The Hypercircle in Mathematical Physics: A Method for the Approximate Solution of Boundary Value ProblemsOriginally published in 1957, this book was written to provide physicists and engineers with a means of solving partial differential equations subject to boundary conditions. The text gives a systematic and unified approach to a wide class of problems, based on the fact that the solution may be viewed as a point in function-space, this point being the intersection of two linear subspaces orthogonal to one another. Using this method the solution is located on a hypercircle in function-space, and the approximation is improved by reducing the radius of the hypercircle. The complexities of calculation are illuminated throughout by simple, intuitive geometrical pictures. This book will be of value to anyone with an interest in solutions to boundary value problems in mathematical physics. |
Contents
Representation of numbers p 7 Representation of functions p | 8 |
Addition subtraction and multiplication by a scalar p 19 Logic | 23 |
Straight lines p 24 Linear dependence of two Fvectors p 24 Linear | 34 |
Conditions to be satisfied by the scalar product p 37 Examples | 43 |
Length and distance in Fspace p 44 Example p 44 Unit Fvectors | 50 |
Orthonormality and linear independence p 50 Orthogonal transforma | 67 |
No closed boxes in functionspacep 76 nspheres | 84 |
Linear subspaces and orthogonality p 98 The vertices V V of two non | 102 |
section p 224 Approximation to stress and warping p 225 Torsion | 233 |
General plan p 241 Use of approximate solutions for the weights of | 269 |
General plan p 270 Use of approximate solutions for the weights of | 280 |
The approximation n 8 p 283 The approximation n 16 p | 286 |
VARIOUS BOUNDARY VALUE PROBLEMS | 292 |
unknown p 292 Splitting the differential equation p 293 The scalar | 299 |
Pyramid Fvectors for threedimensional Pspace p 301 First mixed | 311 |
viscous flow in a channel p 312 Flow problem | 332 |
Summary of formulae for application of the method of the hypercircle | 118 |
Use of nonlinear subspaces p 121 Bounds on S2 obtained with | 124 |
Statement of the Dirichlet problem the Neumann problem and the mixed | 131 |
Case Where the boundary value function f is only piecewise continuous | 137 |
The Greens Fvector p 155 Bounds for the solution p 157 Checks | 163 |
Pyramid functions p 168 Pyramid Fvectors of the first class p | 170 |
Pyramid Fvectors of the second class p 171 Triangulation and poly | 176 |
Summary of plan for the use of pyramid Fvectors p 178 The linear | 185 |
Definition and normalization p 188 Scalar products p 190 Determina | 199 |
Definition and normalization p 200 Scalar products p 202 Determina | 209 |
The torsion problem stated p 214 Torsional rigidity p 216 Transforma | 219 |
Equations of equilibrium and compatibility p 336 Splitting the problem | 345 |
The Greens tensor of elasticity p 347 Bounds at a point in elastic | 354 |
Biharmonic functions and their conjugates p 355 The biharmonic | 366 |
Résuméof available facts p 371 Null Fvectors p 372 Null cones p | 372 |
Orthogonality and orthogonal projection p 373 Orthonormalization | 380 |
The vertices V V of two nonintersecting linear subspaces of finite | 386 |
The separation of two straight lines p 389 The separation of two linear | 392 |
The scalar wave equation p 394 Splitting the problem p 395 Eigenvalue | 402 |
Stationary principles for elastic vibrations p 405 Maxwells equations | 409 |
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Common terms and phrases
approximation biharmonic biharmonic equation boundary conditions boundary value problem calculation centre circle conjugate harmonic function Consider constant coordinates cross-section defined definition Dirichlet integral Dirichlet problem domain elastic equations Euclidean 3-space F-point F-space figures find finding finite first class fixed flow formulae function-space geometry given gives gradient Green’s harmonic function hence hexagonal pyramid F-vectors hollow square hypercircle method hypercircle of class hyperplane of class hypersphere inequality infinite inside integral intersection junction point linear n-space linearly independent lower bound multiply connected n-sphere neighbours Neumann problem normal obtained octant orthogonal linear subspaces orthogonal projection orthonormal P-vector field parameters plane polygon position-vector positive-definite metric pyramid functions radius reduced weights regular hexagon satisfies satisfy scalar product second class shown in Fig side solution square pyramid F-vectors straight line stress strip F-vectors summation symmetry torsion problem torsional rigidity triangle upper bound vanishes vectors lying vertex vertices zero