Polygons: GRE Geometry

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Medium 9781576336922

Square Roots and Powers: PSAT Arithmetic

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Simple Equations: SSAT-ISEE Algebra

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Circles: GED Geometry

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Medium 9781576336830

Real Numbers: Praxis I Arithmetic

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Medium 9781576337783

Sets: SAT Math II Algebra I

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Medium 9781576336403

Medians, Altitudes, Bisectors: GED Geometry

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Medium 9781576336588

Similar Polygons: GRE Geometry

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Medium 9781576336724

Properties of Numbers: SAT Arithmetic

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Medium 9781576337370

Area and Volume: CLEP Algebra II

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Medium 9781576336281

Properties of Numbers: COOP-HSPT Algebra

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Polynomials: CLEP Algebra II

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Medium 9789380298818

14

Parmananda Gupta Laxmi Publications PDF

160

DIFFERENTIAL GEOMETRY AND CALCULUS OF VARIATIONS

∴ The extremals of the functional J[y(x)] can be found by solving Euler’s equation of the transformed functional J1[v(u)]. This is called the principle of invariance of Euler’s equation under coordinates transformations.

In solving practical problems, the change of variables is made directly in the integral representing the functional. We then write and solve the Euler’s equation for the new integral involving new variables. In the extremals so obtained, the new variables are changed to the original variables to get the desired extremals.

Example. Find the extremals of the functional

Sol. Let

J[r(θ)] =

z

z

θ2

θ1

θ2

θ1

r 2 + r ′ 2 dθ , where r = r(θ).

r 2 + r ′ 2 dθ .

Let

x = r cos θ, y = r sin θ.

r=

−1 x 2 + y2 , θ = tan

y x

rx + ry y′ dr dr/ dx

=

= r′ =

= dθ dθ/ dx θ x + θ y y′

x

2

x +y

1 y2

1+ 2 x

2

+

FG − y IJ

H xK

2

y

y′ x + y2

1

1

+ y′

2 x y

1+ 2 x

2

FG IJ

H K

x + yy′ x2 + y2

= xy′ − y x2 + y2

=

Also,

dθ =

xy′ − y dθ dx

. dx = (θx + θy y′)dx = 2 dx x + y2

r2 + r′2 = x2 + y2 +

= (x2 + y2)

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