Mathematical Tools for Physics
13—Vector Calculus 2 413 13.5 The manipulation in the final step of Eq. ( 10 ) seems almosttooobvious. Is it? Well yes, but writ ...
13—Vector Calculus 2 414 13.14 For the vector field in thex-yplane:F~= ( xˆy−yxˆ ) / 2 , use Stokes’ theorem to compute the line ...
13—Vector Calculus 2 415 13.26 Prove the identity∇.(fF~) =f∇.F~+F~.∇f. (b) Apply Gauss’s theorem to∇.(fF~)for a constantF~ to de ...
13—Vector Calculus 2 416 13.31 Derive the analog of Reynolds’ transport theorem for a line integral around a closed loop. d dt ∫ ...
13—Vector Calculus 2 417 13.36 Verify the divergence theorem for the vector field F~=αˆxxyz+βˆy x^2 z(1 +y) +γˆz xyz^2 and for t ...
Complex Variables In the calculus of functions of a complex variable there are three fundamental tools, the same fundamental too ...
14—Complex Variables 419 There are functions that are continuous but with no derivative anywhere. They’re harder* to construct, ...
14—Complex Variables 420 left you get+1. From above and below (θ=±π/ 2 ) you get− 1. The limits aren’t the same, so this functio ...
14—Complex Variables 421 The common formulas for differentiation are exactly the same for complex variables as they are for real ...
14—Complex Variables 422 Just asξkis a point in thekthinterval, so isζka point in thekthinterval along the curveC. How do you ev ...
14—Complex Variables 423 14.3 Power (Laurent) Series The series that concern us here are an extension of the common Taylor or po ...
14—Complex Variables 424 1 /z(z−1)has a zero in the denominator for bothz= 0andz= 1. What is the full behavior near these two po ...
14—Complex Variables 425 The structure of a Laurent series is such that it will converge in an annulus. Examine the absolute con ...
14—Complex Variables 426 A major result is that when a function is analytic at a point (and so automatically in a neighborhood o ...
14—Complex Variables 427 function however says that a function is single valued, so what is this? I’ll leave the answer to this ...
14—Complex Variables 428 Another way to do the integral is to use the residue theorem. There are two poles inside the contour, a ...
14—Complex Variables 429 move across a singularity, you can distort a contour at will. I will push the contourC 1 up, but I have ...
14—Complex Variables 430 ∫ C 1 eikzdz a^4 +z^4 C 1 I’m going to use the same method as before, pushing the contour past some pol ...
14—Complex Variables 431 Do you have to do a lot of algebra to evaluate this denominator? Maybe you will prefer that to the alte ...
14—Complex Variables 432 theorem here is to create a singularity where there is none. Write the sine as a combination of exponen ...
«
16
17
18
19
20
21
22
23
24
25
»
Free download pdf