Mathematical Tools for Physics
10—Partial Differential Equations 293 On this interval 0 < x < athese sine functions are orthogonal, so you take the scala ...
10—Partial Differential Equations 294 Does the exact answer look like this? It doesn’t seem to, but look closer. Ifbathen becau ...
10—Partial Differential Equations 295 As compared to the previous approximation whereab, you can’t as easily tell whether this ...
10—Partial Differential Equations 296 flowingintothis area has the reversed sign, +κ∂T/∂y (26) The total power flow is the integ ...
10—Partial Differential Equations 297 equations are telling you so! It means that the boundary conditions that I specified in Eq ...
10—Partial Differential Equations 298 O 0 b y 0 F 0 a 0 x −F 0 y/κ O b y 0 a −F 0 y/κ x If I can find a solution that equals−F 0 ...
10—Partial Differential Equations 299 the central axis of the box, aboutx=a/ 2. That it is even about this point implies that th ...
10—Partial Differential Equations 300 Now I have to analyze this to see if it makes sense. I’ll look at the same cases that I di ...
10—Partial Differential Equations 301 −L 0 L 2 L x z y V^0 Arrange a set of conducting strips in thex-yplane and with insulation ...
10—Partial Differential Equations 302 The separated solutions are then f(x)g(z) =enπix/L ( Aenπz/L+Be−nπz/L ) (35) The solution ...
10—Partial Differential Equations 303 Now evaluate the integral on the left side. First,m 6 = 0: 〈 emπix/L,V(x,0) 〉 = ∫L −L dxe− ...
10—Partial Differential Equations 304 Having done all the work to get to the answer, what can I learn from it? What does it look ...
10—Partial Differential Equations 305 The original potential on the surface was neither even nor odd, but except for the constan ...
10—Partial Differential Equations 306 z a b x y The equation is still Eq. ( 33 ), but now you have to do the separation of varia ...
10—Partial Differential Equations 307 be represented by a sum of sines inx, and the same statement will hold for theycoordinate. ...
10—Partial Differential Equations 308 Problems 10.1 The specific heat of a particular type of stainless steel (CF8M) is 490 J/kg ...
10—Partial Differential Equations 309 10.7 In the analysis leading to Eq. ( 23 ) the temperature aty =b was set toT 0. If instea ...
10—Partial Differential Equations 310 10.13 Fill in the missing steps in finding the solution, Eq. ( 31 ). 10.14 A variation on ...
10—Partial Differential Equations 311 I picked the axis forθ= 0pointing toward the split between the cylinders. No particular re ...
10—Partial Differential Equations 312 10.24 In the preceding problem suppose that there’s noz-dependence. In problem 15 you foun ...
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