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G , C ≔ y 6 − 3 y 3 , y 5 − 2 y 2 + x , 1 2 x y + 1 2 + 1 2 y 6 − 3 4 x y 4 + 1 2 y 3 − 1 4 x 2 y 5 , − 1 2 x 2 − 3 4 x y 5 + 1 2 x 2 y 3 − x y 2 − 1 2 y 4 + 3 2 y + 1 4 x 3 y 4 , 1 2 y 2 − 1 2 x y 3 , − x y + 1 + 1 2 x 2 y 2 − 1 2 y 3 The TensorArray command also allows you to choose whether or not to sum over the repeated indices of tensorial expressions in the returned Array, and to display the matrix components of the Dirac, Pauli or Gell-Mann matrices - for details see TensorArray . Systems of spacetime coordinates are: X = x1 , x2 , x3 , x4 , Y = y1 , y2 , y3 , y4 The optional argument output=extended computes a transformation matrix C expressing the Groebner basis in terms of the generators J . The output is a sequence (G, C) where G is the Groebner basis and C is a list of lists. Each G[i] is equal to the dot product of C[i] with J . This matrix is not stored, so if it is requested again it must be recomputed together with the Groebner basis.
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Some maple wood has a highly decorative wood grain, known as flame maple, quilt maple, birdseye maple and burl wood. This condition occurs randomly in individual trees of several species and often cannot be detected until the wood has been sawn, though it is sometimes visible in the standing tree as a rippled pattern in the bark. There are 10 species naturally growing in the country, with at least one in each province. Although the idea of the tree as a national symbol originally hailed from the province of Quebec [17] where the sugar maple is significant, today's arboreal emblem of Canada rather refers to a generic maple. [18] The design on the flag is an eleven-point stylization modeled after a sugar maple leaf (which normally bears 23 points). [19]
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One milliliter (ml) of granulated sugar equates to 0.85 grams. To estimate how many grams there are in a ml volume of sugar, Color/Appearance: Unlike most other hardwoods, the sapwood of hard maple lumber is most commonly used rather than its heartwood. Sapwood color ranges from nearly white, to an off-white cream color, sometimes with a reddish or golden hue. The heartwood tends to be a darker reddish brown. Birdseye maple is a figure found most commonly in hard maple, though it’s also found less frequently in other species. Hard maple can also be seen with curly or quilted grain patterns. B μ , ν , γ μ , σ μ , R μ , ν , α , β , U α , β , μ , ν , W α , β , μ , ν , ∂ μ , g μ , ν , ε α , β , μ , ν
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If you want to simplify or contract only some of the factors in a product, you can use the `.` dot product operator instead of the `*` star operator and enclose the sub-expression you want simplified in parenthesis. For example, compare the following output with (131) and the complete simplification above (132) One common operation on tensorial expressions is to sum over some or all of its repeated indices. > TensorArray Ricci α , ρ Ricci β , ν Ricci μ , σ , exploreForeman, Grant (1934). The Five Civilized Tribes. Norman: University of Oklahoma Press. pp.283–284. ISBN 978-0-8061-0923-7. To change the coordinates used by the differential operators d_ , D_ and dAlembertian , for example from X to Y, enter Setup differentiationvariables = Y . The trace of the metric g_ is equal to the dimension of spacetime, which is equal to 4 by default (you can change this value using Setup ) The first attested use of the word was in 1260 as "mapole", and it also appears a century later in Geoffrey Chaucer's Canterbury Tales, spelled as "mapul". [20] The maple is also a symbol of Hiroshima, ubiquitous in the local meibutsu. For water, 1 milliliter equals 1 gram. For other ingredients, the weight varies. 1ml of milk weighs 1.04 grams, 1ml of flour weighs 0.53 grams
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The coordinates of the first system of coordinates you set are used by the d_ , D_ and dAlembertian commands as the default differentiation variables. It often occurs that two products have the same tensors with the same contracted indices, but in each product the contracted indices are represented by different letters, thus obscuring the fact that the two products are mathematically equal. When defining a tensor this way, without passing a tensorial equation or the keyword minimizetensorcomponents , the symmetry indicated is not automatically taken into account. That allows for further abstract manipulations, and when you want the symmetry to be taken into account you can use Simplify >Due to the use of alias , the label of a coordinate system - say X - actually represents the sequence of the four coordinates, and so you can use this label to refer to all the coordinates at once, for example, when specifying the functionality of functions >
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