aarjaneiro/planetmath huggingface dataset:
Brunn-Minkowski inequality

Brunn-Minkowski inequality

Let

A

and

B

be non-empty

compact subsets

of

ℝ

d

.
Then

vol

(

A

+

B

)

1

/

d

≥

vol

(

A

)

1

/

d

+

vol

(

B

)

1

/

d

,

where

A

+

B

denotes the

Minkowski sum

of

A

and

B

, and

vol

⁡

(

S

)

denotes the volume of

S

.

References

1

Jiří Matoušek.

Lectures on Discrete Geometry

, volume 212 of

GTM

.

Springer, 2002.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0999.52006

Zbl
0999.52006.

Title

Brunn-Minkowski inequality

Canonical name

BrunnMinkowskiInequality

Date of creation

2013-03-22 13:58:20

Last modified on

2013-03-22 13:58:20

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

7

Author

bbukh (348)

Entry type

Theorem

Classification

msc 51M16

Generated on Fri Feb 9 22:08:13 2018 by

LaTeXML
Hadwiger-Finsler inequality

Hadwiger-Finsler inequality

In a

triangle

with sides

a

,

b

,

c

and an area

A

the following

inequality

holds:

a

2

+

b

2

+

c

2

≥

(

a

-

b

)

2

+

(

b

-

c

)

2

+

(

c

-

a

)

2

+

4

⁢

A

⁢

3

.

Title

Hadwiger-Finsler inequality

Canonical name

HadwigerFinslerInequality

Date of creation

2013-03-22 12:45:18

Last modified on

2013-03-22 12:45:18

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

5

Author

mathwizard (128)

Entry type

Theorem

Classification

msc 51M16

Related topic

WeizenbocksInequality

Generated on Fri Feb 9 22:08:15 2018 by

LaTeXML
proof of Hadwiger-Finsler inequality

proof of Hadwiger-Finsler inequality

From the

cosines law

we get:

a

2

=

b

2

+

c

2

-

2

⁢

b

⁢

c

⁢

cos

⁡

α

,

α

being the angle between

b

and

c

. This can be transformed into:

a

2

=

(

b

-

c

)

2

+

2

⁢

b

⁢

c

⁢

(

1

-

cos

⁡

α

)

.

Since

A

=

1

2

⁢

b

⁢

c

⁢

sin

⁡

α

we have:

a

2

=

(

b

-

c

)

2

+

4

⁢

A

⁢

1

-

cos

⁡

α

sin

⁡

α

.

Now remember that

1

-

cos

⁡

α

=

2

⁢

sin

2

⁡

α

2

and

sin

⁡

α

=

2

⁢

sin

⁡

α

2

⁢

cos

⁡

α

2

.

Using this we get:

a

2

=

(

b

-

c

)

2

+

4

⁢

A

⁢

tan

⁡

α

2

.

Doing this for all sides of the

triangle

and adding up we get:

a

2

+

b

2

+

c

2

=

(

a

-

b

)

2

+

(

b

-

c

)

2

+

(

c

-

a

)

2

+

4

⁢

A

⁢

(

tan

⁡

α

2

+

tan

⁡

β

2

+

tan

⁡

γ

2

)

.

β

and

γ

being the other angles of the triangle. Now since the halves of the triangle’s angles are less than

π

2

the function

tan

is convex we have:

tan

⁡

α

2

+

tan

⁡

β

2

+

tan

⁡

γ

2

≥

3

⁢

tan

⁡

α

+

β

+

γ

6

=

3

⁢

tan

⁡

π

6

=

3

.

Using this we get:

a

2

+

b

2

+

c

2

≥

(

a

-

b

)

2

+

(

b

-

c

)

2

+

(

c

-

a

)

2

+

4

⁢

A

⁢

3

.

This is the

Hadwiger-Finsler inequality

.

□

Title

proof of Hadwiger-Finsler inequality

Canonical name

ProofOfHadwigerFinslerInequality

Date of creation

2013-03-22 12:45:21

Last modified on

2013-03-22 12:45:21

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

5

Author

mathwizard (128)

Entry type

Proof

Classification

msc 51M16

Generated on Fri Feb 9 22:08:22 2018 by

LaTeXML
antiprism

antiprism

An

antiprism

is a

prismatoid

whose bases are

congruent

and whose lateral faces are congruent

triangles

.

