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To construct a multivariate B-spline, we simply take the Kronecker tensor product between the univariate basis functions constructed for each individual dimension.

What I'd like to know is how do you construct a B-spline derivative for a multivariate function? Is it the same method of using the tensor product, but with the univariate B-spline derivatives?

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Remember that you can "slice" a multivariate function: let one variable vary and fix the rest. –  J. M. Dec 2 '10 at 0:31
Ok, so basically I should set all variables to constants except the dimension I want to slice, take the univariate derivative splines for the constants and sliced dimension and combine them with the tensor product? –  Projectile Fish Dec 2 '10 at 0:38
You'll have to also decide in advance how "thin" your slices would be (discretization) in each dimension. –  J. M. Dec 2 '10 at 0:44
Oh hang on, so do I need to model the function as a 1D slice first and then get the derivative, or is it possible to model it as a multivariate function then take the sliced derivative of the multivariate model? –  Projectile Fish Dec 2 '10 at 0:59
Where else would you get the partial derivatives needed by the B-spline? ;) –  J. M. Dec 2 '10 at 11:47

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