Differential Geometry Part 3 Transformations and the Metric Tensor

Continuing development of differential geometry with vector components and derivation of a formula for the metric tensor in oblique coordinates. The method is easily generalilzed to higher dimensions. Some notes on vector notation and the Einstein summation convention is introduced.

Category:

Education

Tags:

• Differential
• geometry
• vector
• tensor
• contravariant
• covariant
• components
• metric
• Einstein
• summation
• convention

[LokiClock]

What happens when you multiply g_ijk with u^i? It seems like you'd end up with two parameters to a single-index tensor.

[Mathview]

@LokiClock Ah.. ok. This notation takes some practice.... The metric tensor components are an nxn matrix indicated as g_ij . The inverse matrix of g_ij as g^ij with superscript i,j. So when you matrix multiply g_ij by a contravariant vector a^j you get the covariant components of a namely a_i where _i means subscript and ^j means superscript. Derivatives of g_ij give you an n x n x n array of derivatives. They show up in the Christoffel symbols which have 3 indices. Make sense?

[Mathview]

@LokiClock Ah and another thing is ... here multiplication is as in matrix multiplication (inner product) using the g_ij and vectors like A^j. If we were doing tensor product multiplication (exterior product) then you would get objects with three indices.

[Mathview]

@LokiClock hmmm ... ok, of course the metric tensor g is a second rank tensor, so only has two indices. So that's probably not the question. Consider a general tensor H_n,m,k - a third rank tensor having three covariant indices. Then an inner product with a contravariant vector u^k would sum over k. That would produce a new tensor of second rank, say Q_n,m. Transform H_n,m,k to a different coordinate system uses a transformation with 6 indices , leaving three indices on the new H.

[Mathview]

You may enjoy more Mathview videos at our website googleplexMath

Just google this: sites.google.com/site/threesph­ere2011

Mathview video collection organized by topic and logical sequence:

Extra features | References | Book Reviews | Commentary | Recommendations | Visitor comments |

[Mathview]

For more information on the covariant and contravariant components of vectors, check out the Playlist Differential Geometry 1.

BTW in more recent videos you may notice I am using an opposite index notation for the metric tensor. Downstairs indices on g indicating the metric tensor, and upstairs indices for the inverse. Calling the matrix with downstairs indices the metric tensor conforms to more conventional usage in the literature.

[TMSxYouT]

I liked your diagram very much , thank you for your efforts!

[MrSlayer4ever]

very useful...thanks

[LokiClock]

@Mathview Okay, thanks. Does that mean that u_i and u_j aren't considered two descriptions of the same object if g_ij doesn't signify the metric tensor? Like, you might as well write H_ij*u^i=q_j, without this special relationship between u_i and q_j.

[LokiClock]

@Mathview I got that, but what about g_ijk*u^i? Would that be g_k*u_j? g_j*u_k? I understand how, when there's only one index left, the indexing of the result is predictable. But I don't know what to do when G is two orders above a.

[fielsjd]

I was a little thrown when you wrote, after 8:40 or so, c sub 1 squared + c sub 2 squared = u sub 1 * u super 1 + u sub 2 * u super 2.

The superscript on c indicates squaring it, whereas the superscripts on u indicate a choice of components.

Otherwise I am liking the videos.

[FallofDarkness55]

string theory.