## Recent **Others** Questions & Answers

###### Show that if fn(x) - - ?? f(x) in the mean a n d gn(x) ???? g(x) u n i formly i n some interval [a, b], then r fn(x)gn(x) dx - r f(x)g(x) dx. Hint. Us...

##### Show that if fn(x) - - ?? f(x) in the mean a n d gn(x) ???? g(x) u n i formly i n some interval [a, b], then r fn(x)gn(x) dx - r f(x)g(x) dx. Hint. Use Schwarz's inequal ity....

###### Prove that any function hex) E ...

##### Prove that any function hex) E ...

###### Formulate a variat ional problem leading to the Sturm-Liouville equation ( 1 4) subject to the boundary conditions y'(a) = 0, y'(b) = 0, instead of th...

##### Formulate a variat ional problem leading to the Sturm-Liouville equation ( 1 4) subject to the boundary conditions y'(a) = 0, y'(b) = 0, instead of the boundary conditions ( 1 5). Hint. Recall the natural boundary conditions (29) of Sec. 6....

###### Write the Sturm-Liouville equation associated with the quadratic functional J [y] = r (Cly'2 + cy2) dx, where c and Cl > 0 are constants, subject to t...

##### Write the Sturm-Liouville equation associated with the quadratic functional J [y] = r (Cly'2 + cy2) dx, where c and Cl > 0 are constants, subject to the boundary conditions yea) = 0, y(b) = o. Find the corresponding eigenvalues and eigenfunctions....

###### Use the Ritz method to find an approximate solution of the equation 82u 82u - + - = - I 8x2 8y2 inside the square R : - a ?? x ?? a, - a ?? y ?? a, wh...

##### Use the Ritz method to find an approximate solution of the equation 82u 82u - + - = - I 8x2 8y2 inside the square R : - a ?? x ?? a, - a ?? y ?? a, where u vanishes on the boundary of R. Hint. Study the functional J[u] = J L [(::r + (:;r - 2u] dx dy, and choose the two-dimensional generalization of ...

###### Use the Ritz method to find an approximate solution of the problem of minimizing the functional J [y] = J: (y'2 + y2 + 2xy) dx, y(O) = y(2) = 0, and c...

##### Use the Ritz method to find an approximate solution of the problem of minimizing the functional J [y] = J: (y'2 + y2 + 2xy) dx, y(O) = y(2) = 0, and compare the answer with the exact solution....

###### Use the Ritz method to find an approximate solution of the extremum problem associated with the functional J [y] = f (X3yH2 + 1 00xy2 - 20xy) dx, Hint...

##### Use the Ritz method to find an approximate solution of the extremum problem associated with the functional J [y] = f (X3yH2 + 1 00xy2 - 20xy) dx, Hint. Choose the sequence {tpn(x)} to be y( l ) = y'( l ) = o. (x - 1 )2, x(x - 1 )2, x2(x - 1 )2, . . ....

###### Use the Ritz method to find an approximate solution of the problem of minimizing the functional J [y] = J01 (y'2 - y2 - 2xy) dx, y(O) = y( l ) = 0, an...

##### Use the Ritz method to find an approximate solution of the problem of minimizing the functional J [y] = J01 (y'2 - y2 - 2xy) dx, y(O) = y( l ) = 0, and compare the answer with the exact solution. Hint. Choose the sequence {tpn(x)} (see p. 1 95) to be x( l - x), x2( l - x), x3( l - x), . . ....

###### Let the functional J [y] be such that J [y] > - 00 for some admissible function, and let sup J [y] = !J. < + 00 , where sup denotes the least upper bo...

##### Let the functional J [y] be such that J [y] > - 00 for some admissible function, and let sup J [y] = !J. < + 00 , where sup denotes the least upper bound or supremum. By analogy with the treatment given in Sec. 39, define a maximizing sequence, and then state and prove the corresponding version of t...

###### Show that the Lagrangian density It' of the preceding problem is Lorentzinvariant if u transforms like a scalar and if Ao, A l o A 2 , Aa transform li...

##### Show that the Lagrangian density It' of the preceding problem is Lorentzinvariant if u transforms like a scalar and if Ao, A l o A 2 , Aa transform like the components of a vector under Lorentz transformations. Use this fact to derive various conservation laws for the field described by It'....