0.2 Cartesian vectors and tensors: their calculus

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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

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