I am tasked with solving a system of equations using Newton-Raphson method and the first step is to find the Jacobian Matrix. Well, I'm having a really hard time trying to figure out how to do so as most examples do not give me the information as the problem is given to me now. Furthermore, I think fact we have two types of variables of form w_x and c_x just really trips me up.
Can somebody potentially help me get me on track?
Edit: Here is the system of equations, thought I linked it at first. The formatting had it hidden. Woops. https://i.stack.imgur.com/rnGn4.png
Find the parameters of the three point Gauss-Legendre formula $$\int_0^1f(x)dx = w_1f(c_1)+w_2f(c_2)+w_3f(c_3)$$ which is correct for polynomials up to degree 5 by solving $$F_k(w,c)=w_1c_1^k+w_2c_2^k+w_3c_3^k=\frac1{k+1}\text{ for }k=0,1,2,3,4,5.$$
You get $$ \frac{\partial F_k}{∂w_m}=c_m^{\;k}\ \text{ and }\ \frac{\partial F_0}{∂c_m}=0,\;\frac{\partial F_k}{∂c_m}=kw_mc^{k-1} $$ which covers all the entries of the Jacobian.