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## Main Question or Discussion Point

In physics texts, its customary to write (and even to define the gradient as) the following:

[tex]dT = (\nabla T) \cdot dl[/tex]

Working in Cartesian coordinates, we can expand this as follows:

[tex]dT = \frac{\partial T}{\partial x} dx + \frac{\partial T}{\partial y} dy + \frac{\partial T}{\partial z}dz[/tex]

But the above equation is really just a "first-order" approximation for the differential dT, is it not? Working just in the differential dx, a more "complete" way to write this expression would be:

[tex]dT \approx \frac{\partial T}{\partial x} dx + \frac{1}{2} \frac{\partial ^2T}{\partial x^2} (dx)^2 + \frac{1}{6} \frac{\partial ^3T}{\partial x^3} (dx)^3 + ... [/tex]

along with the additional, commensurate expansions in terms of the y and z variables as well.

So while we can define equations in terms of gradient, and work with gradients, etc, is it fair to say that all these gradient expressions are really just giving us answers which are first-order approximations? And that if we desired more precise answers, we would have to delve deeper into the higher-order terms?

[tex]dT = (\nabla T) \cdot dl[/tex]

Working in Cartesian coordinates, we can expand this as follows:

[tex]dT = \frac{\partial T}{\partial x} dx + \frac{\partial T}{\partial y} dy + \frac{\partial T}{\partial z}dz[/tex]

But the above equation is really just a "first-order" approximation for the differential dT, is it not? Working just in the differential dx, a more "complete" way to write this expression would be:

[tex]dT \approx \frac{\partial T}{\partial x} dx + \frac{1}{2} \frac{\partial ^2T}{\partial x^2} (dx)^2 + \frac{1}{6} \frac{\partial ^3T}{\partial x^3} (dx)^3 + ... [/tex]

along with the additional, commensurate expansions in terms of the y and z variables as well.

So while we can define equations in terms of gradient, and work with gradients, etc, is it fair to say that all these gradient expressions are really just giving us answers which are first-order approximations? And that if we desired more precise answers, we would have to delve deeper into the higher-order terms?