Recent content by llarsen

I think the formula J^T J is a little misleading. I will write this out using index notation since matrix format doesn't clarify the difference between upper and lower indices (vectors and covectors). A metric can be represented as a symmetric matrix (or tensor) g_{j k} = g_{k j} (i.e...

One part of the tensor picture is something called differential forms. These are contravariant tensors with an antisymmetric product called the wedge product defined. The wedge product can be written in terms of the tensor product. Both are easy to use. If you want to get a feel for tensors in a...

Above I should have said that the coordinate lines associated with x are (x, y_0, z_0) where y_0 and z_0 are constants. (x,0,0) is the coordinate axis. But there is a coordinate line associated with every choice of y_0 and z_0. However since y_0 and z_0 are constants, when we take the...

Let me try and give you a quick explanation of covariant and contravariant indicies and what they mean. First it is useful to understand first order tensors, by which I mean tensors that have one index. These are called vectors and covectors. Vectors are represented by a contravariant (upper)...

Bacle, I don't know exactly what you do and don't understand about the chain rule. But let me try and explain how it fits into differential. This may seem pretty basic, but sometimes the basics geometric concepts are not explained clearly in mathematical texts, and they can be pretty useful in...

I don't use the Jacobian notation often because I tend to make mistakes more easily. And this happens to be one of those cases. I wrote an example out and you are correct. I was the one that was using the wrong transformation. Anyway, it seems like you are understanding how the process works.

I think my statements above are partially incorrect, which may have led to a little confusion (sorry). I was only paying attention to the form of the Jacobian that you needed to get your transformations correct. I chose \frac{\partial u_i}{\partial x_j} as the Jacobian without really considering...

Almost The statements you make below are mostly right if the Jacobian is defined with terms like du/dx instead of dx/du. Lets use the coordinate system (u_1,u_2,u_3) \rightarrow (x_1,x_2,x_3) since this will allow me to write the Jacobian in the compact index form \frac{\partial u_i}{\partial...

One of the things I was trying to convey in my discussions above is that the chain rule tends to lie at the heart of differential geometry. One of the problems with matrix and tensor notation is that it obscures the chain rule. I realized that my discussion above may be difficult to follow...

The cases I deal with only required real coordinates. Since I don't have to deal with complex change of coordinates, I seldom take time to think about the modifications needed for complex systems, so I won't try to address this question for you. Sometimes the theorems need little modification to...

I made a mistake in my post above. I claimed that when you transform from (d/dx, d/dy, d/dz) to (d/du, d/dv, d/dw) by multiplying the vector (dx/dt, dy/dt, dz/dt) transposed by the jacobian to get: \[ \left( \begin{array}{ccc} \frac{dx}{dt} x_u + \frac{dy}{dt} x_v + \frac{dz}{dt} x_w\\...

I found a short references you might try http://faculty.gg.uwyo.edu/dueker/tensor%20curvilinear%20relativity/tensor%20analysis%20intro.pdf" . It seems to do a good job explaining some of the tensor concepts. There is a longer introduction http://arxiv.org/PS_cache/math/pdf/0403/0403252v1.pdf...

(S*U) is a dot product which results in a scalar k=S*U. The scalar is them multiplied by the vector V. If you want to define a unit vector that points in the direction of V and call it Uv, then you can say that the magnitude is k|V| as you said and the result is k|V|Uv, but this is the same as...

Is 7thSon an Orson Scott Card reference?

In my experience, the e_i notation is used to indicate unit vectors. To define a unit vector you need to define a metric and use the metric to scale the basis vectors at each point in space such that the basis vectors are 1 unit long. It just so happens that in euclidean space, the basis vector...