By Isaac Todhunter

This Elibron Classics e-book is a facsimile reprint of a 1864 variation by means of Macmillan and Co., Cambridge and London.

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This Elibron Classics e-book is a facsimile reprint of a 1863 version through Macmillan and Co. , Cambridge - London.

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This definition implies that T is a linear transformation from a linear space X into a linear space Y iff T n n α i xi i=1 = αi T (xi ) for all xi ∈ X and for all αi ∈ F , i=1 i = 1, . . , n. This result is also known as the principle of superposition. We cite some examples of linear transformations. 6. Let X = Y denote the space of real-valued function of the class t C[a, b]. Let T : X → Y be defined by T [x](t) = a x(s) ds, a ≤ t ≤ b, where integration is in the Riemann sense. Then T is a linear transformation.

4! + + 3 + 4 + · · · ; (b) + 2 + 3 + 4 · · · . 3 32 3 3 3 3 3 3 22 + 1 32 + 1 42 + 1 (c) 1 + 3 + + + ···. 2 + 1 33 + 1 43 + 1 n n+1 n! Hint. (a) sn = n , sn+1 = n+1 ; Ans. Converges; (b) sn = n , sn+1 = 3 3 3 (n + 1)! n2 + 1 (n + 1)2 + 1 . Ans. Diverges; (c) s = , s = ; Ans. Test n n+1 3n+1 n3 + 1 (n + 1)3 + 1 fails. (a) 2 The Concept of Green’s Functions Green’s function for a differential equation is its solution when the forcing term is the Dirac delta function due to a unit point source (or sink) in a given domain.

Consider the 2-D case of a concentrated force F of unit strength applied at the origin x = 0, x = (x, y), positioned at the boundary of a region D of the (x, y)-plane and directed along the z-axis. To measure F let this force be 1 This function was introduced by P. A. M. Dirac in 1926-27. The development of the analysis in this and next two sections has been reiterated by various authors, but it can be traced back to examples and ideas found in Dirac [1926-27; 1947]. 20 2. THE CONCEPT OF GREEN’S FUNCTIONS uniformly distributed over a circular area of radius ε and center at the origin.

### A Treatise On The Differential Calculus with numerous examples by Isaac Todhunter

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