I am in the middle of self studying calculus by reading a textbook, and have just finished the section of lagrange multiplier. I've noticed the examples given in the book always consider a very specific kind of condition as follow:
$$\text{consider a $(m+n)$ variable function $f(x_1,x_2,...,x_m,...,x_{m+n})$}\\\text{with $n$ constraints $\varphi_i(x_1,x_2,...,x_m,...,x_{m+n})=0$, where $i=1,2,...,n$}$$
In the above case, each constraint $\varphi_i$ contains the same $(m+n)$ variables as $f$, and because there are $(m+n)$ variables with $n$ constraints, the degree of freedom (or number of independent variables) is just $m$
This makes me wondering:
- If for all $\varphi_i$, they contain different numbers of variables, less than $(m+n)$, what the degree of freedoms would be in this case ?
- For example, suppose I have a $5$ variables function $$f(x_1,x_2,x_3,x_4,x_5)$$ and $3$ constraints $$\varphi_i(x_1,x_2,x_5)=0\\\varphi_i(x_3,x_4)=0\\\varphi_i(x_4,x_5)=0$$ then what is the dergee of freedom in this case, assuming the existence of all implicit function here ?
- If for all $\varphi_i$, they contain different numbers of variables, and some variables are independent while different from the original $(m+n)$ variables, what the degree of freedoms would be in this case ?
- For example, suppose I have a $5$ variables function $$f(x_1,x_2,x_3,x_4,x_5)$$ and $3$ constraints $$\varphi_i(x_1,x_2,x_5,t_1)=0\\\varphi_i(x_3,x_4,t_2)=0\\\varphi_i(x_4,x_5,t_3)=0$$ then what is the dergee of freedom in this case, assuming the existence of all implicit function here ?