亲爱的志刚，

地图是一个动作。没人需要把张量定义为动作。它可以被定义为一个对象——类似于向量。没有地图没有问题:)

Kosta

亲爱的Zhigang和Amit:

感谢你们有趣的讨论。

让我们考虑一个非常简单的例子。在欧几里德三维空间中运动的粒子。t时刻任意时刻的位置向量x(t)可以写成正则笛卡尔基向量{i，j，k}。如果你现在对x(t)对时间求导你就得到速度v(t)它对相同的基有分量{i，j，k}。我们认为位置矢量和速度矢量都在同一线性空间中。但是要理解我们不能加上位置矢量和速度矢量因为它们有不同的物理维度。

还要注意，当使用曲线坐标时，自然基将具有不同的物理尺寸，因此矢量的分量可能没有正确的物理尺寸，例如在球坐标中。在这里，我们必须使用所谓的物理组件。

If this particle moves in a curved space then position vector is not even defined. However, velocity is defined and at any given time t it lies in the tangent space to the ambient space at the point x(t). So, in this case one cannot even pose the question of adding "position" and velocity vectors.

Let us call the linear space of velocities at a point x, V. Using rate of change of energy or power, velocity and force are dual to each other (similarly force and displacement are dual to each other as Zhigang mentioned). Forces live in the dual space of V, i.e. V*. This is the space of one-forms or covectors. Choosing a basis for V, one has a natural dual basis for V*. These, in general, have different physical dimensions. There is a natural pairing between elements of V and V* and this natural pairing does not require an inner product (or metric). An inner product (metric) induces an isomorphism between V and V*, i.e. one can identify the two linear spaces. Note that, in general, this isometry changes from point to point.

Given two linear spaces V and W an isomorphism is a one-to-one and onto linear map. If you specify the map on a basis for V then the map is uniquely defined. The two vector spaces being isomorphic doesn't mean you can add a vector in V to a vector in W. In this sense, identifying the linear space of forces with that of velocities doesn't mean you can add the two.

I think Cauchy stress is a tensor in the sense of tensors in the notes of Prof. Carlson using the isomorphy between vectors and covectors. One doesn't even need to worry about identifying the linear spaces of forces and velocities. Cauchy stress can be simply represented using a basis of covectors (and is obviously symmetric). However, again, I agree with Zhigang that for mechanics applications one should work with multilinear maps on a collection of linear spaces that are not necessarily identical. The obvious examples are deformation gradient and the first Piola-Kirchhoff stress.

Regards,
Arash