Directional derivative - need some help with $D_{X_u}X_v$
Let's say we have a surface in $mathbb{R}^3$ which is parameterized with only two variables $u$ and $v$ so it can be defined as $X(u,v)=(f_1 (u,v),f_2(u,v),f_3(u,v))$. We calculate the tangent vectors $$X_u=left (frac{partial f_1}{partial u},frac{partial f_2}{partial u} ,frac{partial f_3}{partial u} right )$$ and $$X_v=left (frac{partial f_1}{partial v},frac{partial f_2}{partial v} ,frac{partial f_3}{partial v} right ),$$ which are objects in $mathbb{R}^3$. Now I want to calculate such derivative $D_{X_u} X_v$ and this should be according to my understanding $$D_{X_u} X_v= left(sum_i frac{partial f_i}{partial u} cdot frac{partial }{partial x_i} frac{partial f_1}{partial v},sum_i frac{partial f_i}{partial u} cdot frac{partial }{partial x_i} frac{partial f_2}{partial v},sum_i frac{partial f_i}{partial u} cdot frac{partial }{partial x_i} frac{partial f_3}{partial v}right).$$ Now, according to some lecture notes this expression should be equal to $$frac{partial X}{partial u partial v}=left (frac{partial f_1}{partial upartial v},frac{partial f_2}{partial upartial v} ,frac{partial f_3}{partial upartial v} right )$$ I don't see a connection between those equations.
derivatives differential-geometry
asked Jan 4 at 17:22
HexHex
305
add a comment |
Let's say we have a surface in $mathbb{R}^3$ which is parameterized with only two variables $u$ and $v$ so it can be defined as $X(u,v)=(f_1 (u,v),f_2(u,v),f_3(u,v))$. We calculate the tangent vectors $$X_u=left (frac{partial f_1}{partial u},frac{partial f_2}{partial u} ,frac{partial f_3}{partial u} right )$$ and $$X_v=left (frac{partial f_1}{partial v},frac{partial f_2}{partial v} ,frac{partial f_3}{partial v} right ),$$ which are objects in $mathbb{R}^3$. Now I want to calculate such derivative $D_{X_u} X_v$ and this should be according to my understanding $$D_{X_u} X_v= left(sum_i frac{partial f_i}{partial u} cdot frac{partial }{partial x_i} frac{partial f_1}{partial v},sum_i frac{partial f_i}{partial u} cdot frac{partial }{partial x_i} frac{partial f_2}{partial v},sum_i frac{partial f_i}{partial u} cdot frac{partial }{partial x_i} frac{partial f_3}{partial v}right).$$ Now, according to some lecture notes this expression should be equal to $$frac{partial X}{partial u partial v}=left (frac{partial f_1}{partial upartial v},frac{partial f_2}{partial upartial v} ,frac{partial f_3}{partial upartial v} right )$$ I don't see a connection between those equations.
derivatives differential-geometry
asked Jan 4 at 17:22
HexHex
305
Where did $partial/partial x_i$ come from? What are $x_i$? Note that $f_i$ are functions of $(u,v)$ only.
– Ted Shifrin
Jan 4 at 19:16
yeah, but what probably hinders my imagination is that I treat those there components as $x,y,z$ vector
– Hex
Jan 4 at 19:25
So, start with an easier example. Take one scalar function $f$ on your surface. What is $D_{X_u} f$? Are you given $f$ as a function on $Bbb R^3$ or only on the parametrized surface, hence as a function of $u$ and $v$?
– Ted Shifrin
Jan 4 at 20:44
add a comment |
Let's say we have a surface in $mathbb{R}^3$ which is parameterized with only two variables $u$ and $v$ so it can be defined as $X(u,v)=(f_1 (u,v),f_2(u,v),f_3(u,v))$. We calculate the tangent vectors $$X_u=left (frac{partial f_1}{partial u},frac{partial f_2}{partial u} ,frac{partial f_3}{partial u} right )$$ and $$X_v=left (frac{partial f_1}{partial v},frac{partial f_2}{partial v} ,frac{partial f_3}{partial v} right ),$$ which are objects in $mathbb{R}^3$. Now I want to calculate such derivative $D_{X_u} X_v$ and this should be according to my understanding $$D_{X_u} X_v= left(sum_i frac{partial f_i}{partial u} cdot frac{partial }{partial x_i} frac{partial f_1}{partial v},sum_i frac{partial f_i}{partial u} cdot frac{partial }{partial x_i} frac{partial f_2}{partial v},sum_i frac{partial f_i}{partial u} cdot frac{partial }{partial x_i} frac{partial f_3}{partial v}right).$$ Now, according to some lecture notes this expression should be equal to $$frac{partial X}{partial u partial v}=left (frac{partial f_1}{partial upartial v},frac{partial f_2}{partial upartial v} ,frac{partial f_3}{partial upartial v} right )$$ I don't see a connection between those equations.
