Vector integrals generalize integration to vector-valued functions. Line integrals measure the accumulated effect of a vector field along a curve, surface integrals measure flux through a surface, and volume integrals measure the total effect within a region. Vector integrals are used widely in physics and engineering to calculate quantities such as work, magnetic flux, and fluid flow.
Analysis of the Use of Universal Distribution Factors in SEC Power Gridresearchinventy
Distribution factors have been extensively used in many power system analysis and planning studies. In recent power system studies, the AC distribution factors are insensitive to the operating point and relatively sensitive at certain degree to changes in network topology. These factors are linear approximations of sensitivities of variables with various inputs. This paper presents the calculation of the universal distribution factors (UDF’s) applies them on several practical scenarios of Saudi Electricity Company (SEC) power grid. The results are analyzed and evaluated considering various system conditions of SEC load. The results show that the accuracy of the used approach is acceptable compared with exact method. This is practically beneficial to SEC in computing its grid complex power flows using UDF's at the base case without the need to recalculate UDF’s which save efforts and time.
Mapping Gradex values on the Tensift basin (Morocco)IJERA Editor
The aim of this study is to elaborate the cartography of Gradex parameter used in the Gradex method for estimating flood peaks in order to size hydraulic structures. Map of spatial variation is elaborated using the geostatistical method of kriging. Several reference functions (exponential model, spherical, linear, Gaussian and cubic) were used for modeling the kriging variogram. Cross-validation enabled a comparison between the results of these models and choice of spherical model with anisotropy and trend fit by a second-order polynomial as the most suitable. The use of available series of annual maximum daily rainfall recorded at 23 rainfall stations, distributed over the Tensift basin, led also to develop the cartography of standard prediction errors‟ values associated to the predicted parameter for each point of Tensift basin. These errors vary from acceptable values (16.8%) to very high ones depending on the density of the rainfall stations at the desired site.
Action Trajectory Reconstruction for Controlling of Vehicle Using SensorsIOSR Journals
Abstract: Inertial sensors, such as accelerometers and gyro-scopes, are rarely used by themselves to compute
velocity and position as each requires the integration of very noisy data. The variance and bias in the resulting
position and velocity estimates grow un-bounded in time. This paper proposes a solution to provide a de-biased
and de-noised estimation of position and velocity of moving vehicle actions from accelerometer measurements.
The method uses a continuous wavelet transform applied to the measurements recursively to provide reliable
action trajectory reconstruction. The results are presented from experiments performed with a MEMS accelerometer
and gyroscope.
Keywords: Action trajectory, continuous wavelet transform, inertial measurement unit.
Analysis of the Use of Universal Distribution Factors in SEC Power Gridresearchinventy
Distribution factors have been extensively used in many power system analysis and planning studies. In recent power system studies, the AC distribution factors are insensitive to the operating point and relatively sensitive at certain degree to changes in network topology. These factors are linear approximations of sensitivities of variables with various inputs. This paper presents the calculation of the universal distribution factors (UDF’s) applies them on several practical scenarios of Saudi Electricity Company (SEC) power grid. The results are analyzed and evaluated considering various system conditions of SEC load. The results show that the accuracy of the used approach is acceptable compared with exact method. This is practically beneficial to SEC in computing its grid complex power flows using UDF's at the base case without the need to recalculate UDF’s which save efforts and time.
Mapping Gradex values on the Tensift basin (Morocco)IJERA Editor
The aim of this study is to elaborate the cartography of Gradex parameter used in the Gradex method for estimating flood peaks in order to size hydraulic structures. Map of spatial variation is elaborated using the geostatistical method of kriging. Several reference functions (exponential model, spherical, linear, Gaussian and cubic) were used for modeling the kriging variogram. Cross-validation enabled a comparison between the results of these models and choice of spherical model with anisotropy and trend fit by a second-order polynomial as the most suitable. The use of available series of annual maximum daily rainfall recorded at 23 rainfall stations, distributed over the Tensift basin, led also to develop the cartography of standard prediction errors‟ values associated to the predicted parameter for each point of Tensift basin. These errors vary from acceptable values (16.8%) to very high ones depending on the density of the rainfall stations at the desired site.
