By Eisenhart L.P.
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Extra resources for Transformations of Applicable Conjugate Nets of Curves on Surfaces
1 Correspondence between the i-Plane of Vectors and the Spinor Plane From the foregoing, one notices that each vector x in the i-plane of vectors determines a unique spinor z in the spinor plane as given by z = σ1 x = σ1 (x1 σ1 + x2 σ2 ) = x1 + i x2 . 19) Conversely, each spinor z in the spinor plane determines a unique vector x in the vector plane as follows: σ1 x = z, σ12 x = σ1 z x = σ1 z. 20) Distinction between Vector and Spinor Planes The elements of the vector and spinor planes have different algebraic properties because of two distinct interpretations of i in them, which endow different geometric significance to each of the planes.
K=0 = 1 + A/1! + A /2! + · · · + A /k! 57) if |A| has a definite magnitude. 57) can be shown to be absolutely convergent for all values of A provided |A| has a definite magnitude. So, it may be extended to general multivectors. 57) is completely defined in terms of the basic operations of addition and multiplication (geometric product), which determine all the properties of the function. By using the closure property of the geometric algebra under the operations of addition and multiplication (geometric product) it can be shown that exp( A) is a definite multivector.
13): a (b + c) = a · (b + c) + a ∧ (b + c) = (a · b + a · c) + (a ∧ b + a ∧ c) = (a · b + a ∧ b) + (a · c + a ∧ c) = ab + ac [definition of geometric product] [associative rules for addition] [rearrangement of terms] [definition of geometric product]. 14). 14) are independent of one another because the geometric product is, in general, neither commutative nor anticommutative. In any algebra the associative property is extremely useful in algebraic manipulations. For this purpose we assume that for any three vectors a , b, and c the geometric product is associative: a (bc) = (ab)c = abc.