Pictorial Definition of Partial Derivatives
The following two statements are equivalent:
- A function z = f(x,y) is differentiable at a point
(x0, y0, z0 = f(x0));
- If we infinitely magnify the graph of the function
(which is a possibly undulating
surface) about the point
(x0, y0, z0)
it will look like a flat plane.
And in that case the slope of the plane in the x-direction
is the partial derivative of the
function at that point with respect to x, and slope of the plane in the y-direction
is the partial derivative of the
function at that point with respect to y.
Partial Derivatives.
We saw in an earlier animation that if a function y = f(x) of one
variable results in a smooth curve, then it is differentiable at a point x if and
only if an infinite magnification of the curve at (x, y) is a straight line. Once
you reach infinite magnification all distances are infinitesimal - which means
infinitely small (but not zero! Don't worry about wrapping your mind around
this concept; it requires a kind of Zen acceptance... oooohhhmmmmm...). The slope of this
line is the derivative, which can be nicely represented as the rise of the function
divided by the horizontal run of the function as it changes from x to x + dx, where
dx is an infinitesimal number (mantra time). So the horizontal change is
(x + dx) - x = dx. The vertical rise is f(x + dx) - f(x). So the derivative
is
df/dx = (f(x + dx) - f(x))/dx.
Well, that's fine. But we have a function of two variables, and the
graph is a surface, not a curve. The same basic ideas apply, however.
If the function, z = f(x,y), is differentiable at some point (x,y) in its
domain (where it's defined), then an infinite magnification of the surface at the point
(x,y,z) will result in what looks like a flat, but probably tilted plane. A flat planar
surface is to 3 dimensions what a line is to 2 dimensions. Because it's flat, at any
starting point (x,y,z) on a plane, the slope of the plane in the x-direction will be
a constant, as will the slope in the y-direction. The slope in the x-direction
is the rise in the z-direction resulting from a change in x,
divided by the corresponding change in x horizontally.
Infinitesimally speaking this is
df/dx =
(f(x + dx, y) - f(x, y))/dx.
(Note: that delta above should be a del, but I haven't figured out
how to make a del using HTML, so just pretend.)
The slope in the y-direction
is the rise in the z-direction divided by the corresponding change in y horizontally.
Infinitesimally speaking this is
df/dy =
(f(x, y + dy) - f(x, y))/dy.
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Ok, here's a zoom of the function
z = f(x,y) = cos2(x)sin2(y)
centered on the point (x = 1.0, y = -1.0). You can read the x and y limits
right from the four graphs. The last plot is a 30 times magnification of the first
plot centered at (1,-1). At that magnification the function appears already to
be pretty close to planar, and we can read the two partial derivatives right
off the last plot, at least approximately. Along the y = -0.9 line at the
right front of the plot, when x = 0.9, z = 0.25 approximately, and when x = 1.1,
z = 0.13 approximately. So the approximate slope in the x direction is
(0.13 - 0.25)/(1.1 - 0.9) = -0.12/0.2 = -0.6.
The actual partial derivative of f with respect to x (given a new notation) is:
fx(x,y) = -2cos(x)sin(x)sin2(y).
Therefore,
fx(1,-1) = -0.644,
which is pretty close to the value I got from the graphic. (NOTE!: I'll leave it up
to you to do the same thing for fy(x,y), the partial of f with respect
to y. Just make sure you remember to take values of the sine and cosine using
radians, not degrees!)
In general, if u2+v2 = 1, so that [u,v] is a unit vector in the
x,y-plane, then
ufx(x,y) + vfy(x,y)
is the slope of the function f(x,y) in the direction of [u,v] at the point (x,y).
It turns out that the vector [fx(x,y), fy(x,y)] points in the
direction of maximum slope (increase) of the function f at (x,y). I'll let the reader
figure out that this means the maximum slope of f at (x,y) is
((fx(x,y))2 + (fy(x,y))2)1/2.
The slope in the direction perpendicular to the maximal direction is zero. That is,
in that direction the surface goes off horizontally, parallel to the x,y-plane. Just
imagine walking on a level path on the side of a mountain. The direction of maximum
increase in mountain slope relative to where you are must be perpendicular to the
level path.
These ideas generalize very easily to functions of more than two variables, but in that case
there is no easy way to graph the result (which would be a 3-dimensional "surface" curving
around in a 4-dimensional space; and trust me, you can't imagine 4 space dimensions).
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