## Definition of a Tensor

I am still a physics novice and am learning new things everyday. I've been looking at tensors recently and I'm finding that I can't really understand what they are. Could someone explain in relatively simple words what the definition of a tensor is and why they are so significant? Also, what is the significance of Riemann's metric tensor, which I read about in Michio Kaku's Hyperspace. Examples would be greatly appreciated. Thanks!
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 See this video: http://www.youtube.com/watch?v=f5liqUk0ZTw Take a look at this thread, where the OP asked a similar question. There are a variety of good explanations that you may find helpful. http://www.physicsforums.com/showthread.php?t=623124 Also, you're mixing together two different tensors - the Riemann curvature tensor and the metric tensor.
 Yes, my mistake. I meant the Riemann curvature tensor.

## Definition of a Tensor

What is a tensor? Really, there are two kinds of tensors. One kind of tensor is a linear operator that, for example, maps vectors to other vectors. Hence the matrices representing rotations, reflections, and the like are representations of such tensors.

The other kind of tensor represents a generalization of a vector. From the span of two vectors, you can build a plane. From three vectors, you can build a volume, and so on. Some tensors represent these objects.

For both kinds of tensors, there is a transformation law based on the idea that a change of coordinates should not change the tensor itself--new coordinates may change the components, but the overall object should remain unchanged. It is this common transformation law (of the components) that is why both types of tensors--the linear operators and the generalization of vectors--are typically lumped together.

 Tags definition, riemann, riemann's metric, tensor