Action Trajectory Reconstruction for Controlling of Vehicle Using SensorsIOSR Journals
Abstract: Inertial sensors, such as accelerometers and gyro-scopes, are rarely used by themselves to compute
velocity and position as each requires the integration of very noisy data. The variance and bias in the resulting
position and velocity estimates grow un-bounded in time. This paper proposes a solution to provide a de-biased
and de-noised estimation of position and velocity of moving vehicle actions from accelerometer measurements.
The method uses a continuous wavelet transform applied to the measurements recursively to provide reliable
action trajectory reconstruction. The results are presented from experiments performed with a MEMS accelerometer
and gyroscope.
Keywords: Action trajectory, continuous wavelet transform, inertial measurement unit.
Lid driven cavity flow simulation using CFD & MATLABIJSRD
Steady Incompressible Navier-Stokes equation on a uniform grid has been studied at various Reynolds number using CFD (Computational Fluid Dynamics). Present paper aim is to obtain the stream-function and velocity field in steady state using the finite difference formulation on momentum equations and continuity equation. Reynold number dominates the flow problem. Taylor’s series expansion has been used to convert the governing equations in the algebraic form using finite difference schemes. MATLAB has been used to draw to flow simulations inside the driven-cavity.
Comparision of flow analysis through a different geometry of flowmeters using...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Analysis of green’s function and surface current density for rectangular micr...eSAT Journals
Abstract In this paper, Green’s function and surface current density for planar structure has been calculated. The approach makes use of the popular and rigorously used spectral domain full wave analysis method in conjunction with method of moment as numerical analysis tool. In present approach, boundary conditions are applied at patch metallization, which leads to integral equation with the involvement of green’s function in spectral domain, which includes the effect of dielectric, conductor loss, surface wave modes and space wave radiation. By applying Galerkin’s moment method integral equation are transformed to linear set of equations. Entire domain basis function is used to improve the efficiency of the solution. Keywords: Spectral domain full wave analysis, Green function, Galerkin’s moment method, Entire domain basis function, Surface current density.
Analysis of green’s function and surface current density for rectangular micr...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Prediction of nodes mobility in 3-D space IJECEIAES
Recently, mobility prediction researches attracted increasing interests, especially for mobile networks where nodes are free to move in the threedimensional space. Accurate mobility prediction leads to an efficient data delivery for real time applications and enables the network to plan for future tasks such as route planning and data transmission in an adequate time and a suitable space. In this paper, we proposed, tested and validated an algorithm that predicts the future mobility of mobile networks in three-dimensional space. The prediction technique uses polynomial regression to model the spatial relation of a set of points along the mobile node’s path and then provides a time-space mapping for each of the three components of the node’s location coordinates along the trajectory of the node. The proposed algorithm was tested and validated in MATLAB simulation platform using real and computer generated location data. The algorithm achieved an accurate mobility prediction with minimal error and provides promising results for many applications.
Path Loss Prediction by Robust Regression Methodsijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Comparison of Explicit Finite Difference Model and Galerkin Finite Element Mo...AM Publications
This paper describes Galerkin finite element (FEFLOW) models for the simulation of groundwater flow in twodimensional,
transient, unconfined groundwater flow systems. This study involves validation of FEFLOW model with reported
analytical solutions and also comparison of reported Explicit Finite Difference Model for groundwater flow simulation
(FDFLOW). The model is further used to obtain the space and time distribution of groundwater head for the reported
synthetic test case. The effect of time step size, space discretizations, pumping rates is analyzed on model results.
