Vector Fields and Tensors Differential Geometry Part 4

Vector fields and the concept of parallel transport of a vector are introduced. Cartesian tensors are defined using the concept of tensor product which "glues" two vectors together. This leads us to scalar products of tensors and vectors, non-comutative property, the unit tensor, and the composition of a tensor from tensor products of the Cartesian unit vectors.

Category:

Education

Tags:

• Cartesian
• tensor
• parallel
• transport
• vector
• field
• bundle
• algebra
• product
• composition
• Mathematics
• Education

[omicron8251]

Wondering if that's the tangent space, what's a cotangent space? (i.e.)Tn*Mn.



[Mathview]

@omicron8251 Yes, excellent question. Occasionally, I use the term tangent space loosely to indicate any local linear space at a point in a manifold. The actual meaning should be clear in the context. Officially, the local linear space of differentials in the neighborhood of a point p in a Riemann manifold is called the cotangent space at p. It is the dual of the local vector space of inverse differentials say.. (d/dx, d/dy, d/dz) at p. The metric tensor converts from one to another.

[Mathview]

Very smart question. Tensors as discussed here can be shown to form a natural linear space (vector space.) Tensors of the same rank can be added or subtracted element by element. Tensors can be multiplied by a scalar again element by element. There is a zero tensor. So yes. More info in the playlist titled Mathematical Spaces and in the video Rotation Operators part3.

[Mathview]

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[Mathview]

Completed two new videos continuing the discussion of Vectors and Tensors (Sept 20 2010.) One covers topics of Tensor Algebra and Wedge Product, Exterior Product Spaces, and Differential Forms explained simply. The other is on converting Geometry into Algebra using Tensors. We know mathematics can convert Geometry into Analysis using Analytic Geometry and Calculus. We can also convert Geometry into Algebra using Tensor Products, and create new kinds of Vector Spaces. Comments welcome.

[johann022086]

very nice thanks. but where is the video next to this? Is it this: "How do I get the arc length from the metric tensors".

[Mathview]

Thank you, you are observant. My original intention was put differential forms after this part. But I decided that it would be better at the beginning of a new series on Curved Spaces and Riemannian Manifolds.

I hope to get those videos started in the next week or so. It seems there is interest in this subject, and I am really enjoying it. Thanks for the views.

[bebeheuy]

I am very happy to see the vidoe after you give this Vector fields and the concept of parallel transport of a vector are introduced. Cartesian tensors are defined using the concept of tensor product which "glues" two vectors together

[AntoMelta]

I Really Like The Video From Your Vector fields and the concept of parallel transport of a vector are introduced

[willamricard]

Your Video Is Very Useful Sharing Vector fields and the concept of parallel transport of a vector are introduced. Cartesian tensors are defined using the concept of tensor product which "glues" two vectors together.

[imegatrone]

after i watched this video, my insight is very open because the video is very good to give information Vector Fields and Tensors Differential Geometry Part 4

[Gigantum1000]

HI and thanks a lot for this very easy way of teaching Tensors

[lordborda]

Great!

[SarahStarmer]

I am so glad you made these videos. I kept coming accross the words covarient and contravarient and tensor and it made me feel very ignorant especialy when I looked them up in wickapedia and still didn't understand what they ment.

So thank you so much you have been very helpful.

[danieljimenez1989]

@FanOfJanis I'm soooo jealous!!!!

what was your experience with professor Chandrasekhar?