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 Tensor power series andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 05/13/2008, 07:58 AM I think a nice way to discuss the power series idea where x is a vector is within the context of tensor calculus, or multilinear algebra as its called now. Roughly speaking, tensors are just multi-dimensional arrays that have additional properties. Here is a brief overview of my understanding of tensors. $X = x^i$ is a contravariant 1-tensor (or vector or column vector), this means that it transforms contravariantly (whatever that means), and it has 1 dimension of indeces. As a mixed tensor, this is also called a (1,0)-tensor or a $\left(\begin{tabular}{c}1 \\ 0\end{tabular}\right)$-tensor. $f_i$ is a covariant 1-tensor (or covector or row vector), which means that it is like a function that takes a vector and gives a scalar (0-tensor) as output. As a mixed tensor, this is also called a (0,1)-tensor or a $\left(\begin{tabular}{c}0 \\ 1\end{tabular}\right)$-tensor. $F = f^j_i$ is a mixed variance 2-tensor, or a vector valued function (a matrix), meaning it is both covariant (i) and contravariant (j). This is a combination of both descriptions taken above, because (like a matrix) it is a function that sends a vector (i.e. covariance) to a vector (i.e. contravariance). Although I have not seen anything to support this view of covariance/contravariance, it seems like it doesn't contradict any description either. So one way to think of it is that "contravariant" means "gives a vector" and "covariant" means "takes a vector". This is called a 2-tensor, but to distinguish between each kind of index, this can also be called a (1,1)-tensor. ${\nabla}_i$ is a covariant 1-tensor (known as the gradient), but its different in that it is also an operator, which takes a function (covector), and gives a covector. In order to replace the concept of a derivative (or the Jacobian matrix) we will not be using it as an operator directly, but we will be multiplying it with the tensor product $\otimes$ which means it will operate very much like the Jacobian when used on a (1,1)-tensor. $0^i$ is a contravariant 1-tensor, because it is like a vector, that holds all zeros. $F(X) = \sum_i f^j_i(x^i)$ is the application of a 2-tensor function to a 1-tensor, which is actually a form of tensor contraction or matrix multiplication in this case (but this is only true for linear functions, non-linear functions cannot be written this way). Notice that I don't use the Einstein notation (also known as index notation) because I think it is confusing when you can't see the summation $\Sigma$. For linear functions, this is the power series of that function. However, what we want to do is generalize this so that this works for non-linear functions as well. « Next Oldest | Next Newest »

 Messages In This Thread Tensor power series - by andydude - 05/13/2008, 07:58 AM RE: Tensor power series - by andydude - 05/13/2008, 07:59 AM RE: Tensor power series - by andydude - 05/13/2008, 08:11 AM RE: Tensor power series - by andydude - 05/14/2008, 06:18 AM RE: Tensor power series - by Gottfried - 05/20/2008, 08:39 PM RE: Tensor power series - by andydude - 05/22/2008, 12:58 AM RE: Tensor power series - by andydude - 05/22/2008, 04:11 AM RE: Tensor power series - by andydude - 05/22/2008, 04:36 AM RE: Tensor power series - by bo198214 - 05/24/2008, 10:10 AM RE: Tensor power series - by andydude - 06/04/2008, 08:08 AM

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