An antiprism can be used to construct an

icosahedron

: Let

A

be an antiprism with

regular pentagons

as its bases and

equilateral triangles

as its lateral faces. To each base of

A

, attach a

pentagonal pyramid

whose base is congruent to the bases of

A

and whose lateral faces are equilateral triangles.

For any integer

n

≥

3

, the following are

equivalent

statements about an antiprism

A

:

1.

A

has a base that is an

n

-gon;

2.

A

has

2

⁢

n

+

2

faces;

3.

A

has

2

⁢

n

vertices;

4.

A

has

4

⁢

n

edges.

Title

antiprism

Canonical name

Antiprism

Date of creation

2013-03-22 16:57:02

Last modified on

2013-03-22 16:57:02

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

9

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51M20

Classification

msc 51M04

Generated on Fri Feb 9 22:08:25 2018 by

LaTeXML
cylindroid

cylindroid

At the most general level, a

cylindroid

is simply a

cylinder

that has been deformed in an intentional and

well-defined

way. The barrel die used in certain board role-playing games are sometimes described as being cylindroids in shape.

Most mathematical dictionaries that define the term at all say nothing more than “a cylinder with an elliptical

cross-section

.” (see for example the

Harper-Collins Dictionary of Mathematics

and the

Oxford Concise Dictionary of Mathematics

.) The term is also used for certain conoids, such as Plücker’s conoid.

The term is usually not included in general pocket dictionaries. The

Random House Unabridged Dictionary

defines cylindroid the noun as “a solid having the form of a cylinder, esp[ecially] one with an elliptical, as opposed to a circular,

cross section

,” and the adjective as “resembling a cylinder.”

References

1

S. P. Radzevich, “A Possibility of Application of Pliicker’s Conoid for Mathematical Modeling of Contact of Two Smooth Regular Surfaces in the First Order of Tangency”,

Mathematical and

Computer

Modelling

42

(2005): 999 - 1022

Title

cylindroid

Canonical

name

Cylindroid

Date of creation

2013-03-22 16:41:12

Last modified on

2013-03-22 16:41:12

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 51M20

Classification

msc 14J25

Classification

msc 51M04

Generated on Fri Feb 9 22:08:31 2018 by

LaTeXML
developable surface

developable surface

A

generatrix

of a

ruled surface

is

torsal

, if in each of its points there is one and the same

tangent plane

of the

surface

.

A ruled surface is

torsal

iff it only has

torsal generatrices

.

A surface is

developable

, if one can spread it out on a plane without any stretching or tearing.

K. F. Gauss has proved that a surface is developable if and only if it is a torsal ruled surface.

One may divide the

developable surfaces

into three :

1.

Cylindrical surfaces

2.

Conical surfaces

3.

Tangential surfaces

of a

space curve

; they can be expressed by

r

→

=

γ

→

⁢

(

t

)

+

s

⁢

d

⁢

γ

→

⁢

(

t

)

d

⁢

t

where

r

→

=

γ

→

⁢

(

t

)

is the equation of the space curve,

s

and

t

are parameters.

Figure 1:

Tangential surface of a

circular helix

Title

developable surface

Canonical name

DevelopableSurface

Date of creation

2013-03-22 15:29:29

Last modified on

2013-03-22 15:29:29

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Topic

Classification

msc 51M20

Classification

msc 51M04

Synonym

torsal surface

Related topic

Area2

Related topic

RiemannMultipleIntegral

Defines

developable

Defines

torsal generatrix

Defines

torsal

Defines

tangential surface

Generated on Fri Feb 9 22:08:33 2018 by

LaTeXML
equation of tangent of circle

equation of tangent of circle

We derive the equation of

tangent line

for a circle with radius

r

.  For

simplicity

, we chose for the origin the centre of the circle, when the points

(

x

,

y

)

of the circle satisfy the equation

x

2

+

y

2

=

r

2

.

(1)

Let the point of tangency be

(

x

0

,

y

0

)

.  Then the slope of radius with

end point

(

x

0

,

y

0

)

is

y

0

x

0

, whence, according to the parent entry (

http://planetmath.org/TangentOfCircle

), its opposite

inverse

-

x

0

y

0

is the slope of the

tangent

, being

perpendicular

to the radius.  Thus the equation of the tangent is written as

y

-

y

0

=

-

x

0

y

0

⁢

(

x

-

x

0

)

.

Removing the denominator and the parentheses we obtain from this first

x

0

⁢

x

+

y

0

⁢

y

=

x

0

2

+

y

0

2

,  and then

x

0

⁢

x

+

y

0

⁢

y

=

r

2

(2)

since

(

x

0

,

y

0

)

satisfies (1).

