Mathematical Tools for Physics - Department of Physics - University

13—Vector Calculus 2 334 problem13.17. If you have a loop that encloses the singular line, then you can’t shrink the loop withou ...

13—Vector Calculus 2 335 You can easily verify thatA~=B~×~r/ 2 is a vector potential for the uniform fieldB~. Neither the scalar ...

13—Vector Calculus 2 336 Now for the next two terms, which require some manipulation. Add and subtract the surface that forms th ...

13—Vector Calculus 2 337 Now take a uniform static field 3 : B~=B 0 ˆz with a radially expanding surface z= 0, x^2 +y^2 < R^2 ...

13—Vector Calculus 2 338 For a proof, just write it out and then find the vector identity that will allow you to integrate by pa ...

13—Vector Calculus 2 339 The method is essentially two partial integrals, moving two derivatives fromF~over to~u. Start with the ...

13—Vector Calculus 2 340 This combination of results, the Helmholtz theorem, describes a field as the sum of a gradient and a cu ...

13—Vector Calculus 2 341 Problems 13.1 In the equation (13.4) what happens if you start with a different parametrization forxand ...

13—Vector Calculus 2 342 13.8 Verify Stokes’ theorem for the fieldF~=Axyˆx+B(1 +x^2 y^2 )yˆand for the rectangular loop a < x ...

13—Vector Calculus 2 343 13.24 The same field and surface as the preceding problem, but now the surface integraldA~×F~. Ans:zˆ 2 ...

13—Vector Calculus 2 344 13.31 Derive the analog of the Reynolds transport theorem, Eq. (13.39), for a line integral around a cl ...

13—Vector Calculus 2 345 P V 13.35 Work in a thermodynamic system is calculated fromdW =P dV. Assume an ideal gas, so thatPV=nRT ...

13—Vector Calculus 2 346 13.47 Prove the identity Eq. (13.43). Write it out in index notation first. 13.48 There an analog of St ...

Complex Variables . In the calculus of functions of a complex variable there are three fundamental tools, the same funda- mental ...

14—Complex Variables 348 is the appropriate definition, but for it to exist there are even more restrictions than in the real ca ...

14—Complex Variables 349 Necessarily iffis analytic atz 0 it will also be analytic at every point within the disk ε |z−z 0 |< ...

14—Complex Variables 350 If you have a parametric representation for the values ofx(t)andy(t)along the curve this is ∫t 2 t 1 [ ...

14—Complex Variables 351 Theorder of the pole is the size of the largest negative power. These have respectively first order and ...

14—Complex Variables 352 The ratio test on the second sum is if for large enough positivek, |ak+1||z|k+1 |ak||z|k = |ak+1| |ak| ...

14—Complex Variables 353 important part of the proof is the one that I’ll leave to every book on complex variables ever written. ...