By Alfred North Whitehead

Symbolic Reasoning allied to bland A lgebra. the manager examples of such structures are Hamilton sQ uaternions, Grassmann sC alculus of Extension and Boole sS ymbolic good judgment. Such algebras have an intrinsic worth for separate distinct learn; they are also priceless of a comparative learn, for the sake of the sunshine thereby thrown at the basic concept of symbolic reasoning, and on algebraic symbolism particularly. The comparative research inevitably presupposes a few earlier separate examine, comparability being most unlikely with no wisdom. as a result after the final ideas of the entire topic were mentioned in ebook I. of this quantity, the remainder books of the quantity are dedicated to the separate examine of the Algebra of Symbolic good judgment, and of Grassmann sC alculus of Extension, and of the guidelines excited by them. the assumption of a generalized notion of area has been made in demand, within the trust that the houses and operations eager about it may be made to shape a uniform approach to interpretation of some of the algebras. therefore it truly is was hoping during this paintings to convey the algebras either as structures of symbolism, and in addition as engines for the research of the chances of notion and reasoning attached with the summary common suggestion of area. A usual mode of comparison.between the algebras is hence right away supplied by means of the cohesion of the subject-matters in their interpretation. The particular comparability in their symbolic buildings has been adjourned to the second one quantity, within which it really is meant to house Quaternions, Matrices, and the final concept of Linear A lgebras. This comparative anatomy of the topic used to be originated by means of B. Peirce spaper on Linear Associative A lgebra, and has been carried ahead by way of more moderen investigations in Germany. Firat learn ahead of theN ational Academy of Soienoes in Washington, 1871, and repabliahed within the American magazine of M

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12. 144), respectively. 13. Prove that the Laplacian of a vector-valued function t (r) can be given by Δt = t,i|i . Specify this identity for Cartesian coordinates. 14. 144), respectively. 15. Write out the Laplacian of a vector ﬁeld t (r) in component form in an arbitrary curvilinear coordinate system. 144). 1) where the real variable t belongs to some interval: t1 ≤ t ≤ t2 . 2) over the whole deﬁnition domain. 1) can alternatively be deﬁned by θi = θi (t) , i = 1, 2, 3. 4) Example. Straight line.

4. Write out in full the following expressions for n = 3: (a) δji aj , (b) δij xi xj , ∂fi j dx . 5. Prove that 0 · x = 0, ∀x ∈ En . 6. Prove that a set of mutually orthogonal non-zero vectors is always linearly independent. 7. Prove the so-called parallelogram law: x + y 2 = x 2 + 2x · y + y 2 . 8. Let G = {g 1 , g 2 , . . , g n } be a basis in En and a ∈ En be a vector. Prove that a · g i = 0 (i = 1, 2, . . , n) if and only if a = 0 . 9. Prove that a = b if and only if a · x = b · x, ∀x ∈ En .

Proof. First we prove the suﬃciency. To this end, we map the vector equation Ax = 0 by A−1 . 124) it yields: 0 = A−1 Ax = Ix = x. To prove the necessity we consider a basis G = {g 1 , g 2 , . . , g n } in En . It can be shown that the vectors hi = Ag i (i = 1, 2, . . , n) form likewise a basis of En . Conversely, let these vectors be linearly dependent so that ai hi = 0 , where not all scalars ai (i = 1, 2, . . , n) are zero. Then, 0 = ai hi = ai Ag i = Aa, where a = ai g i = 0 , which contradicts the assumption of the theorem.

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