Question regarding minimization and orthogonality
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Let $f:mathbbRrightarrowmathbbR^n$ be a differentiable mapping with $f'(t)neqtextbf0$ for all $tinmathbbR$, and let $textbfpinmathbbR^n$ be a point not in $f(mathbbR)$.
(a) Show there is a point $textbfq=f(t)$ on the curve $f(mathbbR)$ $textitclosest$ to $textbfp$.
(b) Show that the vector $r=(textbfp-textbfq)$ is orthogonal to the curve at $textbfq$. $textbfHint: $Consider the function $tmapsto|textbfp-f(t)|$ and its derivative.
I'm not sure how to approach this. It appears to be a least squares problem. The hint on (b) seems to imply that the derivative of the distance formula is $0$ at $textbfq$, but I'm not sure how that helps.
real-analysis multivariable-calculus derivatives
asked Aug 30 at 1:21
Atsina
721115
add a comment |Â
up vote
0
down vote
favorite
Let $f:mathbbRrightarrowmathbbR^n$ be a differentiable mapping with $f'(t)neqtextbf0$ for all $tinmathbbR$, and let $textbfpinmathbbR^n$ be a point not in $f(mathbbR)$.
(a) Show there is a point $textbfq=f(t)$ on the curve $f(mathbbR)$ $textitclosest$ to $textbfp$.
(b) Show that the vector $r=(textbfp-textbfq)$ is orthogonal to the curve at $textbfq$. $textbfHint: $Consider the function $tmapsto|textbfp-f(t)|$ and its derivative.
I'm not sure how to approach this. It appears to be a least squares problem. The hint on (b) seems to imply that the derivative of the distance formula is $0$ at $textbfq$, but I'm not sure how that helps.
real-analysis multivariable-calculus derivatives
asked Aug 30 at 1:21
Atsina
721115
For part (a), consider the function $t mapsto operatornamedist(f(t), p)$. This is an $mathbbR$-valued function of a single variable. Is it continuously differentiable? If so, can you use any results about $mathbbR$-valued functions of a single variable to prove the result? Something about finding minima and maxima on compact sets? For part (b), consider Fermat's theorem.
– Xander Henderson
Aug 30 at 1:32
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $f:mathbbRrightarrowmathbbR^n$ be a differentiable mapping with $f'(t)neqtextbf0$ for all $tinmathbbR$, and let $textbfpinmathbbR^n$ be a point not in $f(mathbbR)$.
(a) Show there is a point $textbfq=f(t)$ on the curve $f(mathbbR)$ $textitclosest$ to $textbfp$.
(b) Show that the vector $r=(textbfp-textbfq)$ is orthogonal to the curve at $textbfq$. $textbfHint: $Consider the function $tmapsto|textbfp-f(t)|$ and its derivative.
I'm not sure how to approach this. It appears to be a least squares problem. The hint on (b) seems to imply that the derivative of the distance formula is $0$ at $textbfq$, but I'm not sure how that helps.
real-analysis multivariable-calculus derivatives
asked Aug 30 at 1:21
Atsina
721115
Let $f:mathbbRrightarrowmathbbR^n$ be a differentiable mapping with $f'(t)neqtextbf0$ for all $tinmathbbR$, and let $textbfpinmathbbR^n$ be a point not in $f(mathbbR)$.
(a) Show there is a point $textbfq=f(t)$ on the curve $f(mathbbR)$ $textitclosest$ to $textbfp$.
(b) Show that the vector $r=(textbfp-textbfq)$ is orthogonal to the curve at $textbfq$. $textbfHint: $Consider the function $tmapsto|textbfp-f(t)|$ and its derivative.
I'm not sure how to approach this. It appears to be a least squares problem. The hint on (b) seems to imply that the derivative of the distance formula is $0$ at $textbfq$, but I'm not sure how that helps.
real-analysis multivariable-calculus derivatives
real-analysis multivariable-calculus derivatives
asked Aug 30 at 1:21
Atsina
721115
asked Aug 30 at 1:21
Atsina
721115
asked Aug 30 at 1:21
Atsina
721115
asked Aug 30 at 1:21
Atsina
721115
asked Aug 30 at 1:21
Atsina
721115
721115
For part (a), consider the function $t mapsto operatornamedist(f(t), p)$. This is an $mathbbR$-valued function of a single variable. Is it continuously differentiable? If so, can you use any results about $mathbbR$-valued functions of a single variable to prove the result? Something about finding minima and maxima on compact sets? For part (b), consider Fermat's theorem.
