Mathematical Tools for Physics - Department of Physics - University

10—Partial Differential Equations 254 The sines are orthogonal by the theorem Eq. (5.15), so you can pick out the componentanby ...

10—Partial Differential Equations 255 does one EM wave get through while the other one doesn’t? I won’t solve the whole electrom ...

10—Partial Differential Equations 256 without bound. Terms such aseπz/Lhowever increase withz. This means that the coefficients ...

10—Partial Differential Equations 257 Having done all the work to get to the answer, what can I learn from it? What does it look ...

10—Partial Differential Equations 258 odd. The constant term in Eq. (10.30) is then= 0element of the cosine set, and that’s nece ...

10—Partial Differential Equations 259 whereknmis thek 3 of the preceding equation. What happened to the other term inz, the one ...

10—Partial Differential Equations 260 α=n^2 > 0 d^2 g dφ^2 =−n^2 g =⇒ g(φ) =Acosnφ+Bsinnφ r^2 d^2 f dr^2 +r df dr −n^2 f= 0 = ...

10—Partial Differential Equations 261 Multiply bysinmφ and integrate overφ. The trigonometric functions are orthogonal, so all t ...

10—Partial Differential Equations 262 Problems 10.1 The specific heat of a particular type of stainless steel (CF8M) is 500 J/kg ...

10—Partial Differential Equations 263 there must be zero. That in turn means that the sum over all the rest of the separated ter ...

10—Partial Differential Equations 264 I picked the axis forφ= 0pointing toward the split between the cylinders. No particular re ...

10—Partial Differential Equations 265 10.25 Do problem8.23and now solve it, finding all solutions to the wave equation. Ans:f(x− ...

10—Partial Differential Equations 266 10.35 For the preceding equations show that there are solutions of the formf(r) =Arn, and ...

Numerical Analysis . You could say that some of the equations that you encounter in describing physical systems can’t be solved ...

11—Numerical Analysis 268 The line passing through the two points of the graph is y−f 0 = (x−x 0 )(f 1 −f 0 )/h, x 0 x 1 f 0 =f( ...

11—Numerical Analysis 269 11.2 Solving equations Example:sinx−x/2 = 0 From the first graph, the equation clearly has three real ...

11—Numerical Analysis 270 A milder form of non-convergence can occur if at the root the curvature changes sign and is large, as ...

11—Numerical Analysis 271 As usual, the geometric approach doesn’t indicate the size of the error, so it’s back to Taylor’s seri ...

11—Numerical Analysis 272 Again, you have a more accurate formula by evaluating the derivative between the data points: h= 2k f( ...

11—Numerical Analysis 273 This is more fully explained in section1.6, and it is the basis for the various methods of numerical e ...