CFD and Artificial Neural Networks Analysis of Plane Sudden Expansion FlowsCSCJournals
It has been clearly established that the reattachment length for laminar flow depends on two non-dimensional parameters, the Reynolds number and the expansion ratio, therefore in this work, an ANN model that predict reattachment positions for the expansion ratios of 2, 3 and 5 based on the above two parameters has been developed. The R2 values of the testing set output Xr1, Xr2, Xr3, and Xr4 were 0.9383, 0.8577, 0.997 and 0.999 respectively. These results indicate that the network model produced reattachment positions that were in close agreement with the actual values. When considering the reattachment length of plane sudden-expansions the judicious combination of CFD calculated solutions with ANN will result in a considerable saving in computing and turnaround time. Thus CFD can be used in the first instance to obtain reattachment lengths for a limited choice of Reynolds numbers and ANN will be used subsequently to predict the reattachment lengths for other intermediate Reynolds number values. The CFD calculations concern unsteady laminar flow through a plane sudden expansion and are performed using a commercial CFD code STAR-CD while the training process of the corresponding ANN model was performed using the NeuroShellTM simulator.
Determination of wind energy potential of campus area of siirt universitymehmet şahin
In this study, wind energy potential of Siirt
University campus area is statistically examined by using the mean hourly wind speed data between 2014
and 2015 years which are measured by Vantage Pro2 device, located at the roof of the Engineering Faculty building with 6 m altitude. Weibull distribution
function and Rayleigh distribution function are used
as statistical approach to evaluate the wind data. Weibull distribution function is examined by using two different methods that are maximum likelihood estimation and Rayleigh method. The determination
coefficient (R2) and Root Mean Square Error (RMSE) values of these methods are compared. According the error analysis, it is indicated that the Rayleigh method
gives better results. Wind speed and wind power density are calculated in pursuance of Weibull distribution parameters. The results are evaluated as
monthly and annually. Hence, this preliminary study is made to determine the wind energy potential of Siirt University campus area.
3D Graph Drawings: Good Viewing for Occluded VerticesIJERA Editor
The growing studies show that the human brain can comprehend increasingly complex structures if they are
displayed as objects in three dimensional spaces. In addition to that, recent technological advances have led to
the production of a lot of data, and consequently have led to many large and complex models of 3D graph
drawings in many domains. Good Drawing (Visualization) resolves the problems of the occluded structures of
the graph drawings and amplifies human understanding, thus leading to new insights, findings and predictions.
We present method for drawing 3D graphs which uses a force-directed algorithm as a framework.
The main result of this work is that, 3D graph drawing and presentation techniques are combined available at
interactive speed. Even large graphs with hundreds of vertices can be meaningfully displayed by enhancing the
presentation with additional attributes of graph drawings and the possibility of interactive user navigation.
In the implementation, we interactively visualize many 3D graphs of different size and complexity to support
our method. We show that Gephi Software is capable of producing good viewpoints for 3D graph drawing, by
its built-in force directed layout algorithms.
This project aims at simulating lid driven cavity flow problem using package MATLAB. Steady Incompressible Navier-Stokes equation with continuity equation will be studied at various Reynolds number. The main aim is to obtain the velocity field in steady state using the finite difference formulation on momentum equations and continuity equation. Reynold number is the pertinent parameter of the present study. Taylor’s series expansion has been used to convert the governing equations in the algebraic form using finite difference schemes.
Lid driven cavity flow simulation using CFD & MATLABIJSRD
Steady Incompressible Navier-Stokes equation on a uniform grid has been studied at various Reynolds number using CFD (Computational Fluid Dynamics). Present paper aim is to obtain the stream-function and velocity field in steady state using the finite difference formulation on momentum equations and continuity equation. Reynold number dominates the flow problem. Taylor’s series expansion has been used to convert the governing equations in the algebraic form using finite difference schemes. MATLAB has been used to draw to flow simulations inside the driven-cavity.
