Implicit (multi) differentiation

[Sean Fitzpatrick]

I'd like to add a few more paragraphs and another example to the subsection on implicit differentiation in the section on multivariable chain rule.

Right now the only case covered is that of a level curve f(x,y)=c.

I agree it's cool to see how this makes implicit differentiation as seen in Chapter 2 way more transparent.

But I think the case f(x,y,z)=c is more important, and it's missing.

Here's why I think we need it.

Initially, tangent planes and differentiability ("smoothness", if you like) are developed for graphs z=g(x,y). Later we make the observation about the gradient and say it can be used for tangent planes to level surfaces.

But the real reason we can make sense of tangent planes in this case is implicit differentiation. (More to the point, the implicit function theorem, but I'm not saying we should include that. Maybe I am.)

Implicit differentiation lets us see that a level surface is "locally a graph", and that the definition of tangent plane originally encountered for graphs does indeed match with the one we get from the gradient vector.

[gregory...@gmail.com]

I've had to think hard about this. On one hand, extending implicit differentiation to 3 variables can justify better the "gradient is orthogonal to level surfaces" statement at the end of 12.7. On the other, I wonder if the added content will add value to the book in the student's eyes, or if a seemingly easy-to-apply rule (the gradient of 3 variables is normal to the surface) will get lost within the justification. 

In 12.5, I'm not opposed to extending the examples to level surfaces and labeling the result as the Implicit Function Theorem of Three Variables. The justification is analogous to what is done already with a few small additions. Then an example can be done.

Then in 12.7 ... how does this play out? As stated above, I don't want things to get too complicated. I know that the text lacks the depth that many want to see in a calc book, and I have received feedback about the lack of proofs, etc. But I am very happy with justifying "the gradient is orthogonal to the level surface" via analogy. It's clean, simple, and gives students something they can apply. A longer justification may obfuscate the point. But maybe I'm also missing the point.

[Sean Fitzpatrick]

I think the details aren't too onerous. Suppose f(x,y,z)=K defines z=g(x,y) implicitly.

For the graph z=g(x,y), the tangent plane at (a,b,c) (where c=g(a,b)) is:

z=c+g_x(a,b)(x-a)+g_y(a,b)(y-b),

or

-g_x(a,b)(x-a)-g_y(a,b)(y-b)+(z-c)=0.

Now sub in g_x = -f_x/f_z and g_y=-f_y/f_z, (based on results from 12.5) and multiply through by f_z to get

f_x(a,b,c)(x-a)+f_y(a,b,c)(y-b)+f_z(a,b,c)(z-c)=0

From the section on planes, we can immediately conclude that the normal vector is the gradient of f at (a,b,c).

I think this could almost be done in an aside, except that the plane equation wouldn't fit.

The part you leave out (although I think it's the cool part):

if grad f(a,b,c)=<0,0,0>, we call (a,b,c) a critical point. (That much is covered.)

the value f(a,b,c)=K is then called a critical value. [You might recall that I once objected to using "critical value" as a synonym for "critical point" in one variable.]

If K is not a critical value, we call K a regular value.

A fundamental theorem in differential topology is that if K is a regular value, the level set f(x,y,z)=K is a manifold.

In this context, that just means that a tangent plane is well defined at every point on the surface.

Why? If K is a regular value, then by definition, there are no critical points on the surface f(x,y,z)=K.

That means that at every point, at least one partial derivative is nonzero, and we can assume that the corresponding variable is defined implicitly as a function of the others.

So whether you write z=g(x,y), or y=h(x,z), or x=k(y,z), implicit differentiation will lead to the same tangent plane equation.

I assign this as an exploration every time I teach this material, because I think it's one of the more satisfying results.

Then we play around with some examples: if K is a critical value, what do the level sets look like?

(The family x^2+y^2-z^2=K is a good example, with the cone being the critical set.)

[gregory...@gmail.com]

I agree, the details are not difficult nor are they lengthy; seems like a good way to show the gradient is normal to the surface.

And while it's dear to your heart, I'd leave out the differential topology part. :) 

[Sean Fitzpatrick]

This is now done, at https://github.com/sean-fitzpatrick/APEXCalculusPTX/commit/e0a030f526b282aa1f6860a77be24c60f67d26b1

(Haven't built HTML yet, and I'm going to add a few more things before making a pull request.)