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

 Page 4 / 9
$dV=d{x}_{1}\phantom{\rule{0.166667em}{0ex}}d{x}_{2}\phantom{\rule{0.166667em}{0ex}}d{x}_{3}$

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

show that the set of all natural number form semi group under the composition of addition
what is the meaning
Dominic
explain and give four Example hyperbolic function
_3_2_1
felecia
⅗ ⅔½
felecia
_½+⅔-¾
felecia
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
ok
Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
x*x=2
felecia
2+2x=
felecia
×/×+9+6/1
Debbie
Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22
Naagmenkoma
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Abdullahi
hi mam
Mark
find the value of 2x=32
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
16
Makan
x=16
Makan
use the y -intercept and slope to sketch the graph of the equation y=6x
how do we prove the quadratic formular
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Jeannette has $5 and$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
What is the expressiin for seven less than four times the number of nickels
How do i figure this problem out.
how do you translate this in Algebraic Expressions
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Got questions? Join the online conversation and get instant answers!

#### Get Jobilize Job Search Mobile App in your pocket Now! By Rhodes By Stephanie Redfern By Joli Julianna By David Martin By John Gabrieli By Janet Forrester By OpenStax By By OpenStax By OpenStax