Wikipedia:Reference desk/Archives/Mathematics/2017 April 24

= April 24 =

derivative of constrained function is sum of unconstrained partials?
Hello, while studying neural networks I came upon what was described by the lecturer as a "math trick" to solve a particular type of optimization problem using gradient descent. Basically, when optimizing a neural net that requires two parameters to be equal, you can replace the partial derivative for each constrained parameter by the sum of the partials with respect to each constrained parameter. So if you have $$f(x_{1},x_{2})$$, then the derivative of $$f(x,x)$$ with respect to $$x$$ is the same as the sum of the partial derivatives of $$f(x_{1},x_{2})$$ with respect to $$x_{1}$$ and $$x_{2}$$. So far I have not been able to find a counter example, but I also do not know how to prove it. If anyone has pointers, clues, or proofs please help! Sorry if its obvious, and thanks! Brusegadi (talk) 01:09, 24 April 2017 (UTC)
 * It's a particular case of the multivariable chain rule; it's easier to see what's going on from the more general form $$\frac{d}{dt}(f(x(t),y(t))) = \frac{\partial f}{\partial x}\frac{dx}{dt}+ \frac{\partial f}{\partial y}\frac{dy}{dt}$$. --JBL (talk) 01:52, 24 April 2017 (UTC)
 * Thank you so much! Brusegadi (talk) 02:08, 24 April 2017 (UTC)

How about a slightly different situation involving total derivative of a function G of constrained independent variables xi with constant sum, for instance 1. Can the partial derivative with respect to xi exist by keeping the other xj constant even if only the sum of xi is constant, not every xi? Is this due to fact that dxi is around zero? Thanks.--82.137.9.214 (talk) 23:49, 24 April 2017 (UTC)


 * The partial derivative is a feature of the function, not of the combination of function and constraint. So yes, the partial derivative can exist regardless of what context the function will be used in. You can take the total differential of G, which in the n=2 case is


 * $$dG=\frac{\partial G}{\partial x_1}dx_1 +\frac{\partial G}{\partial x_2}dx_2.$$


 * Then if you impose $$x_1+x_2=k$$ and hence $$dx_1+dx_2=0$$ hence $$dx_2=-dx_1,$$ you get


 * $$dG=\frac{\partial G}{\partial x_1}dx_1+\frac{\partial G}{\partial x_2}\times (-dx_1)=\left(\frac{\partial G}{\partial x_1}-\frac{\partial G}{\partial x_2}\right)dx_1.$$
 * Loraof (talk) 16:16, 25 April 2017 (UTC)
 * But if n>2, not enough information has been provided—for the last step, we need to know how the offset of $$dx_1$$ is distributed among $$dx_2, \, dx_3,$$ etc. Loraof (talk) 16:21, 25 April 2017 (UTC)


 * But can x2 be held constant when taking the partial derivative in respect to x1 as requested by the definition of the partial derivative? Similar question for the other independent variable x1 to be held constant when taking the partial derivative in respect to x2. Isn't the situation a bit stretched because strictly one independent variable cannot be made constant when the partial derivative is taken in respect to the other indepedent variable in such cases where only the sum of independent variables can be constant? Or it is about quasi-constancy of independent variables which is satisfactory in this situation?--82.137.14.76 (talk) 00:17, 26 April 2017 (UTC)


 * The original question does not ask for a partial derivative so it's hard to see the relevance of this query. Maybe you should ask a new question instead, where you can make your hypotheses clear.  --JBL (talk) 01:07, 26 April 2017 (UTC)

Johnson's SU-distribution
Is Johnson's SU-distribution and "shepherd's crook" and Johnson Curve all the same thing? Thanks. Anna Frodesiak (talk) 06:25, 24 April 2017 (UTC)

I want to know because of and  that I did. Anna Frodesiak (talk) 10:24, 24 April 2017 (UTC)

Face value numbers
When we count, for example, coins or banknotes by their face value rather than actual quantity expressed by natural numbers, are those still natural numbers or some other kind? Thanks.--212.180.235.46 (talk) 16:59, 24 April 2017 (UTC)


 * Natural numbers don't include decimals, and most currencies have a sub-denomination (like cents for dollars), which makes it a decimal number. Yen is one that doesn't, so, for that case, I suppose you could use natural numbers for prices, in most cases (with exceptions for buying in quantity, where they might break the price down by tenths of a yen, etc.) StuRat (talk) 17:38, 24 April 2017 (UTC)
 * Perhaps not everyone will agree, but I would consider natural numbers to be dimensionless. Currency is given units of whatever denomination it is, so $5 is not the same as 5 even though there's no decimals being used. --RDBury (talk) 05:21, 25 April 2017 (UTC)
 * Yep as we all know time is money so definitely not dimensionless :) Dmcq (talk) 10:25, 28 April 2017 (UTC)