Remark.

In the equation (2) of the tangent,

x

0

,

y

0

are the

coordinates

of the point of tangency and

x

,

y

the coordinates of an arbitrary point of the tangent line.  But one can of course swap those meanings; then we interprete (2) such that

x

0

,

y

0

are the coordinates of some point

P

outside the circle (1) and

x

,

y

the coordinates of the point of tangency of either of the tangents which may be drawn from

P

to the circle.  If (2) now is again interpreted as an equation of a line (its degree (

http://planetmath.org/AlgebraicEquation

) is 1!), this line must

pass through

both the mentioned points of tangency

A

and

B

(they satisfy the equation!); in a , (2) is now the equation of the

tangent chord

A

⁢

B

of

P

=

(

x

0

,

y

0

)

.  See also

http://mathworld.wolfram.com/Polar.html

polar.

Title

equation of tangent of circle

Canonical name

EquationOfTangentOfCircle

Date of creation

2013-03-22 18:32:16

Last modified on

2013-03-22 18:32:16

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Derivation

Classification

msc 51M20

Classification

msc 51M04

Related topic

Polarising

Generated on Fri Feb 9 22:08:42 2018 by

LaTeXML
parallelotope

parallelotope

A

parallelotope

is the covering term used to denote any element of the series of geometric objects generalizing the

parallelogram

:

\PMlinktofile

View SVGparallelotope.svg

The

hypercube

is formed by moving the

parallelepiped

in a fourth direction, thus creating a 4-dimensional object

bounded

by 8 parallelepipeds (2 for each axis of freedom). The volume of an n-parallelotope can be calculated by taking the product of its lengths in each

dimension

or by taking the

absolute value

of the determinant of the vectors spanning it.

Title

parallelotope

Canonical name

Parallelotope

Date of creation

2013-03-22 17:37:35

Last modified on

2013-03-22 17:37:35

Owner

slider142 (78)

Last modified by

slider142 (78)

Numerical id

13

Author

slider142 (78)

Entry type

Definition

Classification

msc 51M20

Related topic

parallelogram

Related topic

parallelepiped

Related topic

Parallelogram

Generated on Fri Feb 9 22:08:44 2018 by

LaTeXML
ruled surface

ruled surface

A

straight line

g

moving continuously in space sweeps a

ruled surface

.  Formally:  A

surface

S

in

ℝ

3

is a ruled surface if it is connected and if for any point

p

of

S

, there is a line

g

such that

p

∈

g

⊂

S

.

Such a surface may be formed by using two auxiliary curves given e.g. in the parametric forms

r

→

=

a

→

⁢

(

t

)

,

r

→

=

b

→

⁢

(

t

)

.

Using two parameters

s

and

t

we express the
position vector (

http://planetmath.org/PositionVector

) of an arbitrary point of the ruled surface as

r

→

=

a

→

⁢

(

t

)

+

s

⁢

b

→

⁢

(

t

)

.

Here

r

→

=

a

→

⁢

(

t

)

is a curve on the ruled surface and is called or the
of the surface, while

r

→

=

b

→

⁢

(

t

)

is the

director curve

of the surface.  Every position of

g

is a

generatrix

or

ruling

of the ruled surface.

Examples

1.  Choosing the

z

-axis (

r

→

=

c

⁢

t

⁢

k

→

,

c

≠

0

) as the and the

unit circle

(

r

→

=

i

→

⁢

cos

⁡

t

+

j

→

⁢

sin

⁡

t

)
as the director curve we get the

helicoid

(“screw surface”; cf. the

circular helix

)

r

→

=

c

⁢

t

⁢

k

→

+

s

⁢

(

i

→

⁢

cos

⁡

t

+

j

→

⁢

sin

⁡

t

)

=

(

s

⁢

cos

⁡

t

s

⁢

sin

⁡

t

c

⁢

t

)

.

2.  The equation

z

=

x

⁢

y

presents a

hyperbolic paraboloid

(if we rotate the

coordinate system

(

http://planetmath.org/RotationMatrix

) 45 about the

z

-axis using the formulae

x

=

(

x

′

-

y

′

)

/

2

,

y

=

(

x

′

+

y

′

)

/

2

,  the equation gets the form

x

′

⁣

2

-

y

′

⁣

2

=

2

⁢

z

).  Since the position vector of any point of the surface may be written using the parameters

s

and

t

as

r

→

=

(

0

t

0

)

+

s

⁢

(

1

0

t

)

,

we see that it’s a question of a ruled surface with rectilinear

directrix

and director curve.

