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Total and Partial Derivatives, Chain Rule

There is no doubt that a great deal confusion arises from the following question, ``What are the variables of my function?''

For example, suppose we have a three-dimensional space $(x,y,z)$ , in which there is an embedded surface $(x(w,v), y(w,v), z(w,v))$ $\vec{x}(w,v)$ = $\vec{x}(\vec{u})$ where $\vec{u} = (v,w)$ is a vector that lies in the surface, and an embedded curve $(x(s), y(s), z(s)) = \vec{x}(s)$ . Furthermore, suppose there is a curve that lies within the surface $(w(t),v(t)) = \vec{u}(t)$ .

Suppose that $\mathcal{E} = f(x,y,z)$ is a scalar function of $(x,y,z)$ .

Here are some questions that arise in different applications:

1. How does $\mathcal{E}$ vary as a function of position?
2. How does $\mathcal{E}$ vary along the surface?
3. How does $\mathcal{E}$ vary along the curve?
4. How does $\mathcal{E}$ vary along the curve embedded in the surface?

Taylor Series

The behavior of a function near a point is something that arises frequently in physical models. When the function has lower-order continuous partial derivatives (generally, a ``smooth'' function near the point in question), the stock method to model local behavior is Taylor's series expansions around a fixed point.

Taylor's expansion for a scalar function of $n$ variables, $f(x_1, x_2 , \ldots , x_n)$ which has continuous first and second partial derivatives near the point $\vec{\xi} = (\xi_1, \xi_2, \ldots , \xi_n)$ is:

$$\begin{split}& f(\xi_1 , \xi_2 , \ldots , \xi_n) = f(x_1 , x_... ...] +\ldots + \mathcal{O}\left[(\xi_n - x_n)^3\right] \end{split}$$ (12-8)

or in a vector shorthand:

$$f(\vec{x}) = f(\vec{\xi}) + \left.\nabla_{\vec{x}} f\right\vert _... ...dot (\vec{xi} - \vec{x}) +\mathcal{O}\left[\norm {\vec{\xi} - \vec{x}}^3\right]$$ (12-9)

MATHEMATICA$^{\text{\scriptsize {\textregistered }}}$ Example

(notebook Lecture-12)

(html Lecture-12)

(xml+mathml Lecture-12)

Approximating Surfaces Example of turning a function of two variables, $f(x,y)$ into an approximating function of four variables:

1. Start with a function $f(x,y)$ and expand it about a point $(\xi, \eta)$ to second order--use MATHEMATICA$^{\text{\scriptsize {\textregistered }}}$ 's Normal function to convert from series representation to a parabolic representation.
2. The expansion is now a function of four variables--the first two variables are the point the function is expanded around, and the second two are the variable of the parabolic approximation at that point:
   $$f_{\mbox{appx}} (\xi, \eta ; x, y) = f(x,y) + \ensuremath{\left.\... ...ft.\frac{\partial{f}}{\partial{y}}\right\vert _{x,y}} (\eta - y) + \mathcal{Q}$$
   where
   $$\mathcal{Q} \equiv \frac{1}{2}\ensuremath{\left.\frac{\partial^{2... ...frac{\partial^{2}{f}}{\partial{y}^{2}}\right\vert _{x,y}} (\eta - y)(\eta - y)$$
   or
   $$f_{\mbox{appx}} (\xi, \eta, x, y) = f(x,y) + \nabla f \cdot ... ... -y \end{array}\right) + \frac{1}{2} \mathcal{Q}_{\mbox{form}}$$
   where
   $$\mathcal{Q}_{\mbox{form}} \equiv (\xi -x , \eta - y) \left( ... ...) \left( \begin{array}{c} \xi -x\\ \eta -y \end{array}\right)$$

3. Plot the function $f_{\mbox{appx}}(\xi, \eta, x, y)$ for $\vert\xi - x\vert < \delta$ and $\vert\eta - y\vert < \delta$ for a selected number of points $(x,y)$ .

Just a few of many examples of instances where Taylor's expansions are used are:

linearization

Examining the behavior of a model near a point where the model is understood. Even if the model is wildly non-linear, a useful approximation is to make it linear by evaluating near a fixed point.

approximation

If a model has a complicated representation in terms of unfamiliar functions, a Taylor expansion can be used to characterize the `local' model in terms of a simple polynomials.

asymptotics

Even when a system has singular behavior (e.g, the value of a function becomes infinite as some variable approaches a particular value), how the system becomes singular is very important. At singular points, the Taylor expansion will have leading order terms that are singular, for example near $x=0$ ,

$$\frac{\sin(x)}{x^2} = \frac{1}{x} - \frac{x}{6} + \mathcal{O}(x^3)$$ (12-10)

The singularity can be subtracted off and it can be said that this function approaches $\infty$ "linearly" from below with slope -1/6. Comparing this to the behavior of another function that is singular near zero:

$$\frac{\exp(x)}{x} = \frac{1}{x} + 1 + \frac{x}{2} + \frac{x^2}{6} + \mathcal{O}(x^3)$$ (12-11)

shows that the $e^x/x$ behavior is ``more singular.''

Figure 12-1: Behavior of two singular functions near their singular points.

stability

In the expansion of energy about a point is obtained, then the various orders of expansion can be interpreted:

zero-order

The zeroth-order term in a local expansion is the energy of the system at the point evaluated. Unless this term is to be compared to another point, it has no particular meaning (if it is not infinite) as energy is always arbitrarily defined up to a constant.

first-order

The first-order is related to the driving force to change the state of the system. Consider:

$$\Delta E = \ensuremath{\nabla}E \cdot \delta \vec{x} = -\vec{F} \cdot \delta \vec{x}$$ (12-12)

If force exists, the system can decrease it energy linearly by picking a particular change $\delta \vec{x}$ that is anti-parallel to the force.

For a system to be stable, it is a necessary first condition that the forces (or first order expansion coefficients) vanish.

second order

If a system has no forces on it (therefore satisfying the necessary condition of stability), then where the system is stable or unstable depends on whether a small $\delta \vec{x}$ can be found that deceases the energy:

$$\begin{split}\Delta E & = \frac{1}{2} \delta \vec{x} \cdot \e... ...ert _{x_1 , x_2, \ldots x_n}} \delta x_i \delta x_j \end{split}$$ (12-13)

where the summation convention is used above and the point $(x_1, x_2, \ldots ,x_n)$ is one for which $\nabla E$ is zero.

numerics

Derivatives are often obtained numerically in numerical simulations. The Taylor expansion provides a formula to approximate numerical derivatives--and provides an estimate of the numerical error as a function of quantities like numerical mesh size.