Loading...
Please wait, while we are loading the content...
Similar Documents
Boundary Value Problems, Potential Theory, and Conformal Mapping
| Content Provider | Scilit |
|---|---|
| Author | Jeffrey, Alan |
| Copyright Year | 2005 |
| Description | Book Name: Complex Analysis and Applications |
| Abstract | This section provides some physical motivation for the study of conformal mapping that follows. Its purpose is to establish the connection between a conformal mapping and the solution of what is called a two-dimensional boundary value problem for the Laplace equation, which is studied later in some detail. We start by recalling that a two-dimensional harmonic function f(x, y) in a domain D of the (x, y)-plane satisfies the Laplace equation (5.1) Let w f(z) u iv be an analytic function that maps a domain D in the z-plane one-one and conformally onto a domain D∗ in the w-plane. Let us examine what happens to the Laplace Equation (5.1) if the Cartesian coordinates (x, y) are changed to the differentiable curvilinear coordinates variables u u(x, y) and v v(x, y) in the function f(z) u iv, when f(x, y) becomes the function (u, v) f(u(x, y), v(x, y)). From the chain rule for differentiation, we have (5.2) and a further differentiation with respect to x gives (5.3) ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ f x x u u x ∂ ∂ ∂ ∂ u u x 2 ∂ ∂ ∂ ∂ ∂ ∂ x v v x ∂ ∂ ∂ ∂ v v x ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ f x u u x v v x , f f f ( , ) .x y x y ∂ ∂ ∂ ∂ Examination of Equation (5.2) shows that the operation of differentiation of f with respect to x is related to the operation of differentiation of (u, v) with respect to u and v by the linear operator (5.4) In terms of this operator in Equation (5.3) becomes (5.5) The corresponding expression for ∂2f/∂y2 follows directly from Equation (5.5) by replacing x by y whenever it occurs in a partial derivative. |
| Related Links | https://content.taylorfrancis.com/books/download?dac=C2010-0-48358-X&isbn=9780429114748&doi=10.1201/9780203026564-11&format=pdf |
| Ending Page | 530 |
| Page Count | 112 |
| Starting Page | 419 |
| DOI | 10.1201/9780203026564-11 |
| Language | English |
| Publisher | Informa UK Limited |
| Publisher Date | 2005-11-10 |
| Access Restriction | Open |
| Subject Keyword | Book Name: Complex Analysis and Applications Mathematical Physics Differentiation Function Boundary Conformal Mapping Laplace Coordinates Respect |
| Content Type | Text |
| Resource Type | Chapter |