3.  Other ruled surfaces are for example all

cylindrical
surfaces

(plane included),

conical surfaces

,

one-sheeted hyperboloid

(

http://planetmath.org/QuadraticSurfaces

).

Title

ruled surface

Canonical name

RuledSurface

Date of creation

2016-03-03 17:28:55

Last modified on

2016-03-03 17:28:55

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

19

Author

pahio (2872)

Entry type

Topic

Classification

msc 51M20

Classification

msc 51M04

Related topic

EquationOfPlane

Related topic

GraphOfEquationXyConstant

Defines

directrix

Defines

base curve

Defines

director curve

Defines

generatrix

Defines

generatrices

Defines

ruling

Defines

helicoid

Generated on Fri Feb 9 22:08:48 2018 by

LaTeXML
Archimedes’ cylinders in cube

Archimedes’ cylinders in cube

The following problem has been solved by

Archimedes

:

Two distinct circular cylinders are

inscribed

in a cube; the axes thus

intersect

each other perpendicularly.  Determine the volume common to both

cylinders

, when the radius of the base of the cylinders is

r

.

If the solid common to both cylinders is cut with a plane

parallel

to the axes of both cylinders, the figure of

intersection

is a square.  Denote the

distance

of the plane from the center of the cube be

x

.  By the

Pythagorean theorem

, half of the side of the square is

r

2

-

x

2

and the area of the square is

4

⁢

(

r

2

-

x

2

)

2

. Accordingly, we have the

function

A

⁢

(

x

)

:=

4

⁢

(

r

2

-

x

2

)

for the area of the intersection square.  If we let

x

here to grow from

0

to

r

, then half of the given solid is got.  By the volume of the parent entry (

http://planetmath.org/VolumeAsIntegral

), the half volume of the solid is

1

2

⁢

V

=

∫

0

r

4

⁢

(

r

2

-

x

2

)

⁢

𝑑

x

=

4

⁢

/

x

=

0

r

⁡

(

r

2

⁢

x

-

x

3

3

)

=

8

3

⁢

r

3

.

So the volume in the question is

16

3

⁢

r

3

.  It is

2

3

of the volume of the cube.

Title

Archimedes’ cylinders in cube

Canonical name

ArchimedesCylindersInCube

Date of creation

2013-03-22 17:20:51

Last modified on

2013-03-22 17:20:51

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Example

Classification

msc 51M25

Classification

msc 51-00

Synonym

perpendicular cylinders

Synonym

cylinders inscribed in cube

Related topic

SubstitutionNotation

Generated on Fri Feb 9 22:08:50 2018 by

LaTeXML
base and height of triangle

base and height of triangle

Considering the area of a

triangle

, one usually names a side of the triangle to be its

base

.  For expressing the calculation way of the area of the triangle, one then uses the

height

(a.k.a.

altitude

), which means the perpendicular distance of the vertex, to the base side, from the line determined by the base.  In the above two triangles, the heights

h

1

and

h

2

correspond the horizontal . One calls

foot of the height

the projection of the vertex onto the line of the base.

The rule for the calculation reads

area  =  base times height divided by 2

In the below figure, there is the illustration of the rule.  The

parallelogram

A

⁢

B

⁢

C

⁢

D

has been divided by the

diagonal

B

⁢

D

into two triangles, which are

congruent

by the ASA criterion (see the

alternate interior angles

).  Thus the both triangles have the areas half of the area of the parallelogram, which in turn has the common base

A

⁢

B

and the common height

h

with the triangle

A

⁢

B

⁢

D

.

Note.

In an

isosceles triangle

, one sometimes calls the two equal sides the

legs

and the third side the

base

.

Title

base and height of triangle

Canonical name

BaseAndHeightOfTriangle

Date of creation

2013-03-22 18:50:15

Last modified on

2013-03-22 18:50:15

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

14

Author

pahio (2872)

Entry type

Definition

Classification

msc 51M25

Classification

msc 51M04

Classification

msc 51-01

Synonym

base of triangle

Synonym

height of triangle

Related topic

AreaOfAPolygonalRegion

Related topic

HeightOfATriangle

Related topic

Area2

Related topic

ProjectionFormula

Related topic

OrthicTriangle

Defines

base

Defines

height

Defines

foot of height

Defines

foot of altitude

Generated on Fri Feb 9 22:09:07 2018 by

LaTeXML
isepiphanic inequality

isepiphanic inequality

The classical

isepiphanic inequality

36

⁢

π

⁢

V

2

≦

A

3

,

concerns the volume

V

and the area

A

of any solid in

ℝ

3

.  It asserts that the ball has the greatest volume among the solids having a given area.

