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## Main Question or Discussion Point

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

I would presume that it would work like this:

[tex]

\begin{equation}

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

\end{equation}

[/tex]

However, this does not seem to take into account that tensor quantities themselves can act as functions of the spacial 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?

I would presume that it would work like this:

[tex]

\begin{equation}

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

\end{equation}

[/tex]

However, this does not seem to take into account that tensor quantities themselves can act as functions of the spacial 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?