Today we start 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 what we want to generalize.
- Take a look at Table 1, p. 925. How can we use the univariate idea?
- What about other directions? Think about a topo map -- you can set
off in any direction from a point (often): what does the partial
derivative mean in any direction?
- #2, p. 935
So here's the partial with respect to $x$:
\[
f_x(x,y) = \lim_{h \to 0}\frac{f(x+h,y)-f(x,y)}{h}
\]
We think of $y$ as fixed in this derivative. It's not varying. Only $x$ is
varying; is variable.
And here's the partial with respect to $y$:
\[
f_y(x,y) = \lim_{h \to 0}\frac{f(x,y+h)-f(x,y)}{h}
\]
They're easy to compute, actually: e.g., for $f_x$, just imagine that $y$ is a parameter,
and differentiate in the univariate way with respect to $x$.
Figures 4 and 5 on p. 928 give us this idea very nicely.
Problems:
- #5, p. 936
- #10
- #15
- #24
Higher partial derivatives are easy to calculate, as well, since each partial
is again 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.
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
\[
\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}
\]