Talk:Gradients and Directional Derivatives
This page is great. I have two vanishingly picky comments that you could take or leave.
In your example, you use x with a vector arrow to represent position and represent the coordinates as x, y, and z. This isn't wrong of course, but my sense is that it's more standard to either represent the vector as the triple (x,y,z) or, if you want to use x-arrow as the position vector, to use x_1, x_2, and x_3 as the coordinates. I actually am totally happy with your notation; it just may be unusual.
Another notation convention. F tends to be used for vector fields, while f tends to be used for scalar fields, so in your examples at the end, you should replace F with f if you want to follow convention.
I really like the changes that you've made. Can you throw up some quick plots to go with the examples? That should be a short task, and that's what I think it needs.
I made a couple small edits and added two examples at the end.
Terminology police moment: a function isn't "made" equal to a constant, it's "set" equal to one
Can you place that first example on it's own line? Just to set it apart from the text a bit.
Also, can there be an examples of gradient calculations and directional derivative calculations?
I really like this page. I like your first image but am wondering if it is possible to have a topographical map that has a satellite image next to it to show photographically what is happening. I think this will aid in understanding the math of the image.
Is there a reason why you're restricting your discussion of directional derivatives to 2 dimensions and not using 3?
To start out even gentler in your very first paragraph why don't you have a contour map mouseover which says something like "a contour map shows various levels of a surface"?
"A level set is the set of all points
generated when a function is made equal to a constant" where a function has a constant value" or something.
I like your lead in for directional derivatives. Good to give folks an idea where they'll be going.
Hey, so your definition of the gradient operator in the comment below is still a bit off, but this is unimportant to the page, so I can explain this issue to you next time I'm in the office, if you would like. The explanation in the page now is beautiful, except I would rewrite "the gradient vector..." as "the gradient at a given point in space x, is a vector defined in terms of partial derivatives..." This makes it a bit clearer that itself requires an input.
Also, this idea of "steepest ascent" and f representing height, in your more mathematical section just does not work. In this section, you are treating f as a function that takes three input variables, so f takes a point in (x,y,z) space as input, and as output gives temperature, or air pressure, or whatever. The gradient of f would give the direction you go in three-space to increase temperature, or air pressure, or whatever, the fastest. f *cannot* represent height as a function of horizontal coordinates, however, because in this case f is a function that takes two variables as input.
Hey, thanks for the clarification about the gradient's domain. I was confusing the gradient operator, which does take a scalar function and returns a vector, with this gradient function which I now denote , which takes a position vector and returns another vector known as the gradient.
Thanks for continuing to clean up the math. There's one sentence that's continuing to trip you up, which is:
- The gradient thus has a domain of scalar functions, such as a function for height in terms of horizontal position, and a codomain of vectors.
There are two problems with this statement.
- \grad f doesn't have a domain of scalar functions, but a domain of vectors. This can also be more simply worded as "\grad f takes points in R3 as inputs and has vectors as outputs." Of course a point in R3 is itself a vector, but usually we think of the input to \grad f as a point in space and the output as an arrow. This wording helps to clarify that. Also, "domain" and "codomain" are unnecessarily jargon-y. I have a math degree and I think of inputs and outputs, not domains and codomains.
- You are on to something when you say "the gradient has a domain of scalar functions", which is this: The \grad operator as an object in its own right takes scalar functions (also called scalar fields) as input, and gives a vector field as output (specifically, the \grad operator takes f as input and spits out \grad f). So \grad on its own maps scalar fields to vector fields, and \grad f maps vectors to vectors, but neither maps from scalar fields to vectors, which is what your definition states. All of this is just for your edification. This page should probably just focus on \grad f, and not on the \grad operator itself at all.
- The example of "function in terms of horizontal position" is not appropriate. In your definition, you talk about the gradient of a function from R3 to R, but this example is a function from R2 to R. In fact, you may want to make a big point somewhere in this section that the contour map you describe in the Basic Description deals with gradients of functions from R2 to R, while your mathematical discussion focuses on gradients of functions from R3 to R.
Steve Maurer 6/23
First, I second all of Abram's 6/23 comments about what he calls mathematical "details", except that it is pretty important to use terms properly.
