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Say you have a physical simulation, there are "wind current" vectors stored in a 2d space.

example of wind vectors

So you know that the vectors near each other will likely be similar in direction.

Can we capitalize on the "similarity" across the vector field, and use it to write an approximation to the vector field?

So is there an alternative way to represent a vector field (something like a Fourier Transform for vector fields?)

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    An interesting question, to be sure, but I'm not sure if it counts as [tag:multivariable-calculus], and I can't help but think it might be better answered on a more information theory oriented site.2011-12-30
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    I disagree Zhen, I think the question is asking: given a collection of tangent vectors in $\mathbb{R}^2$, is there always a continuous vector field that extends it?2011-12-30
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    Look at the component functions. If the vector field is smooth, then you could perform a Taylor expansion (for instance) on the components, thus approximating the vector field. Sketch the case where one of the components is constant.2011-12-30
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    You can always do an interpolation considering the nearest vectors to your location.2011-12-30
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    Anything you can do to a scalar field, you can do to the components of a vector field... Well, not quite: anything *linear* in the scalar values can be applied to the components of a vector field, and the result will be sensible under change of basis of the vector values. So you can do bilinear interpolation, spline interpolation, componentwise Fourier transforms, and so on...2012-08-26
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    This is an essential problem in numerical fluid dynamics. It's not enough to come up with some smooth interpolation of the given data. One should also take care that "conservation laws" at work in these data are represented in the interpolation. Otherwise during numerical processing the system will be fueled, e.g., with extraneous energy not present in the situation on the ground.2013-02-07

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