# 0.2 Cartesian vectors and tensors: their calculus  (Page 4/9)

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$dV=d{x}_{1}\phantom{\rule{0.166667em}{0ex}}d{x}_{2}\phantom{\rule{0.166667em}{0ex}}d{x}_{3}$

Suppose, however, that it is convenient to describe the position by some other coordinates, say ${\xi }_{1}$ , ${\xi }_{2}$ , ${\xi }_{3}$ . We may ask what volume is to be associated with the three small changes $d{\xi }_{1}$ , $d{\xi }_{2}$ , $d{\xi }_{3}$ .

The change of coordinates must be given by specifying the Cartesian point x that is to correspond to a given set ${\xi }_{1}$ , ${\xi }_{2}$ , ${\xi }_{3}$ , by

${x}_{i}={x}_{i}\left({\xi }_{1},{\xi }_{2},{\xi }_{3}\right)$

Then by partial differentiation the small differences corresponding to a change $d{\xi }_{i}$ are

$d{x}_{i}=\frac{\partial {x}_{i}}{\partial {\xi }_{j}}\phantom{\rule{0.166667em}{0ex}}d{\xi }_{j}$

Let $d{\mathbf{x}}^{\left(j\right)}$ be the vectors with the components $\left(\delta {x}_{i}/\delta {\xi }_{j}\right)d{\xi }_{j}$ for $j=1,2,and3$ . Then the volume element is

$\begin{array}{ccc}dV\hfill & =& d{\mathbf{x}}^{\left(1\right)}•\left(d{\mathbf{x}}^{\left(2\right)}×d{\mathbf{x}}^{\left(3\right)}\right)\hfill \\ & =& {\epsilon }_{ijk}\phantom{\rule{0.166667em}{0ex}}\frac{\partial {x}_{i}}{\partial {\xi }_{1}}d{\xi }_{1}\phantom{\rule{0.166667em}{0ex}}\frac{\partial {x}_{j}}{\partial {\xi }_{2}}d{\xi }_{2}\phantom{\rule{0.166667em}{0ex}}\frac{\partial {x}_{k}}{\partial {\xi }_{3}}d{\xi }_{3}\hfill \\ & =& J\phantom{\rule{0.166667em}{0ex}}d{\xi }_{1}\phantom{\rule{0.166667em}{0ex}}d{\xi }_{2}\phantom{\rule{0.166667em}{0ex}}d{\xi }_{3}\hfill \end{array}$

where

$J=\frac{\partial \left({x}_{1},\phantom{\rule{0.166667em}{0ex}}{x}_{2},\phantom{\rule{0.166667em}{0ex}}{x}_{3}\right)}{\partial \left({\xi }_{1},\phantom{\rule{0.166667em}{0ex}}{\xi }_{2},\phantom{\rule{0.166667em}{0ex}}{\xi }_{3}\right)}={\epsilon }_{ijk}\phantom{\rule{0.166667em}{0ex}}\frac{\partial {x}_{i}}{\partial {\xi }_{1}}\phantom{\rule{0.166667em}{0ex}}\frac{\partial {x}_{j}}{\partial {\xi }_{2}}\phantom{\rule{0.166667em}{0ex}}\frac{\partial {x}_{k}}{\partial {\xi }_{3}}$

is called the Jacobian of the transformation of variables

## Vector fields

When the components of a vector or tensor depend on the coordinates we speak of a vector or tensor field. The flow of a fluid is a perfect realization of a vector field for at each point in the region of flow we have a velocity vector $\mathbf{v}\left(\mathbf{x}\right)$ . If the flow is unsteady then the velocity depends on the time as well as position, $\mathbf{v}=\mathbf{v}\left(\mathbf{x},t\right)$ .