Comparision of flow analysis through a different geometry of flowmeters using...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Analysis of green’s function and surface current density for rectangular micr...eSAT Journals
Abstract In this paper, Green’s function and surface current density for planar structure has been calculated. The approach makes use of the popular and rigorously used spectral domain full wave analysis method in conjunction with method of moment as numerical analysis tool. In present approach, boundary conditions are applied at patch metallization, which leads to integral equation with the involvement of green’s function in spectral domain, which includes the effect of dielectric, conductor loss, surface wave modes and space wave radiation. By applying Galerkin’s moment method integral equation are transformed to linear set of equations. Entire domain basis function is used to improve the efficiency of the solution. Keywords: Spectral domain full wave analysis, Green function, Galerkin’s moment method, Entire domain basis function, Surface current density.
Analysis of green’s function and surface current density for rectangular micr...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Prediction of nodes mobility in 3-D space IJECEIAES
Recently, mobility prediction researches attracted increasing interests, especially for mobile networks where nodes are free to move in the threedimensional space. Accurate mobility prediction leads to an efficient data delivery for real time applications and enables the network to plan for future tasks such as route planning and data transmission in an adequate time and a suitable space. In this paper, we proposed, tested and validated an algorithm that predicts the future mobility of mobile networks in three-dimensional space. The prediction technique uses polynomial regression to model the spatial relation of a set of points along the mobile node’s path and then provides a time-space mapping for each of the three components of the node’s location coordinates along the trajectory of the node. The proposed algorithm was tested and validated in MATLAB simulation platform using real and computer generated location data. The algorithm achieved an accurate mobility prediction with minimal error and provides promising results for many applications.
Path Loss Prediction by Robust Regression Methodsijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Comparison of Explicit Finite Difference Model and Galerkin Finite Element Mo...AM Publications
This paper describes Galerkin finite element (FEFLOW) models for the simulation of groundwater flow in twodimensional,
transient, unconfined groundwater flow systems. This study involves validation of FEFLOW model with reported
analytical solutions and also comparison of reported Explicit Finite Difference Model for groundwater flow simulation
(FDFLOW). The model is further used to obtain the space and time distribution of groundwater head for the reported
synthetic test case. The effect of time step size, space discretizations, pumping rates is analyzed on model results.
CFD and Artificial Neural Networks Analysis of Plane Sudden Expansion FlowsCSCJournals
It has been clearly established that the reattachment length for laminar flow depends on two non-dimensional parameters, the Reynolds number and the expansion ratio, therefore in this work, an ANN model that predict reattachment positions for the expansion ratios of 2, 3 and 5 based on the above two parameters has been developed. The R2 values of the testing set output Xr1, Xr2, Xr3, and Xr4 were 0.9383, 0.8577, 0.997 and 0.999 respectively. These results indicate that the network model produced reattachment positions that were in close agreement with the actual values. When considering the reattachment length of plane sudden-expansions the judicious combination of CFD calculated solutions with ANN will result in a considerable saving in computing and turnaround time. Thus CFD can be used in the first instance to obtain reattachment lengths for a limited choice of Reynolds numbers and ANN will be used subsequently to predict the reattachment lengths for other intermediate Reynolds number values. The CFD calculations concern unsteady laminar flow through a plane sudden expansion and are performed using a commercial CFD code STAR-CD while the training process of the corresponding ANN model was performed using the NeuroShellTM simulator.
Determination of wind energy potential of campus area of siirt universitymehmet şahin
In this study, wind energy potential of Siirt
University campus area is statistically examined by using the mean hourly wind speed data between 2014
and 2015 years which are measured by Vantage Pro2 device, located at the roof of the Engineering Faculty building with 6 m altitude. Weibull distribution
function and Rayleigh distribution function are used
as statistical approach to evaluate the wind data. Weibull distribution function is examined by using two different methods that are maximum likelihood estimation and Rayleigh method. The determination
coefficient (R2) and Root Mean Square Error (RMSE) values of these methods are compared. According the error analysis, it is indicated that the Rayleigh method
gives better results. Wind speed and wind power density are calculated in pursuance of Weibull distribution parameters. The results are evaluated as
monthly and annually. Hence, this preliminary study is made to determine the wind energy potential of Siirt University campus area.
