Superposition principle for functions with multiple variables

[shivaniits]

need help understanding "Superposition Principle"..!

hello everyone..
if we have a function y=f(x) then in-order to prove linearity we try to justify according to superposition principle as :
let x1 and x2 be two inputs then f(x1+x2)=f(x1)+f(x2)
please correct me if i am wrong upto here..
now what if we have more than two variables..let's say we have three variables two independent and one dependent
now we have function z=g(x,y)..now in-order to prove linearity for function involving more than two variables can i say this that for g(x,y) to be linear g(x1+x2,y1+y2)=g(x1,y1)+g(x2,y2)..??
and if this isn't the correct way for proving linearity in functions involving more than two variables..then please justify the correct method along with examples.

 

[Office_Shredder]

If you have a function of multiple variables, you typically want what's called multilinearity - that the function is linear in each variable. For example, g(x₁+x₂,y) = g(x₁,y) + g(x₂,y) and g(x,y₁+y₂) = g(x,y₁) + g(x,y₂). In this case you should be able to figure out what g(x₁+x₂,y₁+y₂) is equal to (it's not what you wrote).

What your g is satisfying is that it is linear in the single input (x,y), which may be what you're looking for.

 

[shivaniits]

If you have a function of multiple variables, you typically want what's called multilinearity - that the function is linear in each variable. For example, g(x₁+x₂,y) = g(x₁,y) + g(x₂,y) and g(x,y₁+y₂) = g(x,y₁) + g(x,y₂). In this case you should be able to figure out what g(x₁+x₂,y₁+y₂) is equal to (it's not what you wrote).

.

hello..
i am understanding a little bit now but if i have to say linearity of functions involving more than two variables then i can't always refer to superposition principle or is there any superposition involving more than two variables..!
and if i have to consider the linearity among differential equations as in linear differential equation then what would be method to justify this..can this multi-linearity principle also holds for differential equation..?

 

[Office_Shredder]

I don't understand what your question is, can you give a specific example?

 

[shivaniits]

i mean we mention differential equation to be linear..as linear differential equation..

and for the example if we take this LDE dy/dt+(x^2)*y=0
it is LDE as for the dependent variable and its deriavtive is in first degree and are not multipled together..please let me know i am wrong..!
then can we apply the superposition principle on this one to justify its linearity
for this one if i have y1 for x1 and y2 for x2 then if i input x1+x2 will i get y as y1+y2..?? acc. to superposition principle..can i really justify its linearity with superposition principle of f(x1+x2)=f(x1)+f(x2)...?