Help on differentiation of tensor fields

[erkokite]

Given a tensor field T(x,y,z), how would I go about differentiating it wrt spatial coordinates?

I would presume that it would work like this:

$$\begin{equation} \frac{\partial T}{\partial x} = \lim_{h\to 0}\frac{T(x+h,y,z)-T(x,y,z)}{h} \end{equation}$$

However, this does not seem to take into account that tensor quantities themselves can act as functions of the spatial coordinates. My understanding is that a tensor field can, in some instances, act like a field that returns functions (or maps). If T returns tensors that are dependent on x, y, or z, wouldn't this have to be taken into account? Or is that another type of differentiation?

 

[pat_connell]

In other words, do I need to differentiate the tensors themselves, with respect to their spatial coordinates?

 

[tacman]

You are correct in your understanding that a tensor field can act as a field that returns functions or maps. In this case, the differentiation would involve the chain rule, as the tensor field is a composite function. For example, if we have a tensor field T(x,y,z) that returns a tensor T_{ij}(x,y,z), then the differentiation would be:

\begin{equation}
\frac{\partial T_{ij}}{\partial x} = \frac{\partial T}{\partial x}\frac{\partial x}{\partial x} + \frac{\partial T}{\partial y}\frac{\partial y}{\partial x} + \frac{\partial T}{\partial z}\frac{\partial z}{\partial x}
\end{equation}

where the partial derivatives of the spatial coordinates are taken into account. This is known as the covariant derivative, and it takes into account the fact that the tensor field itself can vary with respect to the spatial coordinates. In general, the covariant derivative of a tensor field T(x,y,z) is given by:

\begin{equation}
\nabla T = \frac{\partial T}{\partial x^i}dx^i
\end{equation}

where $dx^i$ are the differentials of the spatial coordinates. This allows for the differentiation of a tensor field with respect to the spatial coordinates, taking into account the varying nature of the tensor field itself. I hope this helps with your understanding of differentiating tensor fields.