# 0.2 Cartesian vectors and tensors: their calculus  (Page 8/9)

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## Irrotational vector fields

The vector field $\mathbf{a}$ is irrotational if its curl vanishes everywhere. By Stokes' theorem the circulation around any closed curve also vanishes. Also, an irrotational vector field can be expressed as the gradient of a scalar.

$\left(\begin{array}{c}\nabla ×\mathbf{a}=0\\ \underset{C}{\oint }\mathbf{a}•\mathbf{t}ds=0\\ \mathbf{a}=\nabla \varphi \end{array}}\mathbf{a}\phantom{\rule{5pt}{0ex}}\text{an irrotational field}$

The velocity field of motions where the viscous effects are insignificant compared to inertial effects and the flow is initially irrotational can be approximated as an irrotational velocity field.

## Solenoidal vector fields

A solenoidal vector field is defined as one in which the divergence vanishes. This implies that the flux across a closed surface must also vanish. A vector identity states that the divergence of the curl of a vector is zero. Thus a continuously differentiable solenoidal vector field has the following three equivalent characteristics.

$\left(\begin{array}{c}\nabla •\mathbf{a}=0\\ \underset{S}{\int \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\int \phantom{\rule{-11.66656pt}{0ex}}◯}\mathbf{a}•\mathbf{n}dS=0\\ \mathbf{a}=\nabla ×\mathbf{A}\end{array}}\mathbf{a}\phantom{\rule{5pt}{0ex}}\text{a solenoidal field}$

The velocity field of motions where the effects of compressibility are insignificant can be approximated as a solenoidal vector field. The surface integral of velocity vanishing over any closed surface means that the net volumetric flow across closed surfaces is zero. Incompressible flow fields can be expressed as the curl of a vector potential. Two-dimensional, incompressible flows have only one nonzero component of the vector potential and this is identified as the stream function.

## Helmholtz' representation

We found that an irrotational vector is the gradient of a scalar potential and a solenoidal vector is the curl of a vector potential. Here we show that any vector field with sufficient continuity is divisible into irrotational and solenoidal parts, and so is expressible in terms of a scalar and a vector potential. The fundamental problem in the analysis of a vector field is the determination of these potentials and their expression in terms of the essential characteristics of the vector, namely divergence, curl, discontinuities, and boundary values. For when the potentials are known the vector itself can be determined by differentiation. The following analysis is taken from H. B. Phillips, Vector Analysis , John Wiley&Sons, 1933. The following nomenclature will differ somewhat in that the vector is expressed as the negative of the gradient of a scalar. Also, the vector field of interest will be denoted as $mathbfF$ . Bold face capital letters will also be used for other vector quantities. Also the equations have the $4\pi$ factor of electromagnetism in mks units rather than the factors ${\epsilon }_{o}$ and ${\epsilon }_{o}\phantom{\rule{0.277778em}{0ex}}{c}^{2}$ of the SI units.

Let $V$ be a region of space where the vector field $F$ has piecewise continuous second derivatives, ${S}_{1}$ be surfaces of discontinuity of $F$ , and $S$ be the bounding surface of $V$ . The Helmholtz's theorem states that $F$ can be expressed in terms of the potentials.

$\begin{array}{c}\mathbf{F}=-\nabla \varphi +\nabla ×\mathbf{A}\hfill \\ \text{where}\hfill \\ \varphi \left({\mathbf{x}}_{P}\right)=\underset{\text{V}}{\phantom{\rule{0.277778em}{0ex}}\int \int \int \phantom{\rule{0.277778em}{0ex}}}\frac{\rho dV}{r}+\underset{{S}_{1}}{\phantom{\rule{0.277778em}{0ex}}\int \int \phantom{\rule{0.277778em}{0ex}}}\frac{\sigma dS}{r}-\frac{1}{4\pi }\underset{S}{\int \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\int \phantom{\rule{-11.66656pt}{0ex}}◯}\frac{\mathbf{n}•\mathbf{F}dS}{r}\hfill \\ \mathbf{A}\left({\mathbf{x}}_{P}\right)=\underset{V}{\phantom{\rule{0.277778em}{0ex}}\int \int \int \phantom{\rule{0.277778em}{0ex}}}\frac{\mathbf{I}dV}{r}+\underset{{S}_{1}}{\phantom{\rule{0.277778em}{0ex}}\int \int \phantom{\rule{0.277778em}{0ex}}}\frac{\mathbf{J}dS}{r}-\frac{1}{4\pi }\underset{S}{\int \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\int \phantom{\rule{-11.66656pt}{0ex}}◯}\frac{\mathbf{n}×\mathbf{F}dS}{r}\hfill \\ \nabla •\mathbf{F}=-{\nabla }^{2}\varphi =4\pi \rho \hfill \\ \nabla ×\mathbf{F}=\nabla ×\left(\nabla ×\mathbf{A}\right)=4\pi \mathbf{I}\hfill \\ \mathbf{n}•\Delta \mathbf{F}=4\pi \sigma \hfill \\ \mathbf{n}×\Delta \mathbf{F}=4\pi \mathbf{J}\hfill \\ r=\left|{\mathbf{x}}_{P}-{\mathbf{x}}_{Q}\right|\text{where}{\mathbf{x}}_{Q}\text{is}\text{coordinate}\text{of}\text{integrand}\hfill \end{array}$

The vectors $\mathbf{I}$ and $\mathbf{J}$ are not arbitrary. They are subject to the equation of continuity $\nabla •\mathbf{A}=0$ . The effect of this condition is to make $\mathbf{I}$ and $\mathbf{J}$ behave like space and surface currents of something which is nowhere created or destroyed. In the electromagnetic field they usually represent currents of electricity. In hydrodynamic fields they represent vorticity. If $\mathbf{A}$ is everywhere solenoidal, the following three equations must then be everywhere satisfied.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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