derivatives differential-geometry
asked Jan 4 at 17:22
HexHex
305
Let's say we have a surface in $mathbb{R}^3$ which is parameterized with only two variables $u$ and $v$ so it can be defined as $X(u,v)=(f_1 (u,v),f_2(u,v),f_3(u,v))$. We calculate the tangent vectors $$X_u=left (frac{partial f_1}{partial u},frac{partial f_2}{partial u} ,frac{partial f_3}{partial u} right )$$ and $$X_v=left (frac{partial f_1}{partial v},frac{partial f_2}{partial v} ,frac{partial f_3}{partial v} right ),$$ which are objects in $mathbb{R}^3$. Now I want to calculate such derivative $D_{X_u} X_v$ and this should be according to my understanding $$D_{X_u} X_v= left(sum_i frac{partial f_i}{partial u} cdot frac{partial }{partial x_i} frac{partial f_1}{partial v},sum_i frac{partial f_i}{partial u} cdot frac{partial }{partial x_i} frac{partial f_2}{partial v},sum_i frac{partial f_i}{partial u} cdot frac{partial }{partial x_i} frac{partial f_3}{partial v}right).$$ Now, according to some lecture notes this expression should be equal to $$frac{partial X}{partial u partial v}=left (frac{partial f_1}{partial upartial v},frac{partial f_2}{partial upartial v} ,frac{partial f_3}{partial upartial v} right )$$ I don't see a connection between those equations.
derivatives differential-geometry
derivatives differential-geometry
asked Jan 4 at 17:22
HexHex
305
asked Jan 4 at 17:22
HexHex
305
asked Jan 4 at 17:22
HexHex
305
asked Jan 4 at 17:22
HexHex
305
asked Jan 4 at 17:22
HexHex
305
305
Where did $partial/partial x_i$ come from? What are $x_i$? Note that $f_i$ are functions of $(u,v)$ only.
– Ted Shifrin
Jan 4 at 19:16
yeah, but what probably hinders my imagination is that I treat those there components as $x,y,z$ vector
– Hex
Jan 4 at 19:25
So, start with an easier example. Take one scalar function $f$ on your surface. What is $D_{X_u} f$? Are you given $f$ as a function on $Bbb R^3$ or only on the parametrized surface, hence as a function of $u$ and $v$?
– Ted Shifrin
Jan 4 at 20:44
add a comment |
Where did $partial/partial x_i$ come from? What are $x_i$? Note that $f_i$ are functions of $(u,v)$ only.
– Ted Shifrin
Jan 4 at 19:16
yeah, but what probably hinders my imagination is that I treat those there components as $x,y,z$ vector
– Hex
Jan 4 at 19:25
So, start with an easier example. Take one scalar function $f$ on your surface. What is $D_{X_u} f$? Are you given $f$ as a function on $Bbb R^3$ or only on the parametrized surface, hence as a function of $u$ and $v$?
– Ted Shifrin
Jan 4 at 20:44
Where did $partial/partial x_i$ come from? What are $x_i$? Note that $f_i$ are functions of $(u,v)$ only.
– Ted Shifrin
Jan 4 at 19:16
Where did $partial/partial x_i$ come from? What are $x_i$? Note that $f_i$ are functions of $(u,v)$ only.
– Ted Shifrin
Jan 4 at 19:16
yeah, but what probably hinders my imagination is that I treat those there components as $x,y,z$ vector
– Hex
Jan 4 at 19:25
yeah, but what probably hinders my imagination is that I treat those there components as $x,y,z$ vector
– Hex
Jan 4 at 19:25
So, start with an easier example. Take one scalar function $f$ on your surface. What is $D_{X_u} f$? Are you given $f$ as a function on $Bbb R^3$ or only on the parametrized surface, hence as a function of $u$ and $v$?
– Ted Shifrin
Jan 4 at 20:44
So, start with an easier example. Take one scalar function $f$ on your surface. What is $D_{X_u} f$? Are you given $f$ as a function on $Bbb R^3$ or only on the parametrized surface, hence as a function of $u$ and $v$?
– Ted Shifrin
Jan 4 at 20:44
add a comment |
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Where did $partial/partial x_i$ come from? What are $x_i$? Note that $f_i$ are functions of $(u,v)$ only.
– Ted Shifrin
Jan 4 at 19:16
yeah, but what probably hinders my imagination is that I treat those there components as $x,y,z$ vector
– Hex
Jan 4 at 19:25
So, start with an easier example. Take one scalar function $f$ on your surface. What is $D_{X_u} f$? Are you given $f$ as a function on $Bbb R^3$ or only on the parametrized surface, hence as a function of $u$ and $v$?
– Ted Shifrin
Jan 4 at 20:44