SolitaryRoad.com

Website owner: James Miller

[ Home ] [ Up ] [ Info ] [ Mail ]

CHRISTOFFEL SYMBOLS OF THE FIRST AND SECOND KIND

Christoffel symbols. Christoffel symbols are shorthand notations for various functions associated with quadratic differential forms. The differential form is usually the first fundamental quadratic form of a surface. Each Christoffel symbol is essentially a triplet of three indices, i, j and k, where each index can assume values from 1 to 2 for the case of two variables, or from 1 to n in the case of a quadratic form in n variables. Associated with different combinations of values of the three indices are different functions.

Christoffel symbols of the first kind. For the quadratic differential form in two variables

the Christoffel symbols of the first kind are defined as

where the indices i, j and k can each assume the values of either 1 or 2.

Examples

Other notations. Other notations, instead of [i j, k], are used. They include C_{ij}^{k} and Γ_{ijk} and

Meaning for the First Fundamental Quadratic Form. Let us examine the meaning of these Christoffel symbols for the first fundamental quadratic form

3) Edu^{2} + 2Fdudv + Gdv^{2} .

Let S be a simple surface element defined by the one-to-one mapping

x = x(u, v)

y = y(u, v)

z = z(u, v)

of a region R of the uv-plane into xyz-space. Let

be the position vector to point P on the surface. Let us use subscript notation to denote partial differentiation i.e.

The first fundamental quadratic form is

where:

The functions that correspond to the different possible combinations of values of the indices i, j and k of the Christoffel symbol are [as can be confirmed by plugging the index values into 1)]:

The equivalence of the E, F and G expressions in terms of the is easily confirmed by taking the indicated partial derivatives of

The source of the Christoffel symbol notation becomes clear with a slight change in notation. If we use an integer subscript notation for the partial derivatives where

then

The Christoffel symbols are symmetric in i and j, which means that

[i j, k] = [j i, k] .

Also, by definition, g_{ij} = g_{ji}.

Christoffel symbols of the second kind. For the quadratic differential form in two variables

the Christoffel symbols of the second kind are defined as

Γ_{i} _{j }^{k} = A_{k1}[i j, 1] + A_{k2}[i j, 2]

where

1] the indices i, j and k can each assume the values of either 1 or 2,

2] A_{ki} = C_{ki}/Δ

where C_{ki} is the cofactor of g_{ki} in the determinant

3] [i j, k] are the Christoffel symbols of the first kind.

For the first fundamental quadratic form

3) Edu^{2} + 2Fdudv + Gdv^{2} .

the functions that correspond to the different possible combinations of values of the indices i, j and k are

Also,

The symbols in this case are also given by the following formulas:

The Christoffel symbols of the second kind correspond to the coefficients in the following equations of Gauss, from which they have their origin:

is the unit normal to the surface at point P and L,M, N are the second fundamental coefficients.

Quadratic form in n variables. For a quadratic form in n variables

as might be encountered in differential geometry investigations in n-dimensional space, we
remark that Christoffel symbols are also defined. Here the indices i, j and k can each assume
values 1, 2, ... , n. It is assumed that g_{ij} = g_{ji} for all i and j. The general definition for n variables
reduces to formula 2) above for the case of two variables.

References.

1. Graustein. Differential Geometry.

2. Lipschutz. Differential Geometry. Chapter 9.

3. James/James. Mathematics Dictionary.

4. Eisenhart. A Treatise on the Differential Geometry of Curves and Surfaces.

More from SolitaryRoad.com:

Jesus Christ and His Teachings

Way of enlightenment, wisdom, and understanding

America, a corrupt, depraved, shameless country

On integrity and the lack of it

The test of a person's Christianity is what he is

Ninety five percent of the problems that most people have come from personal foolishness

Liberalism, socialism and the modern welfare state

The desire to harm, a motivation for conduct

On Self-sufficient Country Living, Homesteading

Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.

Theory on the Formation of Character

People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest

Cause of Character Traits --- According to Aristotle

We are what we eat --- living under the discipline of a diet

Avoiding problems and trouble in life

Role of habit in formation of character

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Intellectual disparities among people and the power in good habits

Tools of Satan. Tactics and Tricks used by the Devil.

The Natural Way -- The Unnatural Way

Wisdom, Reason and Virtue are closely related

Knowledge is one thing, wisdom is another

My views on Christianity in America

The most important thing in life is understanding

We are all examples --- for good or for bad

Television --- spiritual poison

The Prime Mover that decides "What We Are"

Where do our outlooks, attitudes and values come from?

Sin is serious business. The punishment for it is real. Hell is real.

Self-imposed discipline and regimentation

Achieving happiness in life --- a matter of the right strategies

Self-control, self-restraint, self-discipline basic to so much in life

SolitaryRoad.com

Website owner: James Miller

[ Home ] [ Up ] [ Info ] [ Mail ]