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You can formulate the question also like this: What is the easiest way of calculating directed derivative of a function if its values are evaluated in a cartesian grid?

a) fit a (spline) surface, differentiate exactly

b) differentiate numerically in cartesian coordinates, project to angle

c) something else

The function at hand is a displacement vector ($u : \mathbb{R}^2 \to \mathbb{R}^2$ ) and I need the circumferential strain tensor component $$\varepsilon_{\theta\theta} = \frac{\partial u_\theta}{\partial \theta}$$. (Also the radial strain would is somewhat interesting.)

background: In 2d strain tensor has four components: $\varepsilon_{i,j}$, where i and j is either taken from {x,y} (cartesian) or {$\theta$,r} (polar). The first index defines the direction of displacement and the second index the differentiation direction, i.e. $$\varepsilon_{i,j} = \frac{\partial u_i}{\partial j} $$

u here is the 2d displacement vector. In cartesian coordinates above is easy, in polar not so much.

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    Not exaclty clear what you ask. Can it be more specific? What is $\Theta$, and in what meaning are you using $\nabla$?2012-09-27
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    ok, I added a bit more details. Basically I meant the angular derivative of the angular component. Also, added the word "tensor" where it was missing...2012-09-27
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    The verb corresponding to the noun "derivative" is "differentiate", not "derivate".2012-09-27
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    @joriki fixed, thanks.2012-09-28

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