For a ball with radius

r

, we have

V

=

4

3

⁢

π

⁢

r

3

,

A

=

4

⁢

π

⁢

r

2

,

whence it follows the equality

36

⁢

π

⁢

V

2

=

A

3

.

Cf. the

isoperimetric inequality

and the

isoperimetric problem

.

References

1

Patrik Nordbeck

:

Isoperimetriska problemet eller Varför ser man så  få  fyrkantiga träd?

Examensarbete.  Lund University (1995).

Title

isepiphanic inequality

Canonical name

IsepiphanicInequality

Date of creation

2013-03-22 19:19:08

Last modified on

2013-03-22 19:19:08

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Theorem

Classification

msc 51M25

Classification

msc 51M16

Generated on Fri Feb 9 22:09:09 2018 by

LaTeXML
solid of revolution

solid of revolution

Let

y

=

f

⁢

(

x

)

be a curve for

x

in an

interval

[

a

,

b

]

satisfying

f

⁢

(

x

)

>

0

for

x

in

(

a

,

b

)

. We may construct a corresponding

solid of revolution

, say

𝒱

=

{

(

x

,

y

,

z

)

:

x

∈

[

a

,

b

]

⁢

and

⁢

y

2

+

z

2

≤

f

⁢

(

x

)

2

}

. Intuitively, it is the solid generated by rotating the

surface

y

≤

f

⁢

(

x

)

about the

x

-axis.

The

interior

of a

surface of revolution

is always a solid of revolution. These include

•

the interior of a

cylinder

of radius

c

>

0

and height

h

with

f

⁢

(

x

)

=

c

for

0

≤

x

≤

h

,

•

the interior of a sphere of radius

R

>

0

with

f

⁢

(

x

)

=

R

2

-

x

2

for

-

R

≤

x

≤

R

, and

•

the interior of a (right,

circular

) cone of base radius

R

>

0

and height

h

with

f

⁢

(

x

)

=

R

⁢

x

/

h

for

0

≤

x

≤

h

.

Let

Γ

be a

simple closed curve

with parametrization

α

⁢

(

t

)

=

(

X

⁢

(

t

)

,

Y

⁢

(

t

)

)

for

t

in an interval

[

a

,

b

]

satisfying

Y

⁢

(

t

)

≥

0

for

t

in

[

a

,

b

]

.
By the

Jordan curve theorem

, we may choose the set of points,

𝒮

, ”inside” the curve,
i.e. let

𝒮

be the

bounded

connected component

of the two connected components
found in

ℝ

2

∖

Γ

.
Another sort of solid of revolution is given by

𝒱

=

{

(

x

,

y

,

z

)

:

x

=

X

⁢

(

t

)

⁢

for some

⁢

t

⁢

in

⁢

[

a

,

b

]

⁢

and

⁢

y

2

+

z

2

=

s

2

⁢

for some

⁢

s

⁢

such that

⁢

(

x

,

s

)

∈

𝒮

∪

Γ

}

.
Intuitively, it is the solid generated by rotating

𝒮

∪

Γ

about the

x

-axis.

Some examples of this sort of solid of revolution include

•

the interior of a torus of

minor radius

r

>

0

and major radius

R

>

r

with

α

⁢

(

t

)

=

(

r

⁢

cos

⁡

t

,

r

⁢

sin

⁡

t

+

R

)

for

0

≤

t

≤

2

⁢

π

,

•

a shell of a sphere with inner radius

r

>

0

and outer radius

R

>

r

with

α

⁢

(

t

)

=

{

(

R

⁢

cos

⁡

π

⁢

t

,

R

⁢

sin

⁡

π

⁢

t

)

if

⁢

t

∈

[

0

,

1

]

(

r

⁢

(

1

-

t

)

+

R

⁢

(

t

-

2

)

,

0

)

if

⁢

t

∈

[

1

,

2

]

(

-

r

⁢

cos

⁡

π

⁢

t

,

r

⁢

sin

⁡

π

⁢

t

)

if

⁢

t

∈

[

2

,

3

]

(

r

⁢

(

4

-

t

)

+

R

⁢

(

t

-

3

)

,

0

)

if

⁢

t

∈

[

3

,

4

]

.

