Functional Dependence Jacobian – Problem 1 – Jacobian – Engineering Mathematics 1


Hi Friends,So here in this video we are gonna to learn a new concept called as functional dependence now let’s say you is function of x and y and v is also function of x and y and if we want to understand the relationship between u and v that is whether you and we are dependent on each other or they are independent then such scenario is called as functional dependency and to find out the functional dependence we use one condition that condition is low of uv baidu of xy that is Jacobian of uv with respect to xy now if this condition is equal to 0 or if this dacovian or if this Jacobian is equal to 0 then we say yes you and we are functionally dependent on each other but if we are not getting 0 then we say no they are not functionally dependent and if they are functionally dependent then we can definitely find out the relationship between U and V so let us understand this by using one example so similarly we can see the functional dependency between more variables also now let’s say there is u v and w where u is function of x y z v is also function of x y z and w is also function of XYZ now if i want to understand the functional dependence between UV and w then i can use the condition as Jacobian of u v w with respect to x y and z and if this condition is equal to 0 then i will say yes there is function dependency between you V and W that is we can find out that one variable is a function of other variable so here we can find out that U is function of V or V is function of U so let us understand this concept by using one example so the example is if u is equal to X upon Y and V is equal to Y upon X then verify whether you we are functionally dependent so to check whether they are functionally dependent or not we have to find out the Jacobian of U V with respect to XY and it is given by the formula dou u by dou X dou u by dou Y dou V by dou X and dou V by dou Y so your too far to will get dou u by dou X as 1 upon Y dou u by dou Y as minus X upon y square dou V by dou X as minus y upon X square and dou V by dou Y as 1 upon X now if we will multiply the terms then we will get 1 upon Y X minus X upon Y square into y upon X square so here if will cancel the terms x at x + y + y then again we’ll get 1 upon Y X minus 1 upon Y X which is equal to 0 so as I explained you that if we are getting the Jacobian equal to 0 then we say yes there is a dependency between the two functions U and V so yes here the dependency exists and let’s find the dependency between you and now there is no fixed method to find out the dependency between U and V but by using the values of U and V but by making use of the algebra and sub mathematical simplification we can definitely find out the relationship between U and V so here if you observe then V is given as Y by X now if well write this as 1 upon X by y then you can say that X by Y is nothing but U so it clearly indicates that V is equal to 1 by U or I will say u V is equal to 1 and this is nothing but a functional dependence between U and V thank you