3D Graph Drawings: Good Viewing for Occluded VerticesIJERA Editor
The growing studies show that the human brain can comprehend increasingly complex structures if they are
displayed as objects in three dimensional spaces. In addition to that, recent technological advances have led to
the production of a lot of data, and consequently have led to many large and complex models of 3D graph
drawings in many domains. Good Drawing (Visualization) resolves the problems of the occluded structures of
the graph drawings and amplifies human understanding, thus leading to new insights, findings and predictions.
We present method for drawing 3D graphs which uses a force-directed algorithm as a framework.
The main result of this work is that, 3D graph drawing and presentation techniques are combined available at
interactive speed. Even large graphs with hundreds of vertices can be meaningfully displayed by enhancing the
presentation with additional attributes of graph drawings and the possibility of interactive user navigation.
In the implementation, we interactively visualize many 3D graphs of different size and complexity to support
our method. We show that Gephi Software is capable of producing good viewpoints for 3D graph drawing, by
its built-in force directed layout algorithms.
This project aims at simulating lid driven cavity flow problem using package MATLAB. Steady Incompressible Navier-Stokes equation with continuity equation will be studied at various Reynolds number. The main aim is to obtain the velocity field in steady state using the finite difference formulation on momentum equations and continuity equation. Reynold number is the pertinent parameter of the present study. Taylor’s series expansion has been used to convert the governing equations in the algebraic form using finite difference schemes.
Similar to Introduction To Vector Integrals.pptx (20)
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
2. 1
INTRODUCTION TO VECTOR INTEGRALS
Vector integrals are a powerful tool in
mathematics and physics.
They extend the concept of integration to
vector-valued functions.
Vector integrals allow us to calculate
quantities such as work, flux, and
circulation.
3. 2
NOTATION AND DEFINITIONS
The symbol ∫ represents the integral sign in
vector integrals.
The integrand is a vector-valued function,
denoted as F(r).
The integration is performed over a path or
a region in space.
4. 3
LINE INTEGRALS
Line integrals are vector integrals
performed along a curve or path.
They measure the accumulated effect of a
vector field along the curve.
Line integrals can be evaluated using
different methods, such as
parameterization or Green's theorem.
5. 4
SURFACE INTEGRALS
Surface integrals are vector integrals
performed over a surface.
They measure the flux or flow of a vector
field through the surface.
Surface integrals can be evaluated using
different methods, such as
parameterization or the divergence
theorem.
6. 5
VOLUME INTEGRALS
Volume integrals are vector integrals
performed over a region in space.
They measure the total effect of a vector
field within the region.
Volume integrals can be evaluated using
different methods, such as
parameterization or the divergence
theorem.
7. 6
APPLICATIONS OF VECTOR INTEGRALS
Vector integrals are widely used in physics,
engineering, and mathematics.
They are used to calculate work done by a
force, magnetic flux through a surface, and
fluid flow in a region.
Vector integrals are essential in
understanding concepts like potential
energy, electromagnetism, and fluid
dynamics.
8. 7
VECTOR INTEGRATION TECHNIQUES
Various techniques can be used to evaluate
vector integrals, such as line and surface
parameterization.
Green's theorem, Stokes' theorem, and the
divergence theorem are powerful tools for
simplifying vector integrals.
Computer software and numerical methods
can also be employed for complex vector
integrals.
9. 8
SUMMARY
Vector integrals extend integration to
vector-valued functions, allowing us to
calculate quantities like work, flux, and
circulation.
Line integrals measure the effect of a
vector field along a curve, while surface
integrals measure flux through a surface.
Volume integrals measure the total effect of
a vector field within a region.