Title

solid of revolution

Canonical name

SolidOfRevolution

Date of creation

2013-03-22 17:19:57

Last modified on

2013-03-22 17:19:57

Owner

nkirby (11104)

Last modified by

nkirby (11104)

Numerical id

10

Author

nkirby (11104)

Entry type

Definition

Classification

msc 51M25

Related topic

SurfaceOfRevolution2

Generated on Fri Feb 9 22:09:16 2018 by

LaTeXML
volume as integral

volume as integral

The volume of a

solid of revolution

(

http://planetmath.org/VolumeOfSolidOfRevolution

) can be obtained from

V

=

∫

a

b

π

⁢

[

f

⁢

(

x

)

]

2

⁢

𝑑

x

,

where the integrand is the area of the

intersection

disc of the solid of revolution and a plane

perpendicular

to the

axis of revolution

at a certain value of

x

.  This volume formula may be generalized to an analogous formula containing instead of the area

π

⁢

[

f

⁢

(

x

)

]

2

a more general intersection area

A

⁢

(

t

)

obtained from a given solid by cutting it with a set of

parallel planes

determined by the

parameter

t

on a certain axis.  One must assume that the

function

t

↦

A

⁢

(

t

)

is continuous on an

interval

[

a

,

b

]

where

a

and

b

correspond to the “ends” of the solid.  If the

t

-axis forms an angle (

http://planetmath.org/AngleBetweenTwoLines

)

ω

with the

normal line

of those planes, then we have the volume formula of the form

V

=

∫

a

b

A

⁢

(

t

)

⁢

𝑑

t

⁢

cos

⁡

ω

.

Title

volume as integral

Canonical name

VolumeAsIntegral

Date of creation

2013-03-22 17:20:44

Last modified on

2013-03-22 17:20:44

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Topic

Classification

msc 51M25

Classification

msc 51-00

Related topic

Volume

Related topic

VolumeOfSolidOfRevolution

Related topic

RiemannMultipleIntegral

Related topic

ExampleOfRiemannTripleIntegral

Generated on Fri Feb 9 22:09:18 2018 by

LaTeXML
volume of ellipsoid

volume of ellipsoid

Let us determine the volume of the

ellipsoid

x

2

a

2

+

y

2

b

2

+

z

2

c

2

=

1

.

Suppose

-

a

≦

x

≦

a

. When we cut the ellipsoid with a plane

parallel

to the

y

⁢

z

-plane, that is, let

x

be , we get the

ellipse

y

2

b

2

+

z

2

c

2

=

1

-

x

2

a

2

,

i.e.

y

2

b

2

⁢

(

1

-

x

2

a

2

)

+

z

2

c

2

⁢

(

1

-

x

2

a

2

)

=

1

,

with the semiaxes

b

1

:=

b

⁢

1

-

x

2

a

2

,

c

1

:=

c

⁢

1

-

x

2

a

2

.

The area of this ellipse is

π

⁢

b

1

⁢

c

1

(see

area of plane region

), and thus we have the function

A

⁢

(

x

)

:=

π

⁢

b

⁢

c

⁢

(

1

-

x

2

a

2

)

expressing the area cut of the ellipsoid by

parallel planes

. By the volume formula of the parent entry (

http://planetmath.org/VolumeAsIntegral

) we can calculate the volume of the ellipsoid as

V

=

∫

-

a

a

A

⁢

(

x

)

⁢

𝑑

x

=

π

⁢

b

⁢

c

⁢

∫

-

a

a

(

1

-

x

2

a

2

)

⁢

𝑑

x

=

π

⁢

b

⁢

c

⁢

/

x

=

-

a

a

⁡

(

x

-

x

3

3

⁢

a

2

)

=

4

3

⁢

π

⁢

a

⁢

b

⁢

c

.

The special case

a

=

b

=

c

=

r

of a sphere is the well-known

expression

4

3

⁢

π

⁢

r

3

.

Title

volume of ellipsoid

Canonical

name

VolumeOfEllipsoid

Date of creation

2013-03-22 17:20:41

Last modified on

2013-03-22 17:20:41

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Result

Classification

msc 51M25

Synonym

ellipsoid volume

Related topic

Ellipsoid

Related topic

SubstitutionNotation

Related topic

SqueezingMathbbRn

Generated on Fri Feb 9 22:09:20 2018 by

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