Expectations II

A contour plot of the function. Pretty respectable looking hills- maybe somewhere in the Ozarks- if I say so myself.

As a further example of yesterday’s post, I was discussing multivariable calculus with a student who had never taken it, and mentioned the gradient.  Putting our discussion into the framework of this post, here is what he wanted out of such a high dimensional analogue of the derivative of a function f: \mathbb{R}^2 \to \mathbb{R} (note to impressionable readers: the function defined below is not quite the gradient):
1. Name the answer: Call the gradient D.
2. Describe the answer:  should be a function from \mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}^3, which takes a point in the domain, a direction in the domain, and returns the vector in the range.  The idea being that if you had a map, knew where you are and in which direction you wished to travel, then the gradient should tell you what 3-dimensional direction you would head off in.

Certainly there is such a function, though in some sense we are making it too complicated.  As an example we have some pictures of the beautiful hills formed by the function

f(x,y) = \sin{3y} + \sin{(4x + 5y)} - x^2 - y^2 + 4.

The (actual) gradient of this function is

\nabla f(x,y) = \left(4\cos (4x + 5y) - 2x, 3\cos(3y) - 2y + 5\cos(4x + 5y)\right).
Plugging in a point in the plane will give a single vector, and then taking the dot product of this vector with a direction will give a rate of change for f at that point, in that direction.  Specifically, if we start walking north at unit speed from the origin, the gradient will be (4,8), and I take the dot product of this with (0,1) to find that I will be climbing at 8m/s (depending on our units!)

Now the correct answer from my student’s point of view would be that the answer is (0,1,8), since this is the direction in 3 dimensions that one would travel, and that the correct definition for would have
Df(x,y) \cdot v = \left(x,y,\nabla(x,y) \cdot v \right).

The graph of the indicated function, including the vector of the "pseudo-gradient" we discuss.

Of course there are more sophisticated examples of this.  Suppose a function u: \mathbb{R}^n \to \mathbb{R} is harmonic.  That is to say, \Delta u := \sum_{j = 1}^n \frac{\partial^2 u}{\partial x_j^2} = 0.  Notice that in order to write down this equality, we already named our solution u.  But just working from this equation, we can deduce a number of qualities that any solution must have: u is infinitely differentiable and, restricted to any compact set, attains its maximum and minimum on the boundary of that set.  Such properties quickly allow us to narrow down the possibilities for solutions to problems.

A picture of some hills that might be shaped like the function we're looking at. In the Ozarks of all places!

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