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Chapter 10 - Learning objectives
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<h2>Chapter 10 - Derivatives of MV functions</h2>
<h4>Learning objectives</h4>


<h3>Partial differentiation</h3>

<p>The <i>partial derivative</i> of a function, $f(x,y,u,v,w....)$
$$\frac{\del f}{\del y}\equiv \left(\frac{\del f}{\del y} \right)_{x,u,v,w...
}$$


This notation means:
<ul>
<li>To take the "partial derivative"--denoted by $\del/\del y$  instead $d/dy$
--of a MV function, with respect to one of its independent variables (for example, with respect to $y$)...

<li>Consider that all the other independent variables are being held constant,
<li>Use all your single-variable derivative knowledge, while treating all the other variable symbols as constants.
</ul>

<p>If $f=f(x,y)$, then $\del f/\del y$ is the slope function for an $x=k$ trace of the surface $f(x,y)$. That is:
$$\left.\frac{\del f}{\del y}\right|^{x=k}\equiv \frac{df(k,y)}{dy}.$$


<p>More notation:
<ul>
<li>$f_x \equiv \frac{\del f}{\del x}$,
<li>$f_{xy} \equiv \frac{\del f}{\del y}\frac{\del f}{\del x}$.
</ul>



<h3>Second-order partial derivatives</h3>
<p>Graphical interpretations
<ul>
<li>$f_x$ and $f_y$ are the <b>slope</b> functions for $y=k$ and $x=k$ traces of the $f(x,y)$ surface.
<li>$f_{xx}$ and $f_{yy}$ are <b>concavity</b> functions for $y=k$ and $x=k$ traces.
<li>$f_{xy}$ and $f_{yx}$ are <b>twist</b> functions for the $f(x,y)$ surface.
</ul>

<p><b>Know how</b> to evaluate the sign of any second-order partial derivative at a particular point on a contour plot of the function. (See the last page, "Mixed Partials" of our <a href="https://people.goshen.edu/~paulmr/physix/213/mvhandouts/10.2-3.partials.pdf">Partial derivatives handout</a>.

<p>Clairaut's theorem:
<ul><li>the order of partial differentiation does not matter
$$f_{xy} = f_{yx}.$$
<li>This follows from<ul><li>the property of a function that there is one and only one "output" value of the function $f(x,y)$ for any point $(x,y)$ (function input) in the function's domain, and<li> Holds for functions which are continuous, and have continuous partial derivatives.
</ul>
</ul>

<h3>Linear model (and chain rule)</h3>
<p>If you have some information about a continuous function at a particular  point $(x_0,y_0)$ in its domain...
<ul><li>You know the height of the surface, $f(x_0,y_0)$,
<li>and the partial derivatives at $(x_0,y_0)$:
$$f_x=\left.\frac{\del f}{\del x}\right|^{(x,y)=(x_0,y_0)}
\ \ \ \text{and}\ \ \
f_x=\left.\frac{\del f}{\del y}\right|^{(x,y)=(x_0,y_0)},
$$
</ul>

then you can find the height of the surface, $f(x,y)$ at some nearby location $(x,y)$ by means of the linear approximation:

  $$f(x,y) \approx f(x_0,y_0)+f_x (x-x_0)+f_y (y-y_0).$$

This is the equation of a <b>tangent plane</b>: A plane which is tangent to the surface $f(x,y)$ at the point $(x_0,y_0,f(x_0,y_0))$.


<p>Re-arranging this last equation just a bit, and  re-writing it in terms of changes  (that is,  $\Delta y = y-y_0$; $\Delta x = x-x_0$; and $\Delta z = f(x,y)-f(x_0,y_0)$):

  $$\begineq f(x,y) - f(x_0,y_0) &\approx f_x (x-x_0)+f_y (y-y_0)\\
  \Delta z &\approx f_x \Delta x+f_y \Delta y\\
  \frac{\Delta z}{\Delta t}
  &\approx \frac{\del f}{\del x}
  \frac{\Delta x}{\Delta t}+\frac{\del f}{\del y} \frac{\Delta y}{\Delta t}\\
  \frac{dz}{dt}
  &= \frac{\del f}{\del x}\frac{dx}{dt}+\frac{\del f}{\del y} \frac{dy}{dt}\\
\endeq
$$

This is the <b>chain rule</b> for a function $f(x,y)$ if $x=x(t)$ and $y=y(t)$ are each functions of a parameter $t$.


<h3>The gradient, and directional derivatives</h3>

<p>The <b>gradient 
of a function</b>,  $\myv \grad f(x,y)$ is a vector in the $xy$ plane which has  components which are the partial derivatives wrt each of the independent variables:
$$\myv\grad f =\langle \frac{\del f}{\del x}, \frac{\del f}{\del y} \rangle.$$
<ul><li>It <b>points in the direction of <i>steepest increase</i></b> of the surface $f$.
<li>It is <b>perpendicular to the contour line</b> at that point.
<li>Its <b>magnitude is the slope of the surface</b> [meters of rise / meters of run...]  in that steepest direction.
</ul>

<p>The <b>directional derivative</b> or slope of a surface in a particular direction, specified in the $xy$ plane by the unit vector $\uv u$, is given by
$$D_{\uv u}f = \myv \grad f \cdot \uv u.$$

<h3>Optimization</h3>
<p>...Finding and characterizing maxima and minima and other critical points.

<h4>Critical points of a surface, $f(x,y)$.</h4>
<p>Look for points $(x,y)$ that simultaneously satisfy:
$$\frac {\del f}{\del x}=0
\ \ \ \ \text{and} \ \ \ \ \frac {\del f}{\del y}=0.$$
You can look at a graph of the surface (Geogebra) to classify your critical points as maxima or minima (absolute or local), or neither (for example, saddle points and inflection points).
<p>Or you can evaluate the Determinant of the Hessian matrix for an analytical test of the critical point type.

<h4>Critical points of a surface along a path $g(x,y)=k$</h4>
<p>...or critical points, subject to a "constraint" $g(x,y)=k$.

<p>There are two ways to find the critical points along a path:
<ol><li><b>Parameterize  the path</b>:
<ol><li>Find $x(t)$ and $y(t)$ such that $g(x(t),y(t))=k$.
<li>Solve $df(x(t), y(t))/dt = 0$ to find one or more critical <i>parameters</i> $t_1, t_2,....$.
<li>Substitute the critical parameters into your expressions for $x$ and $y to find the critical points $(x(t_1),y(t_1))$, $(x(t_2),y(t_2)$,...
</ol>
<li>or, use <b>Lagrange multipliers</b>:  <ol>
<li>At a critical point, the two normal vectors must be parallel:  $$\myv \grad f = \lambda\myv \grad g.$$
<li>Write down a system of equations: one for each component of the gradient equation, together with the constraint equation:
$$\left(\myv \grad f\right)_x = \lambda\left(\myv \grad g\right)_x$$
$$\left(\myv \grad f\right)_y = \lambda\left(\myv \grad g\right)_y$$
$$g(x,y) = k$$

<li>solve the  system of equations for the the coordinates $(x,y)$ and the parameter $\lambda$.  (There may be multiple solution sets).

</ol>
</ol>

<h4>Using CoCalc...</h4>
<p>You should be able to
<ul><li>Make a contour plot of a function $f(x,y)$ over some particular domain.
<li>Plot a path on top of your contour plot.
<li>Examine the plot to determine where the path is at a local maximum / minimum / saddle point or something else.
</ul>
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