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Mathematical Expression Editor
We introduce partial derivatives and the gradient vector.
Given a function , it is often useful to differentiate with respect to a single variable
and hold the other variables as constants. One way to think of a function of several
variables is as a “machine” with lots of knobs:
One way to try and understand the machine above would be to hold all but one of
the knobs constant, and see what happens when you “wiggle” a single knob. As a
explicit example, let Here is our “machine” and the variables and are the “knobs.”
Fixing , allows us to focus our attention to all points on the surface where the -value
is ,
We can now focus our attention on the curve
and differentiate this curve purely with respect to . In a similar way, we could fix
and differentiate with respect to .
Given a function , the partial derivative of with respect to the th variable is
denoted: This means that one should take the single-variable derivative with respect
to of while treating all other variables as constants.
The following interactive let’s you see whats going on with partial derivatives:
Let . Compute:
Compute
There are several different notations for the partial derivative. We’ll mainly be using
these:
Let . Compute:
Compute:
We have shown how to compute a partial derivative, but it may still not be clear
what a partial derivative means. Given , measures the rate at which changes as only
varies: is held constant.
Imagine standing in a rolling meadow, then beginning to walk due east. Depending
on your location, you might walk up, sharply down, or perhaps not change elevation
at all. This is similar to measuring : you are moving only east (in the -direction) and
not north/south at all. Going back to your original location, imagine now walking
due north (in the -direction). Perhaps walking due north does not change your
elevation at all. This is analogous to : does not change with respect to . We can
see that and do not have to be the same, or even similar, as it is easy to
imagine circumstances where walking east means you walk downhill, though
walking north makes you walk uphill. The next example helps us visualize
this.
Let . Find and .
Write with me and Thus and .
Whenever we do a computation in mathematics, we should ask ourselves, “What
does this mean?”
Let . What is the meaning of
First note that . If , this means if one “stands” on the surface at the point and
moves parallelorthogonal to the -axis (so only the -value changes, not the -value), then the instantaneous rate
of change in is . Increasing the -value will increasedecrease the -value; decreasing the -value will increasedecrease the -value.
If , this means if one “stands” on the surface at the point and moves parallelorthogonal to the -axis (so only the -value changes, not the -value), then the instantaneous rate
of change in is . Increasing the -value will increasedecrease the -value; decreasing the -value will increasedecrease the -value.
Finally, since the magnitude of is greater than the magnitude of at , the surface is
“steeper” in the -direction than in the -direction.
Estimating partial derivatives
Functions of several variables, especially ones that map can be described by a table
of values or level curves. In either case we can estimate partial derivatives by looking
at Let’s do an example to make this more clear.
Let be a differentiable function
described by the following table of values:
Estimate .
To estimate , we examine the change in between and and then between and .
We will then average these estimates to find our answer. To start look between and :
Now examine the change in between and :
Now if we average these values together, we see:
Let be a differentiable function described by the following table of values:
Estimate .
Work as we did in the example above, finding two estimates and taking their
averages.
We can also estimate partial derivatives by looking at level curves.
Let be described by the level curves below:
The height of the level curve is marked on the curve, and we are given a point .
Estimate .
To estimate , we examine the change between the level curve that is on
and the nearest level curve found by traveling on a line parallel to the -axis.
Starting at and moving to the left on a line parallel to the -axis, we see
We also should examine the change between the closest level curve when moving to the right:
Now if we average these values together, we see:
Let be described by the level curves below:
The height of the level curve is marked on the curve, and we are given a point .
Estimate .
Work as we did in the example above, finding two estimates and taking
their averages.
Combining partial derivatives
While a function only has one second derivative. However, functions have second
partial derivatives and functions have second partial derivatives! Don’t run off yet,
things get better.
Let be continuous on an open set .
The second pure partial derivative of with respect to then is
The second pure partial derivative of with respect to then is
Moreover, there is also the notion of a mixed partial derivative, and The
notation is ambiguous, it does not state which derivative should be taken first. As
we will see, in practice this is not too much of a problem.
Consider: Find six first and second partial derivatives.
Notice how above . The next theorem states that it is not a coincidence.
Mixed Partial Derivatives Let be a function where are continuous on an
open set . Then for each point in , . A similar result is true for functions
.
Finding and independently and comparing the results provides a convenient way of
checking our work.
The gradient vector
Given a function , we often want to work with all of first partial derivatives
simultaneously. In this case, we will work with the vector: As we will see, for
functions of several variables, this vector will play the role that the derivative did for
functions of a single variable. This vector is called the gradient vector.
Let be a function whose first partial derivatives exist, the gradient is a
vector-valued function of variables.
The upside-down triangle in the notation for the gradient sometimes called a del. It
is also known as a nabla. You can think of the as the vector: and hence when one
writes: , you are literally distributing the across the vector, just as a scalar acts on a
vector.
The gradient is defined at points in the domain where the partial derivatives are
defined. The gradient of a function of two variables lives in . The gradient of a
function of three variables lives in . Generally the gradient of a function of variables
lives in . We can see this in the interactive below.
The gradient at each point is a vector pointing in the -plane.
Try your hand at some casual computations.
Let , compute:
Above, note that is a vector whose components are functions of
and , hence it is a vector-valued function. We can evaluate functions at actual points
in their domain. For instance, if we compute:
And now in three variables.
Let , compute:
Let . Compute:
This is just your first taste of the gradient vector. Much more will be coming
soon.