0.2 Cartesian vectors and tensors: their calculus  (Page 2/9)

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$\begin{array}{c}\frac{\mathbf{x}\left(s+ds\right)-\mathbf{x}\left(s\right)}{ds}=\stackrel{˙}{\mathbf{x}}\left(s\right)+O\left(ds\right)\hfill \\ \text{and}\hfill \\ \frac{\mathbf{x}\left(s+ds\right)-2\phantom{\rule{0.166667em}{0ex}}\mathbf{x}\left(s\right)+\mathbf{x}\left(s-ds\right)}{d{s}^{2}}=\stackrel{¨}{\mathbf{x}}\left(s\right)+O\left(d{s}^{2}\right)\hfill \end{array}$

Thus, in the limit when the points are coincident, the plane reaches a limiting position defined by the first two derivatives $\stackrel{˙}{\mathbf{x}}\left(s\right)$ and $\stackrel{¨}{\mathbf{x}}\left(s\right)$ . This limiting plane is called the osculating plane and the curve appears to lie in this plane in the intermediate neighborhood of the point. To prove this statement: (1) A plane is defined by the two vectors, $\stackrel{˙}{\mathbf{x}}\left(s\right)$ and $\stackrel{¨}{\mathbf{x}}\left(s\right)$ , if they are not co-linear. (2) The coordinates of the three points on the curve in the previous two equations are a linear combination of $\mathbf{x}\left(s\right),\stackrel{˙}{\mathbf{x}}\left(s\right)$ and $\stackrel{¨}{\mathbf{x}}\left(s\right)$ , thus they line in the plane.

Now $\stackrel{˙}{\mathbf{x}}=\mathbf{\tau }$ so $\stackrel{¨}{\mathbf{x}}=\stackrel{˙}{\mathbf{\tau }}$ and since $\mathbf{\tau }•\mathbf{\tau }=1$ ,

$\begin{array}{c}\frac{\mathbf{d}\left(\mathbf{\tau }•\mathbf{\tau }\right)}{ds}=0=\stackrel{˙}{\mathbf{\tau }}•\mathbf{\tau }+\mathbf{\tau }•\stackrel{˙}{\mathbf{\tau }}=2\phantom{\rule{0.166667em}{0ex}}\mathbf{\tau }•\stackrel{˙}{\mathbf{\tau }}\hfill \\ \mathbf{\tau }•\stackrel{˙}{\mathbf{\tau }}=0\hfill \end{array}$

so that the vector $\stackrel{˙}{\mathbf{\tau }}$ is at right angles to the tangent. Let $1/\rho$ denote the magnitude of $\stackrel{˙}{\mathbf{\tau }}$ .

$\begin{array}{c}\stackrel{˙}{\mathbf{\tau }}•\stackrel{˙}{\mathbf{\tau }}=\frac{1}{{\rho }^{2}}\hfill \\ \text{and}\hfill \\ \nu =\rho \phantom{\rule{0.166667em}{0ex}}\stackrel{˙}{\mathbf{\tau }}\hfill \end{array}$

Then $\mathbf{\nu }$ is a unit normal and defines the direction of the so-called principle normal to the curve.

To interpret $\rho$ , we observe that the small angle $d\theta$ between the tangents at $s$ and $s+ds$ is given by

$\begin{array}{c}cosd\theta =\tau \left(s\right)•\tau \left(s+ds\right)\\ 1-\frac{1}{2}d{\theta }^{2}+...=\tau •\tau +\tau •\stackrel{˙}{\tau }\phantom{\rule{0.166667em}{0ex}}ds+\frac{1}{2}\tau •\stackrel{¨}{\tau }\phantom{\rule{0.166667em}{0ex}}d{s}^{2}+...\\ =1-\frac{1}{2}\stackrel{˙}{\tau }•\stackrel{˙}{\tau }\phantom{\rule{0.166667em}{0ex}}d{s}^{2}+...\end{array}$

since $\tau •\stackrel{˙}{\tau }=0$ and so $\tau •\stackrel{¨}{\tau }+\stackrel{˙}{\tau }•\stackrel{˙}{\tau }=0$ . Thus,

$\rho =\left|\frac{ds}{d\theta }\right|$

is the reciprocal of the rate of change of the angle of the tangent with arc length, i.e., $\rho$ is the radius of curvature. Its reciprocal $1/\rho$ is the curvature, $\kappa \equiv \left|d\theta /ds\right|=1/\rho$ .

