Notational confusion of multivariable derivatives: Difference between revisions

Revision as of 07:31, 16 July 2018

I think there's several different confusions that arise from multivariable derivative notation:

The thing where $\frac{\partial w}{\partial t}$ can mean two different things on LHS and RHS when $t$ is used as both an initial and intermediate variable. (See Folland for details.)
The thing where if $f (x, y) = (x^{2}, y^{2})$ then $\frac{\partial f}{\partial x} (x, x)$ feels like it might be $(2 x, 2 x)$ even though it's actually $(2 x, 0)$ . (Example from Tao.) See also [1]
The thing where the total derivative for $n = m = 1$ "should" be a function but people treat it as a number. Refer to "Appendix A: Perorations of Dieudonne" (p. 337) in Pugh's Real Mathematical Analysis.
Total derivative vs derivative matrix

@@ Line 3: / Line 3: @@
 * The thing where <math>\frac{\partial w}{\partial t}</math> can mean two different things on LHS and RHS when <math>t</math> is used as both an initial and intermediate variable. (See Folland for details.)
 * The thing where if <math>f(x,y) = (x^2,y^2)</math> then <math>\frac{\partial f}{\partial x}(x,x)</math> feels like it might be <math>(2x,2x)</math> even though it's actually <math>(2x,0)</math>. (Example from Tao.) See also [https://issarice.com/mathematics-and-notation]
-* The thing where the total derivative for <math>n=m=1</math> "should" be a function but people treat it as a number.
+* The thing where the total derivative for <math>n=m=1</math> "should" be a function but people treat it as a number. Refer to "Appendix A: Perorations of Dieudonne" (p. 337) in Pugh's ''Real Mathematical Analysis''.
 * Total derivative vs derivative matrix