# 0.2 Cartesian vectors and tensors: their calculus

 Page 1 / 9

## Topics covered in this chapter

• Tensor functions of time-like variable
• Curves in space
• Line integrals
• Surface integrals
• Volume integrals
• Change of variables with multiple integrals
• Vector fields
• The vector operator -gradient of a scalar
• The divergence of a vector field
• The curl of a vector field
• Green's theorem and some of its variants
• Stokes' theorem
• The classification and representation of vector fields
• Irrotational vector fields
• Solenoidal vector fields
• Helmholtz' representation
• Vector and scalar potential

Reading assignment: Chapter 3 of Aris

## Tensor functions of time-like variable

In the last chapter, vectors and tensors were defined as quantities with components that transform in a certain way with rotation of coordinates. Suppose now that these quantities are a function of time. The derivatives of these quantities with time will transform in the same way and thus are tensors of the same order. The most important derivatives are the velocity and acceleration.

$\begin{array}{c}\mathbf{v}\left(t\right)=\stackrel{˙}{\mathbf{x}}\left(t\right),{v}_{i}=\frac{d{x}_{i}}{dt}\hfill \\ \mathbf{a}\left(t\right)=\stackrel{¨}{\mathbf{x}}\left(t\right),{a}_{i}=\frac{{d}^{2}{x}_{i}}{d{t}^{2}}\hfill \end{array}$

The differentiation of products of tensors proceeds according to the usual rules of differentiation of products. In particular,

$\begin{array}{c}\frac{d}{dt}\left(\mathbf{a}•\mathbf{b}\right)=\frac{d\mathbf{a}}{dt}•\mathbf{b}+\mathbf{a}•\frac{d\mathbf{b}}{dt}\hfill \\ \frac{d}{dt}\left(\mathbf{a}×\mathbf{b}\right)=\frac{d\mathbf{a}}{dt}×\mathbf{b}+\mathbf{a}×\frac{d\mathbf{b}}{dt}\hfill \end{array}$

## Curves in space

The trajectory of a particle moving is space defines a curve that can be defined with time as parameter along the curve. A curve in space is also defined by the intersection of two surfaces, but points along the curve are not associated with time. We will show that a natural parameter for both curves is the distance along the curve.

The variable position vector $\mathbf{x}\left(t\right)$ describes the motion of a particle. For a finite interval of $t$ , say $a\le t\le b$ , we can plot the position as a curve in space. If the curve does not cross itself (i.e., if $\mathbf{x}\left(t\right)\ne \mathbf{x}\left({t}^{\text{'}}\right),a\le t<{t}^{\text{'}}\le b$ ) it is called simple ; if $\mathbf{x}\left(a\right)=\mathbf{x}\left(b\right)$ the curve is closed. The variable $t$ is now just a parameter along the curve that may be thought of as the time in motion of the particle. If $t$ and ${t}^{\text{'}}$ are the parameters of two points, the cord joining them is the vector $\mathbf{x}\left({t}^{\text{'}}\right)-\mathbf{x}\left(t\right)$ . As $t\to {t}^{\text{'}}$ this vector approaches $\left({t}^{\text{'}}-t\right)\stackrel{˙}{\mathbf{x}}\left(t\right)$ and so in the limit is proportional to $\stackrel{˙}{\mathbf{x}}\left(t\right)$ . However the limit of the cord is the tangent so that $\stackrel{˙}{\mathbf{x}}\left(t\right)$ is in the direction of the tangent. If ${v}^{2}=\stackrel{˙}{\mathbf{x}}\left(t\right)•\stackrel{˙}{\mathbf{x}}\left(t\right)$ we can construct a unit tangent vector $\mathbf{\tau }$ .

$\mathbf{\tau }=\stackrel{˙}{\mathbf{x}}\left(t\right)/v=\mathbf{v}/v$

Now we will parameterize a curve with distance along the curve rather than time. If $\mathbf{x}\left(t\right)$ and $\mathbf{x}\left(t+dt\right)$ are two very close points,

$\mathbf{x}\left(t+dt\right)=\mathbf{x}\left(t\right)+dt\phantom{\rule{0.166667em}{0ex}}\stackrel{˙}{\mathbf{x}}\left(t\right)+O\left(d{t}^{2}\right)$

and the distance between them is

$\begin{array}{ccc}d{s}^{2}& =& \left\{\mathbf{x}\left(t+dt\right)-\mathbf{x}\left(t\right)\right\}•\left\{\mathbf{x}\left(t+dt\right)-\mathbf{x}\left(t\right)\right\}\hfill \\ & =& \stackrel{˙}{\mathbf{x}}\left(t\right)•\stackrel{˙}{\mathbf{x}}\left(t\right)\phantom{\rule{0.166667em}{0ex}}d{t}^{2}+O\left(d{t}^{3}\right)\hfill \end{array}$

The arc length from any given point $t=a$ is therefore

$s\left(t\right)={\int }_{a}^{t}{\left[\stackrel{˙}{\mathbf{x}}\left({t}^{\text{'}}\right)•\stackrel{˙}{\mathbf{x}}\left({t}^{\text{'}}\right)\right]}^{1/2}d{t}^{\text{'}}$

$s$ is the natural parameter to use on the curve, and we observe that

$\frac{d\mathbf{x}}{ds}=\frac{d\mathbf{x}}{dt}\frac{dt}{ds}=\frac{\stackrel{˙}{\mathbf{x}}\left(t\right)}{v}=\tau$

A curve for which a length can be so calculated is called rectifiable . From this point on we will regard $s$ as the parameter, identifying $t$ with $s$ and letting the dot denote differentiation with respect to $s$ . Thus

$\mathbf{\tau }\left(s\right)=\stackrel{˙}{\mathbf{x}}\left(s\right)$

is the unit tangent vector. Let $\mathbf{x}\left(s\right),\mathbf{x}\left(s+ds\right)$ , and $\mathbf{x}\left(s-ds\right)$ be three nearby points on the curve. A plane that passes through these three points is defined by the linear combinations of the cord vectors joining the points. This plane containing the points must also contain the vectors

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
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
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
Got questions? Join the online conversation and get instant answers!