A second normal to the curve may be taken to form a right-hand system with $\tau$ and $\nu$ . This is called the unit binormal,

$\mathbf{\beta }=\tau ×\nu$

Line integrals

If $F\left(x\right)$ is a function of position and $C$ is a curve composed of connected arcs of simple curves, $\mathbf{x}=\mathbf{x}\left(t\right),a\le t\le b$ or $\mathbf{x}=\mathbf{x}\left(s\right),a\le s\le b$ , we can define the integral of $F$ along $C$ as

$\begin{array}{c}{\int }_{C}F\left(\mathbf{x}\right)dt={\int }_{a}^{b}F\left[\mathbf{x}\left(t\right)\right]dt\hfill \\ \text{or}\hfill \\ {\int }_{C}F\left(\mathbf{x}\right)ds={\int }_{a}^{b}F\left[\mathbf{x}\left(t\right)\right]{\left\{\stackrel{˙}{\mathbf{x}}\left(t\right)•\stackrel{˙}{\mathbf{x}}\left(t\right)\right\}}^{1/2}dt\hfill \end{array}$

Henceforth, we will assume that the curve has been parameterized with respect to distance along the curve, $s$ .

The integral is from $a$ to $b$ . If the integral is in the opposite direction with opposite limits, then the integral will have the same magnitude but opposite sign. If $\mathbf{x}\left(a\right)=\mathbf{x}\left(b\right)$ , the curve $C$ is closed and the integral is sometimes written

${\oint }_{C}F\left[\mathbf{x}\left(s\right)\right]ds$

If the integral around any simple closed curve vanishes, then the value of the integral from any pair of points $a$ and $b$ is independent of path. To see this we take any two paths between $a$ and $b$ , say ${C}_{1}$ and ${C}_{2}$ , and denote by $C$ the closed path formed by following ${C}_{1}$ from $a$ to $b$ and ${C}_{2}$ back from $b$ to $a$ .

$\begin{array}{ccc}\hfill {\oint }_{C}F\phantom{\rule{0.166667em}{0ex}}ds& =& {\left[\left[{\int }_{a}^{b}F\phantom{\rule{0.166667em}{0ex}}ds\right]\right]}_{{C}_{1}}+{\left[\left[{\int }_{b}^{a}F\phantom{\rule{0.166667em}{0ex}}ds\right]\right]}_{{C}_{2}}\hfill \\ & =& {\left[\left[{\int }_{a}^{b}F\phantom{\rule{0.166667em}{0ex}}ds\right]\right]}_{{C}_{1}}-{\left[\left[{\int }_{a}^{b}F\phantom{\rule{0.166667em}{0ex}}ds\right]\right]}_{{C}_{2}}\hfill \\ & =& 0\hfill \end{array}$

If $\mathbf{a}\left(\mathbf{x}\right)$ is any vector function of position, $\mathbf{a}•\tau$ is the projection of $\mathbf{a}$ tangent to the curve. The integral of $\mathbf{a}•\tau$ around a simple closed curve $C$ is called the circulation of $a$ around $C$ .

${\oint }_{C}\mathbf{a}•\mathbf{\tau }\phantom{\rule{0.166667em}{0ex}}ds={\oint }_{C}{a}_{i}\left[{x}_{1}\left(s\right),{x}_{2}\left(s\right),{x}_{3}\left(s\right)\right]\phantom{\rule{0.166667em}{0ex}}{\tau }_{i}\phantom{\rule{0.166667em}{0ex}}ds$

We will show later that a vector whose circulation around any simple closed curve vanishes is the gradient of a scalar.

Surface integrals

Many types of surfaces and considered in transport phenomena. Most often the surfaces are the boundaries of volumetric region of space where boundary conditions are specified. The surfaces could also be internal boundaries where the material properties change between two media. Finally the surface itself may the subject of interest, e.g. the statics and dynamics of soap films.

A proper mathematical treatment of surfaces requires some definitions. A closed surface is one which lies within a bounded region of space and has an inside and outside. If the normal to the surface varies continuously over a part of the surface, that part is called smooth . The surface may be made up of a number of subregions, which are smooth and are called piece-wise smooth . A closed curve on a surface, which can be continuously shrunk to a point, is called reducible . If all closed curves on a surface are reducible, the surface is called simply connected . The sphere is simply connected but a torus is not.

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