Mathematical Tools for Physics - Department of Physics - University

15—Fourier Analysis 374 enough do you have a real chance to discern what note you’re hearing. This is a reflection of the facts ...

15—Fourier Analysis 375 This example has four main peaks in the frequency spectrum. The real part ofgis an even function and the ...

15—Fourier Analysis 376 In the last line I interchanged the order of integration, and in the preceding line I had to be sure to ...

15—Fourier Analysis 377 The explicit values ofω±are ω+= −ib+ √ −b^2 + 4km 2 m and ω−= −ib− √ −b^2 + 4km 2 m Let ω′= √ −b^2 + 4km ...

15—Fourier Analysis 378 15.6 Sine and Cosine Transforms Return to the first section of this chapter and look again at the deriva ...

15—Fourier Analysis 379 This is the Fourier Sine transform. For a parallel calculation leading to the Cosine transform, see prob ...

15—Fourier Analysis 380 Problems 15.1 Invert the Fourier transform,g, in Eq. (15.7). 15.2 What is the Fourier transform ofeik^0 ...

15—Fourier Analysis 381 15.13 Show that iff(t)is real then the Fourier transform satisfiesg(−ω) =g*(ω). What are the properties ...

15—Fourier Analysis 382 15.22 Repeat the calculations leading to Eq. (15.21), but for the boundary conditionsu′(0) = 0 = u′(L), ...

Calculus of Variations . The biggest step from derivatives with one variable to derivatives with many variables is from one to t ...

16—Calculus of Variations 384 conservation of energy, the expression for the time to slide down a curve was Eq. (13.6). x y ∫ dt ...

16—Calculus of Variations 385 And the travel time for light through an optical system is ∫ dt= ∫ d` v = ∫b a dx √ 1 +y′^2 v(x,y) ...

16—Calculus of Variations 386 plus terms of higher order inδyandδy′. Put this into Eq. (16.6), and δI= ∫b a dx [ ∂F ∂y δy+ ∂F ∂y ...

16—Calculus of Variations 387 The differential change in the function depends linearly on the changed~rin the coordinates. It is ...

16—Calculus of Variations 388 16.3 Brachistochrone Now for a tougher example, again from the introduction. In Eq. (16.2), which ...

16—Calculus of Variations 389 Make the substitution(y−a)^2 =zin the first half of the integral and(y−a) =asinθin the second half ...

16—Calculus of Variations 390 16.4 Fermat’s Principle Fermat’s principle of least time provides a formulation of geometrical opt ...

16—Calculus of Variations 391 At this point pick a form for the index of refraction that will make the integral easy and will st ...

16—Calculus of Variations 392 Now pull out a vector identity from problem9.36, ∇. ( f~g ) =∇f.~g+f∇.~g and apply it to the previ ...

16—Calculus of Variations 393 Let the potential atr=abeVaand atr=bit isVb. An example function that satisfies these conditions i ...