I have an equation I'm trying to solve and believe I am missing a term. I'm not sure if it is called covariance or not, but I'm having trouble finding any information on it which makes me think it is called something else. I've flipped through a couple of PDE textbooks and searched on mathworld but I may just not be looking for the right thing. If anyone could point me in the right direction or give me a reference I would appreciate it.
Anyways the equation is simple enough.
Q = sum(Ri), Ri is a value for a pathway and there are i pathways
for each pathway: Ri = F(xi)*(1-xi)*C where C is just come constant
so for pathway 1, R1 = F(x1)*(1-x1)*C
now I am trying to solve for dQ/dxj (where j can have all the same values of i) and I believe I have this part right as it is just
dQ/dxj = sum(dRi/dxj)
R1 obviously changes with x1, but other Ri's can also vary with changes in x1, so basically dRj/dxi = 0 isn't necessarily true for i <> j (<> is does not equal)
Here is the part where I believe I am missing something, because when I plug in test cases (with simple equations for dF(xi)) I don't get the answer I should:
dR1/dx1 = dF(x1)/dx1*(1-x1)*C + d(1-x1)/dx1* F(x1)*C , where d(1-x1)/dx1 = -1
similarly....
dR2/dx1 = dF(x2)/dx1*(1-x2)*C + d(1-x2)/dx1*F(x2)*C , neither term goes away because x2's value may change if x1's changes
On the dR1/dx1 term (and possibly other terms like dR2/dx1) I feel like I am missing a term .... something like (dR/dF)(dF/dx1)F(x1).
Hopefully that makes sense and someone can give me some help. Thanks in advance.
Anyways the equation is simple enough.
Q = sum(Ri), Ri is a value for a pathway and there are i pathways
for each pathway: Ri = F(xi)*(1-xi)*C where C is just come constant
so for pathway 1, R1 = F(x1)*(1-x1)*C
now I am trying to solve for dQ/dxj (where j can have all the same values of i) and I believe I have this part right as it is just
dQ/dxj = sum(dRi/dxj)
R1 obviously changes with x1, but other Ri's can also vary with changes in x1, so basically dRj/dxi = 0 isn't necessarily true for i <> j (<> is does not equal)
Here is the part where I believe I am missing something, because when I plug in test cases (with simple equations for dF(xi)) I don't get the answer I should:
dR1/dx1 = dF(x1)/dx1*(1-x1)*C + d(1-x1)/dx1* F(x1)*C , where d(1-x1)/dx1 = -1
similarly....
dR2/dx1 = dF(x2)/dx1*(1-x2)*C + d(1-x2)/dx1*F(x2)*C , neither term goes away because x2's value may change if x1's changes
On the dR1/dx1 term (and possibly other terms like dR2/dx1) I feel like I am missing a term .... something like (dR/dF)(dF/dx1)F(x1).
Hopefully that makes sense and someone can give me some help. Thanks in advance.