By Olga Waelder
This interesting paintings makes the hyperlink among the rarified global of maths and the down-to-earth one inhabited via engineers. It introduces and explains classical and smooth mathematical systems as utilized to the true difficulties confronting engineers and geoscientists. Written in a way that's comprehensible for college students around the breadth in their reports, it lays out the principles for getting to know tough and occasionally complicated mathematical equipment. mathematics examples and figures absolutely aid this process, whereas all vital mathematical options are precise. Derived from the author's lengthy adventure educating classes in utilized arithmetic, it's in line with the lectures, routines and classes she has utilized in her classes.
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Extra resources for Mathematical Methods for Engineers and Geoscientists
Therefore, we have to fit a weighted average considering information from location points of the regionalized variable to obtain an estimation of a certain value at a fixed point (x0 , y0 ) or, in short, x0 . This basic idea can be explained using a simple example. Let [z (x1 , y1 ) , z (x2 , y2 )] be a regionalized variable that can, for example, be the result of two drillings from an oil field. Or think about measurements of, for instance, temperature or soil parameters. However, we are looking for weights, wi , i = 1, 2, to predict at the point (x0 , y0 ).
1 Prediction of a Value: Creating, Refining, or Changing Measurement Grids 39 (3-12). For example, a mean polygon can be constructed in this way. One minor disadvantage might be the unconventional parametric representation of B-splines. 9 B-Splines (2D-Case) Here the basic idea corresponds to an iterated bilinear interpolation. We begin with some definitions. Let b00 , b01 , b10 , b11 be four points in E 3 . A hyperbolic paraboloid allows the following parametric representation: 1 1 z (u, v) = ∑ ∑ bi j B1i (u)B1j (v) , mit i=0 j=0 k = 0, 1; B1k (w) = (1 − w)1−k wk , w = u, v In matrix form this corresponds to z (u, v) = 1 − u u b00 b01 b10 b11 1−v v Now the common rule for constructing B-curves reads as follows: Let bi j N i, j=0 = (xi j , yi j , zi j )T N i, j=0 be (N + 1)(N + 1) three-dimensional data points (measurements on a square grid) and (u, v) ∈ R2 be two parameters.
5). The common rule for two-dimensional polynomial regression reads as follows: Let the following data be given: z1 = z (x1 , y1 ) , . . , zN = z (xN , yN ). A functional relation of the form K z (x, y) = L ∑ ∑ akl · xk · yl , (K + 1) (L + 1) ≤ N k=0 l=0 is searched for in such a way that the following necessary condition is fulfilled: N F(akl : k = 0 . . K, l = 0 . . L) = ∑ i=1 K L ∑∑ k=0 l=0 2 akl · xik · yli − zi → min akl :k=0 ... K, l=0 ... L Fig. 2 36 3 Some Real Problems and Their Solutions This condition leads to ⎧ N K L ∗ ∗ ⎨ ∂F = ∑ 2 ∑ ∑ akl xik yli − zi xik yli = 0 ∗ ∗ ∂a i=1 k=0 l=0 ⎩ ∗ kl k = 0 .