Is this list of conditions enough for the existence for the envelope for the given family?
Suppose $f(x,y,c)=0$ be represented by the family of surfaces. When does the envelop exist for the family of surfaces?
My attempt:-
For finding the envelope, I need to consider two equations $f(x,y,c)=0$ and $f_c(x,y,c)=0.$ For finding the envelope we need to find a relation between $x$ and $y$. Right?. For that, I apply implicit function theorem. For applying implicit function theorem I need the function $F=(f,f_c)$ must be first continuously differentiable. If jacobian $frac{partial(f,f_c)}{partial(x,y)} neq 0$, we can write $x=x(c)$ and $y=y(c)$. If either $x$ or $y$ invertible. We can write $y=y(x)$.
So I found two conditions:- 1. $F(x,y,c)=(f,f_a)$ must be continuously differentiable up to first order.
jacobian $frac{partial(f,f_c)}{partial(x,y)} neq 0$
Either $x$ or $y$ functions are invertible.
Is this list of conditions enough for the existence of the envelope for the given family? Or can we find the envelope with fewer conditions than this?
Please help me.
implicit-function-theorem envelope
asked yesterday
Unknown x
2,50011026
add a comment |
Suppose $f(x,y,c)=0$ be represented by the family of surfaces. When does the envelop exist for the family of surfaces?
My attempt:-
For finding the envelope, I need to consider two equations $f(x,y,c)=0$ and $f_c(x,y,c)=0.$ For finding the envelope we need to find a relation between $x$ and $y$. Right?. For that, I apply implicit function theorem. For applying implicit function theorem I need the function $F=(f,f_c)$ must be first continuously differentiable. If jacobian $frac{partial(f,f_c)}{partial(x,y)} neq 0$, we can write $x=x(c)$ and $y=y(c)$. If either $x$ or $y$ invertible. We can write $y=y(x)$.
So I found two conditions:- 1. $F(x,y,c)=(f,f_a)$ must be continuously differentiable up to first order.
jacobian $frac{partial(f,f_c)}{partial(x,y)} neq 0$
Either $x$ or $y$ functions are invertible.
Is this list of conditions enough for the existence of the envelope for the given family? Or can we find the envelope with fewer conditions than this?
Please help me.
implicit-function-theorem envelope
asked yesterday
Unknown x
2,50011026
add a comment |
Suppose $f(x,y,c)=0$ be represented by the family of surfaces. When does the envelop exist for the family of surfaces?
My attempt:-
For finding the envelope, I need to consider two equations $f(x,y,c)=0$ and $f_c(x,y,c)=0.$ For finding the envelope we need to find a relation between $x$ and $y$. Right?. For that, I apply implicit function theorem. For applying implicit function theorem I need the function $F=(f,f_c)$ must be first continuously differentiable. If jacobian $frac{partial(f,f_c)}{partial(x,y)} neq 0$, we can write $x=x(c)$ and $y=y(c)$. If either $x$ or $y$ invertible. We can write $y=y(x)$.
So I found two conditions:- 1. $F(x,y,c)=(f,f_a)$ must be continuously differentiable up to first order.
jacobian $frac{partial(f,f_c)}{partial(x,y)} neq 0$
Either $x$ or $y$ functions are invertible.
Is this list of conditions enough for the existence of the envelope for the given family? Or can we find the envelope with fewer conditions than this?
Please help me.
implicit-function-theorem envelope
asked yesterday
Unknown x
2,50011026
Suppose $f(x,y,c)=0$ be represented by the family of surfaces. When does the envelop exist for the family of surfaces?
My attempt:-
For finding the envelope, I need to consider two equations $f(x,y,c)=0$ and $f_c(x,y,c)=0.$ For finding the envelope we need to find a relation between $x$ and $y$. Right?. For that, I apply implicit function theorem. For applying implicit function theorem I need the function $F=(f,f_c)$ must be first continuously differentiable. If jacobian $frac{partial(f,f_c)}{partial(x,y)} neq 0$, we can write $x=x(c)$ and $y=y(c)$. If either $x$ or $y$ invertible. We can write $y=y(x)$.
So I found two conditions:- 1. $F(x,y,c)=(f,f_a)$ must be continuously differentiable up to first order.
jacobian $frac{partial(f,f_c)}{partial(x,y)} neq 0$
Either $x$ or $y$ functions are invertible.
Is this list of conditions enough for the existence of the envelope for the given family? Or can we find the envelope with fewer conditions than this?
Please help me.
implicit-function-theorem envelope
implicit-function-theorem envelope
asked yesterday
Unknown x
2,50011026
asked yesterday
Unknown x
2,50011026
asked yesterday
Unknown x
2,50011026
asked yesterday
Unknown x
2,50011026
asked yesterday
Unknown x
2,50011026
2,50011026
add a comment |
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