# 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

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