Partial Differentiation Without Tears - Comments

[andrewkirk]

andrewkirk submitted a new PF Insights post

Partial Differentiation Without Tears

Continue reading the Original PF Insights Post.

 

[stevendaryl]

Some of the ambiguity with functions could be cleared up using the lambda calculus (or just the lambda notation), which makes explicit what arguments a function takes. There is always a confusion in an expression such as $f(x)$ as to whether you mean the function, or whether you mean the value of $f$ at a particular point $x$. The lambda calculus makes this clear. Your notation $x\mapsto x^2$ is basically equivalent (I think) to the lambda calculus, which would indicate that function by $\lambda x . x^2$. I think that the only reason people don't use the lambda notation is that it's a lot of trouble, and usually it's not necessary.

 

[Greg Bernhardt]

Thanks andrewkirk!

 

[robphy]

Especially with respect to thermodynamics, it would be good to be explicit about what variables are held fixed.
http://www.compadre.org/per/document/servefile.cfm?ID=11819&DocID=2669 [PDF]
"Representations of Partial Derivatives in Thermodynamics" Thompson, Manogue, Roundy, Mountcastle

 

[Greg Bernhardt]

Nice work Andrew!

 

[Orodruin]

This issue is particularly important when teaching students variational calculus and Lagrange mechanics. In particular when the base space is multi-dimensional and the Euler-Lagrange equations involve a partial derivative with respect to the base space coordinates of a partial derivative with respect to the arguments of the Lagrangian. This has confused many a student.

 

[anorlunda]

I have a suggestion. Start with images to explain all the differentiation concepts on the first day before even simple total derivative is explained. Once students grasp what they are trying to express, then they can better focus on how.

Using this one image, I think the following concepts can be explained in just seconds.

1. Partial derivatives: The rate of change of altitude as we walk toward the summit versus the rate of change of altitude as we walk parallel to the summit.
2. Total derivative: Imagine the image reduced to a 2D profile choosing a planar slice through the mountain. Then height Y represents altitude, and it varies only with the horizontal variable X. The slope is the total derivative.
3. Covariant derivative: Consider the color shading as another independent variable. Now imagine walking on the mountain following a contour with constant shade of green. The rate of change of altitude is now the derivative covariant with color.