Notational confusion of multivariable derivatives - Revision history

IssaRice: Redirected page to calculus:Notational confusion of multivariable derivatives

2018-11-03T03:43:42Z

Redirected page to calculus:Notational confusion of multivariable derivatives

← Older revision		Revision as of 03:43, 3 November 2018
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	~~I think there's several different confusions that arise from multivariable derivative notation:~~		#redirect [[calculus:Notational confusion of multivariable derivatives]]

	* 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 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 [~~https~~:~~//books.google.com/books?id=2NVJCgAAQBAJ&lpg=PR1&pg=PA357 "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==~~

	* [[Summary table of multivariable derivatives]]

IssaRice: /* The derivative as a linear transformation in the several variable case and a number in the single-variable case */

2018-07-20T06:20:40Z

The derivative as a linear transformation in the several variable case and a number in the single-variable case

← Older revision		Revision as of 06:20, 20 July 2018
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	==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 [https://books.google.com/books?id=2NVJCgAAQBAJ&lpg=PR1&pg=PA357 "Appendix A: Perorations of Dieudonne"] (p. 337) in Pugh's ''Real Mathematical Analysis''.

	==Total derivative versus derivative matrix==		==Total derivative versus derivative matrix==

IssaRice at 06:16, 20 July 2018

2018-07-20T06:16:43Z

← Older revision		Revision as of 06:16, 20 July 2018
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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]
	* Total derivative vs derivative matrix

	==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]]

IssaRice at 06:09, 20 July 2018

2018-07-20T06:09:00Z

← Older revision		Revision as of 06:09, 20 July 2018
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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]
			* Total derivative vs derivative matrix

			==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 vs derivative matrix

	==See also==		==See also==

	* [[Summary table of multivariable derivatives]]		* [[Summary table of multivariable derivatives]]

IssaRice at 07:31, 16 July 2018

2018-07-16T07:31:51Z

← Older revision		Revision as of 07:31, 16 July 2018
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 <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 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		* Total derivative vs derivative matrix

IssaRice at 07:26, 16 July 2018

2018-07-16T07:26:05Z

← Older revision		Revision as of 07:26, 16 July 2018
Line 4:		Line 4:
	* 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 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.
			* Total derivative vs derivative matrix

	==See also==		==See also==

	* [[Summary table of multivariable derivatives]]		* [[Summary table of multivariable derivatives]]

IssaRice at 07:25, 16 July 2018

2018-07-16T07:25:14Z

← Older revision		Revision as of 07:25, 16 July 2018
Line 1:		Line 1:
	I think there's several different confusions that arise from multivariable derivative notation:		I think there's several different confusions that arise from multivariable derivative notation:

	* 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 a 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 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.

IssaRice: Created page with "I think there's several different confusions that arise from multivariable derivative notation: * The thing where $\frac{\partial w}{\partial t}$ can mean two diff..."

2018-07-16T07:24:30Z

Created page with "I think there's several different confusions that arise from multivariable derivative notation: * The thing where <math>\frac{\partial w}{\partial t}</math> can mean two diff..."

New page

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

* 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 a 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.

==See also==

* [[Summary table of multivariable derivatives]]