Fourier series

CONTENTS
9.3
9.4
9.5
Rayleigh–Ritz method
Exercises
Hints and answers
327
329
332
10
10.1
Vector calculus
Differentiation of vectors
334
334
Composite vector expressions; differential of a vector
10.2
10.3
10.4
10.5
10.6
10.7
Integration of vectors
Space curves
Vector functions of several arguments
Surfaces
Scalar and vector fields
Vector operators
339
340
344
345
347
347
Gradient of a scalar field; divergence of a vector field; curl of a vector field
10.8
Vector operator formulae
354
Vector operators acting on sums and products; combinations of grad, div and
curl
10.9
10.10
10.11
10.12
Cylindrical and spherical polar coordinates
General curvilinear coordinates
Exercises
Hints and answers
357
364
369
375
11
11.1
Line, surface and volume integrals
Line integrals
377
377
Evaluating line integrals; physical examples; line integrals with respect to a
scalar
11.2
11.3
11.4
11.5
Connectivity of regions
Green’s theorem in a plane
Conservative fields and potentials
Surface integrals
383
384
387
389
Evaluating surface integrals; vector areas of surfaces; physical examples
11.6
Volume integrals
396
Volumes of three-dimensional regions
11.7
11.8
Integral forms for grad, div and curl
Divergence theorem and related theorems
398
401
Green’s theorems; other related integral theorems; physical applications
11.9
Stokes’ theorem and related theorems
406
Related integral theorems; physical applications
11.10 Exercises
11.11 Hints and answers
409
414
12
12.1
415
415
Fourier series
The Dirichlet conditions
ix