Notational confusion of multivariable derivatives: Difference between revisions
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* 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 <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 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 derivative as a linear transformation in the several variable case and a number in the single-variable case== | ==The derivative as a linear transformation in the several variable case and a number in the single-variable case== | ||
* 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''. | * 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 versus derivative matrix== | |||
Technically the total derivative at a point is a linear transformation, whereas the derivative matrix is a matrix so an array of numbers arranged in a certain order. However, there is a one-to-one correspondence between linear transformations <math>\mathbf R^n \to \mathbf R^m</math> and <math>m</math> by <math>n</math> matrices, so many books call the total derivative a matrix or equate the two. | |||
A similar confusion exists in the teaching of linear algebra, where sometimes only matrices are mentioned. | |||
==See also== | ==See also== | ||
* [[Summary table of multivariable derivatives]] | * [[Summary table of multivariable derivatives]] | ||
Revision as of 06:16, 20 July 2018
I think there's several different confusions that arise from multivariable derivative notation:
- The thing where can mean two different things on LHS and RHS when is used as both an initial and intermediate variable. (See Folland for details.)
- The thing where if then feels like it might be even though it's actually . (Example from Tao.) See also [1]
The derivative as a linear transformation in the several variable case and a number in the single-variable case
- The thing where the total derivative for "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 versus derivative matrix
Technically the total derivative at a point is a linear transformation, whereas the derivative matrix is a matrix so an array of numbers arranged in a certain order. However, there is a one-to-one correspondence between linear transformations and by matrices, so many books call the total derivative a matrix or equate the two.
A similar confusion exists in the teaching of linear algebra, where sometimes only matrices are mentioned.