I am also concerned about the lead sentence, because the vectors in the main picture are not gradients, but rather negative gradients. (At least that is true if your interpretation of the contour plot is correct, but how do you know - only because the earth has many more mountains than anti-mountains (negative mountains). Actually, it does have anti-mountains, but they fill up with water so we don't see them.
But then I decided it is ok to ignore this point in the intro sentence, so long as it is corrected later, which you only half do.
I also want to second the concern about phrases like "3D function". You mean a function from 2 variables to 1 variable, so that the graph is a surface in 3D. But there are also functions from 1 variable to 2 variables, whose graph also requires 3 dimensions, but it is a curve in 3D. So your 3D term is ambiguous and just not used.
The section that uses partial derivatives probably should have partial derivatives explained, or have a link. Indeed, all the ideas in that section deserve fuller explanations. Brendan, clearly you have learned multvariate well, because you have internalized a lot of the concepts. But your readers may not have.
Brendan, this writing continues to be exemplary for clearly articulating mathematical ideas in everyday language and for fluently connecting images, vernacular language, and mathematical language.
Just a few mathematical details.
- Before the Drexel folks work on the animation, please note that a directional derivative is NOT a vector. It is a scalar. D.D. is not represented by an arrow. The phrase "magnitude of the directional derivative" has meaning only insofar as D.D. can be negative or positive, so magnitude would refer to the absolute value.
- It would actually be great to have a more complete explanation of directional derivatives, esp. a slightly more formal definition.
- The gradient doesn't have a domain of scalars, but a domain of vectors (the "input" to the gradient of f is the point in space, not the value of f at that point).
Two writing points:
- When you say what the gradient intuitively means, I would suggest pointing out that a reason for this is given in the following section on directional derivatives.
- If your more mathematical section has examples, be really careful about the fact that you are defining the gradient of a function from R3 -> R, not R2 -> R. Therefore, analogies suggesting that f represents, height, say, as a function of horizontal coordinates aren't consistent with the presented definition. It would make more sense to have f represent something like temperature as a function of point in space.
This page could use an applet as well: it could replace the animation I have towards the end that depicts the directional derivative of a gradient and different paths. I imagine the user would change the direction of the path, and the applet would show the magnitude of the directional derivative for this given path.
Oh, one other thing. Your explanation of a couple of the pictures states that the directional derivative represents a component of the gradient. This is not actually true, because a component of a gradient is itself still a vector, while a directional derivative is a scalar. I feel like your basic idea for visualizing directional derivative is a great, and there should be some way to modify your images/videos so that they stay just as useful, but are also accurate. Let me know if you want some help generating ideas, but you seem to be great at developing visualizations on your own.
I too say yay for this page.
Overall, I really like the way your writing blends mathematical description with intuitive description, so that I feel like each mathematical statement has a visual interpretation that actually aids understanding of this page.
Occasionally your writing lacks a little bit of precision, which may be what concerns Gene (I would respectfully separate precision from your general clarity, which is great). In your basic description, for instance, the statement
"These curves are curves with constant z-component, and when representing a landscape, they show constant height"
could be made more precise by saying,
"These curves are curves with constant z-component. For example, if z-represents the altitude on a landscape and x, and y represent North and South position [or something better than that], then a contour curve represents a set of points that are all at the same height," and illustrating this with a topo map.
If you are worried about the basic description getting too long, you could include only that example in this section, and everything involving equations in the more mathematical description.
The more mathematical section is quite clear. The one thing is that multivariable calculus students can often get confused about the number of dimensions involved in gradient problems, because you can't actually graph show the graph of f:R3 -> R. It may help to do something like use a contextualized example (say, f:R3 -> R represents temperature), and to point out that the actual value of f is *not* indicated anywhere on the graph of a level set.
yay for this much needed page!
My only thought is that you might want two sentence helper pages for domain and codomain, since people tend to mix those up.
A level set is the set of all points
generated when a function is made equal to a constant where a function has a particular constant value. (Or something?)
representing a map???
You have a good start on important topic, Brendan, but you could be a bit clearer in your writing sometimes. I'll take a further look later.