Day 8 in Mat329

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Today:

Announcements
- Reminder: stow those rectangles....
- Your quizzes are returned.
  1. We can learn something.... what?
  2. Definition -- three things, which are?
  3. We do have to worry about some points -- which?
  4. Just a few of you haven't learned the essense of my haiku....
- Assignment 14.2 is returned: graded 5, 14, 17.
  First things first: don't be a limit dropper!
  - I'd like you to justify your actions. So on #5, mention that $f(x,y)$ is a polynomial, hence continuous everywhere on the plane, hence limits are easy.
  - #14: Again, in the end, after conjugates, it's a polynomial. We do need to be careful: it's only the limits that are equal, not the original and the reduced functions.... They're equal on the original domain.
  - #17: here you have a composition of continuous functions, so you need to invoke the theorem stated in the text, on page.... 922.
  As the semester goes on, I will start taking off points for "failure to justify" -- F2J.
Last time we began section 14.3, on partial derivatives.
Here is what I call "the most important definition in calculus" -- the limit definition of the derivative function, $f'(x)$: \[ f'(x) = \lim_{h \to 0}\frac{f(x+h)-f(x)}{h} \]
This is generalized into "partial" derivatives \[ f_x(x,y) = \lim_{h \to 0}\frac{f(x+h,y)-f(x,y)}{h} \] \[ f_y(x,y) = \lim_{h \to 0}\frac{f(x,y+h)-f(x,y)}{h} \]

Where we think of one variable as fixed, and the other as varying.
We consider each as a slope in a particular direction. But there are an infinite number of directions from which one can depart, of course. We are considering only two.
As we saw last time, partials are actually easy to compute, provided one has a formula. If one has only data, then it comes down to approximating with a finite difference.
We saw that higher partial derivatives are easy to calculate, as well, since each partial derivative is a multivariate function in its own right. So do it again!
We think of second derivatives in the univariate world as saying something about curvature, and the same is true in the bivariate case.
- #53, p. 937 (note the notation, p. 930)
The world is governed by partial differential equations, such as the heat equation, the wave equation, Laplace's equation, etc.
- Allow me to motivate the heat equation, for an insulated rod; then we'll consider
- #75, p. 937
There is an important theorem in this section: Clairaut's theorem: Suppose $f$ is defined on a disk $D$ that contains the point $(a,b)$. If the functions $\frac{\partial^2 f}{\partial x \partial y}$ and $\frac{\partial^2 f}{\partial y \partial x}$ are both continuous on $D$, then \[ \frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x} \]
- #74, p. 937 (Clairaut's theorem tells us that we don't need to consider $f_{yx}$).
A little exercise that will lead us into section 14.4. (Some mathematica, as well.)
So on to section 14.4: tangent planes and linear approximations.
Just like derivatives are closely linked to tangent lines, which kiss a curve (osculate); about the slopes of those linear approximations (tangent lines) to a univariate function; so are partial derivatives are about slopes of univariate functions obtained by slicing multivariate functions with planes. If the function is differentiable at a point, then there is a tangent plane, which kisses the surface.
An example from our author's TEC animations.
- Here's a Mathematica file for similar problems which I stole from someone at the Air Force Academy....

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