– Xander Henderson
Aug 30 at 1:32
add a comment |Â
For part (a), consider the function $t mapsto operatornamedist(f(t), p)$. This is an $mathbbR$-valued function of a single variable. Is it continuously differentiable? If so, can you use any results about $mathbbR$-valued functions of a single variable to prove the result? Something about finding minima and maxima on compact sets? For part (b), consider Fermat's theorem.
– Xander Henderson
Aug 30 at 1:32
For part (a), consider the function $t mapsto operatornamedist(f(t), p)$. This is an $mathbbR$-valued function of a single variable. Is it continuously differentiable? If so, can you use any results about $mathbbR$-valued functions of a single variable to prove the result? Something about finding minima and maxima on compact sets? For part (b), consider Fermat's theorem.
– Xander Henderson
Aug 30 at 1:32
For part (a), consider the function $t mapsto operatornamedist(f(t), p)$. This is an $mathbbR$-valued function of a single variable. Is it continuously differentiable? If so, can you use any results about $mathbbR$-valued functions of a single variable to prove the result? Something about finding minima and maxima on compact sets? For part (b), consider Fermat's theorem.
– Xander Henderson
Aug 30 at 1:32
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
This is false. Let $$ p = (0,0) $$ and
$$ f(t) = left( ; e^-t cos t , ; e^-t sin t ; right) $$
answered Aug 30 at 2:16
Will Jagy
97.9k595196
Could you elaborate on why there isn't a value of $t$ for which $f(t)$ is closest to $p$?
– Atsina
Aug 30 at 6:05
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
This is false. Let $$ p = (0,0) $$ and
$$ f(t) = left( ; e^-t cos t , ; e^-t sin t ; right) $$
answered Aug 30 at 2:16
Will Jagy
97.9k595196
Could you elaborate on why there isn't a value of $t$ for which $f(t)$ is closest to $p$?
– Atsina
Aug 30 at 6:05
add a comment |Â
up vote
2
down vote
accepted
This is false. Let $$ p = (0,0) $$ and
$$ f(t) = left( ; e^-t cos t , ; e^-t sin t ; right) $$
answered Aug 30 at 2:16
Will Jagy
97.9k595196
Could you elaborate on why there isn't a value of $t$ for which $f(t)$ is closest to $p$?
– Atsina
Aug 30 at 6:05
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
This is false. Let $$ p = (0,0) $$ and
$$ f(t) = left( ; e^-t cos t , ; e^-t sin t ; right) $$
answered Aug 30 at 2:16
Will Jagy
97.9k595196
This is false. Let $$ p = (0,0) $$ and
$$ f(t) = left( ; e^-t cos t , ; e^-t sin t ; right) $$
answered Aug 30 at 2:16
Will Jagy
97.9k595196
answered Aug 30 at 2:16
Will Jagy
97.9k595196
answered Aug 30 at 2:16
Will Jagy
97.9k595196
answered Aug 30 at 2:16
Will Jagy
97.9k595196
97.9k595196
Could you elaborate on why there isn't a value of $t$ for which $f(t)$ is closest to $p$?
– Atsina
Aug 30 at 6:05
add a comment |Â
Could you elaborate on why there isn't a value of $t$ for which $f(t)$ is closest to $p$?
– Atsina
Aug 30 at 6:05
Could you elaborate on why there isn't a value of $t$ for which $f(t)$ is closest to $p$?
– Atsina
Aug 30 at 6:05
Could you elaborate on why there isn't a value of $t$ for which $f(t)$ is closest to $p$?
– Atsina
Aug 30 at 6:05
add a comment |Â
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For part (a), consider the function $t mapsto operatornamedist(f(t), p)$. This is an $mathbbR$-valued function of a single variable. Is it continuously differentiable? If so, can you use any results about $mathbbR$-valued functions of a single variable to prove the result? Something about finding minima and maxima on compact sets? For part (b), consider Fermat's theorem.
– Xander Henderson
Aug 30 at 1:32