Some More Vector Calculus

We investigate the partial derivative approach to the change of scale formula for the functon space integral and we investigate the vector calculus approach to the directional derivative on the function space and prove relationships among the Wiener integral and the Feynman integral about the directional derivative of a Fourier transform.

Download Full-text

Vector Calculus and Differential Geometry

Fundamentals of Grid Generation ◽

10.1201/9780138755287-4 ◽

2020 ◽

pp. 65-88

Author(s):

Patrick Knupp ◽

Stanly Steinberg

Keyword(s):

Differential Geometry ◽

Vector Calculus

Download Full-text

Review of Vector Calculus

Parameter Estimation and Inverse Problems ◽

10.1016/b978-0-12-804651-7.00020-1 ◽

2019 ◽

pp. 363-369

Author(s):

Richard C. Aster ◽

Brian Borchers ◽

Clifford H. Thurber

Keyword(s):

Vector Calculus

Download Full-text

The Derivation and Application of Fundamental Solutions for Unsteady Stokes Equations

Journal of Mechanics ◽

10.1017/jmech.2015.70 ◽

2015 ◽

Vol 31 (6) ◽

pp. 683-691 ◽

Cited By ~ 1

Author(s):

C-H. Hsiao ◽

D.-L. Young

Keyword(s):

Stokes Equations ◽

Three Dimensional ◽

Fundamental Solutions ◽

Mass Source ◽

Reynolds Number Flow ◽

Singularity Method ◽

Vector Calculus ◽

Unsteady Stokes Flow ◽

Number Flow ◽

Unsteady Stokes Equations

AbstractIn this paper, two formulations in explicit form to derive the fundamental solutions for two and three dimensional unsteady unbounded Stokes flows due to a mass source and a point force are presented, based on the vector calculus method and also the Hörmander’s method. The mathematical derivation process for the fundamental solutions is detailed. The steady fundamental solutions of Stokes equations can be obtained from the unsteady fundamental solutions by the integral process. As an application, we adopt fundamental solutions: an unsteady Stokeslet and an unsteady potential dipole to validate a simple case that a sphere translates in Stokes or low-Reynolds-number flow by using the singularity method instead by the traditional method which in general limits to the assumption of oscillating flow. It is concluded that this study is able to extend the unsteady Stokes flow theory to more general transient motions by making use of the fundamental solutions of the linearly unsteady Stokes equations.

Download Full-text