Associated with any vector field $\mathbf{a}\left(\mathbf{x}\right)$ are its trajectories , which is the name given to the family of curves everywhere tangent to the local vector $\mathbf{a}$ . They are solutions of the simultaneous equations

$\frac{d\mathbf{x}}{ds}=\mathbf{a}\left(\mathbf{x}\right);\text{that}\text{is}\frac{d{x}_{i}}{ds}={a}_{i}\left({x}_{1},\phantom{\rule{0.166667em}{0ex}}{x}_{2},\phantom{\rule{0.166667em}{0ex}}{x}_{3}\right)$

Where $s$ is a parameter along the trajectory. (It will be arc length if $\mathbf{a}$ is always a unit vector.) Streamlines of a steady flow are a realization of these trajectories. For a time dependent vector field the trajectories will also be time dependent. If $C$ is any closed curve in the vector field and we take the trajectories through all points of $C$ , they describe a surface known as the vector tube of the field. For flow fields, it is called a stream tube .

## The vector operator $\nabla$ -gradient of a scalar

The symbol $\nabla$ (enunciated as "del") is used for the symbolic vector operator whose ${i}^{th}$ component is $\delta /\delta {x}_{i}$ . Thus if $\nabla$ operates on a scalar function of position $\varphi \left(\mathbf{x}\right)$ it produces a vector $\nabla \varphi$ with components $\delta \varphi /\delta {x}_{i}$ .

$\frac{\partial \varphi }{\partial {\overline{x}}_{j}}=\frac{\partial \varphi }{\partial {x}_{i}}\frac{\partial {x}_{i}}{\partial {\overline{x}}_{j}}={l}_{ij}\phantom{\rule{0.166667em}{0ex}}\frac{\partial \varphi }{\partial {x}_{i}}$
$\text{grad}\phantom{\rule{0.166667em}{0ex}}\varphi =\nabla \varphi ={\varphi }_{,i}={\mathbf{e}}_{\left(1\right)}\phantom{\rule{0.166667em}{0ex}}\frac{\partial \varphi }{\partial {x}_{1}}+{\mathbf{e}}_{\left(2\right)}\phantom{\rule{0.166667em}{0ex}}\frac{\partial \varphi }{\partial {x}_{2}}+{\mathbf{e}}_{\left(3\right)}\phantom{\rule{0.166667em}{0ex}}\frac{\partial \varphi }{\partial {x}_{3}}$

since ${\overline{a}}_{j}={l}_{ij}\phantom{\rule{0.166667em}{0ex}}{a}_{i}$ so $\nabla \varphi$ is a vector. The suffix notation, $i$ for the partial derivative with respect to ${x}_{i}$ is a very convenient one and will be taken over for the generalization of this operation that must be made for non-Cartesian frame of reference. The notation "grad" for $\nabla$ is often used and referred to as the gradient operator. Thus grad $\varphi$ is the vector which is the gradient of the scalar. The gradient operator can also operate on higher order tensors and the operation raises the order by one. Thus the gradient of a vector $\mathbf{a}$ is grad $\mathbf{a}$ , $\nabla \mathbf{a}$ , or in component notation ${a}_{i,j}$ . $\nabla$ is sometimes written $\delta /\delta \mathbf{x}$ and can be expanded as the vector operator

The suffix notation, $i$ for the partial derivative with respect to ${x}_{i}$ is a very convenient one and will be taken over for the generalization of this operation that must be made for non-Cartesian frame of reference. The notation "grad" for $\nabla$ is often used and referred to as the gradient operator. Thus grad $\varphi$ is the vector which is the gradient of the scalar. The gradient operator can also operate on higher order tensors and the operation raises the order by one. Thus the gradient of a vector $\mathbf{a}$ is grad $\mathbf{a}$ , $\nabla \mathbf{a}$ , or in component notation ${a}_{i,j}$ . $\nabla$ is sometimes written $\delta /\delta \mathbf{x}$ and can be expanded as the vector operator

where we get a research paper on Nano chemistry....?
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
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da
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Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
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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
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Alexandre
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Rafiq
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Damian
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LITNING
scanning tunneling microscope
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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
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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..
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what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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