Download Volume 2, Methods of Mathematical Physics by R.; Hilbert, D. Courant PDF

Download Volume 2, Methods of Mathematical Physics by R.; Hilbert, D. Courant PDF

By R.; Hilbert, D. Courant

Because the first quantity of this paintings got here out in Germany in 1937, this publication, including its first quantity, has remained regular within the box. Courant and Hilbert's remedy restores the traditionally deep connections among actual instinct and mathematical improvement, supplying the reader with a unified method of mathematical physics. the current quantity represents Richard Courant's ultimate revision of 1961.

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Extra info for Volume 2, Methods of Mathematical Physics

Example text

These functions form a complete integral of the differential equation u2(1 (10) + p 2 + q2) = 1. , the surface obtained by eliminating a from (11) (5 2 + (y - w(a))2+ u2 = 1, - a + w’(a)(y - w ( a ) ) = 0, yields another solution. Every such envelope is a tubular surface whose axis is y = ~ ( 5 ) . The total two-parameter family (9) also has another envelope consisting of the planes u = 1 and u = - 1 ; this is intuitively clear and can be confirmed analytically by eliminating a and b from (5 + (y - b)2 + u 2 - = 1, 2 - u = o , z-b=O.

The first derivatives 7 ~ = , uy are immediately obt8ainedfrom (9), (10): hence (11) In order to eliminate the arbitrary function w,we differentiate once more: <;F:NEI~AIIINFORMATION ON V A R I E T Y O F SOLUTIONS 11 as the desired differential eqiiation for all clcvelopxhle surfaces except cylinders perpendicwlal. to the x , pplanc. , that all solutions of the corresponding differential equations belong to the given sets of functions. cz , . '~ , ' ' ' , Tn), identically satisfied in t . If we set t = 1 / ~ ,, we ~ have hence 71 is represented by with sonic function w.

Thus, the function u depends on two arbitrary functions in a rather involved way. The first derivatives 7 ~ = , uy are immediately obt8ainedfrom (9), (10): hence (11) In order to eliminate the arbitrary function w,we differentiate once more: <;F:NEI~AIIINFORMATION ON V A R I E T Y O F SOLUTIONS 11 as the desired differential eqiiation for all clcvelopxhle surfaces except cylinders perpendicwlal. to the x , pplanc. , that all solutions of the corresponding differential equations belong to the given sets of functions.

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