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Is there a notation for the operator $(fracpartialpartial u_x,fracpartialpartial u_y)^T$?
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I know that the operator $nabla$ denotes
$$
nabla = beginpmatrix fracpartialpartial x \ fracpartialpartial y endpmatrix
$$
Is there some kind of similar notation to denote
$$
A = beginpmatrix fracpartialpartial u_x \ fracpartialpartial u_y endpmatrix
$$
where $u_x = fracpartialpartial x u$, likewise $u_y$. Here $u$ denotes some differentiable function in $Omega subset mathbbR^2$.
calculus notation vector-analysis
asked Aug 1 at 9:22
user8469759
1,4271513
add a comment |Â
up vote
0
down vote
favorite
I know that the operator $nabla$ denotes
$$
nabla = beginpmatrix fracpartialpartial x \ fracpartialpartial y endpmatrix
$$
Is there some kind of similar notation to denote
$$
A = beginpmatrix fracpartialpartial u_x \ fracpartialpartial u_y endpmatrix
$$
where $u_x = fracpartialpartial x u$, likewise $u_y$. Here $u$ denotes some differentiable function in $Omega subset mathbbR^2$.
calculus notation vector-analysis
asked Aug 1 at 9:22
user8469759
1,4271513
Like the gradient on the fibre of the tangent bundle?
– Troy Woo
Aug 1 at 9:28
1
I'm not exactly sure what you're asking, but the directional derivative along a vector field $mathbf v = (v_x, v_y)$ is usually denoted $(mathbf v cdot nabla)$, which means $v_x fracpartialpartial x + v_y fracpartialpartial y$. This is used to define the material derivative in continuum mechanics.
– Rahul
Aug 1 at 9:30
@Rahul, No, I'm looking for a shorthand for $$A = fracpartialpartial x fracpartialpartial v_x + fracpartialpartial y fracpartialpartial v_y$$
– user8469759
Aug 1 at 9:35
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I know that the operator $nabla$ denotes
$$
nabla = beginpmatrix fracpartialpartial x \ fracpartialpartial y endpmatrix
$$
Is there some kind of similar notation to denote
$$
A = beginpmatrix fracpartialpartial u_x \ fracpartialpartial u_y endpmatrix
$$
where $u_x = fracpartialpartial x u$, likewise $u_y$. Here $u$ denotes some differentiable function in $Omega subset mathbbR^2$.
calculus notation vector-analysis
asked Aug 1 at 9:22
user8469759
1,4271513
I know that the operator $nabla$ denotes
$$
nabla = beginpmatrix fracpartialpartial x \ fracpartialpartial y endpmatrix
$$
Is there some kind of similar notation to denote
$$
A = beginpmatrix fracpartialpartial u_x \ fracpartialpartial u_y endpmatrix
$$
where $u_x = fracpartialpartial x u$, likewise $u_y$. Here $u$ denotes some differentiable function in $Omega subset mathbbR^2$.
calculus notation vector-analysis
asked Aug 1 at 9:22
user8469759
1,4271513
edited Aug 1 at 17:06
asked Aug 1 at 9:22
user8469759
1,4271513
asked Aug 1 at 9:22
user8469759
1,4271513
asked Aug 1 at 9:22
user8469759
1,4271513
1,4271513
Like the gradient on the fibre of the tangent bundle?
– Troy Woo
Aug 1 at 9:28
1
I'm not exactly sure what you're asking, but the directional derivative along a vector field $mathbf v = (v_x, v_y)$ is usually denoted $(mathbf v cdot nabla)$, which means $v_x fracpartialpartial x + v_y fracpartialpartial y$. This is used to define the material derivative in continuum mechanics.
– Rahul
Aug 1 at 9:30
@Rahul, No, I'm looking for a shorthand for $$A = fracpartialpartial x fracpartialpartial v_x + fracpartialpartial y fracpartialpartial v_y$$
– user8469759
Aug 1 at 9:35
add a comment |Â
Like the gradient on the fibre of the tangent bundle?
– Troy Woo
Aug 1 at 9:28
1
I'm not exactly sure what you're asking, but the directional derivative along a vector field $mathbf v = (v_x, v_y)$ is usually denoted $(mathbf v cdot nabla)$, which means $v_x fracpartialpartial x + v_y fracpartialpartial y$. This is used to define the material derivative in continuum mechanics.
– Rahul
Aug 1 at 9:30
@Rahul, No, I'm looking for a shorthand for $$A = fracpartialpartial x fracpartialpartial v_x + fracpartialpartial y fracpartialpartial v_y$$
– user8469759
Aug 1 at 9:35
Like the gradient on the fibre of the tangent bundle?
– Troy Woo
Aug 1 at 9:28
Like the gradient on the fibre of the tangent bundle?
– Troy Woo
Aug 1 at 9:28
1
1
I'm not exactly sure what you're asking, but the directional derivative along a vector field $mathbf v = (v_x, v_y)$ is usually denoted $(mathbf v cdot nabla)$, which means $v_x fracpartialpartial x + v_y fracpartialpartial y$. This is used to define the material derivative in continuum mechanics.
– Rahul
Aug 1 at 9:30
I'm not exactly sure what you're asking, but the directional derivative along a vector field $mathbf v = (v_x, v_y)$ is usually denoted $(mathbf v cdot nabla)$, which means $v_x fracpartialpartial x + v_y fracpartialpartial y$. This is used to define the material derivative in continuum mechanics.
– Rahul
Aug 1 at 9:30
@Rahul, No, I'm looking for a shorthand for $$A = fracpartialpartial x fracpartialpartial v_x + fracpartialpartial y fracpartialpartial v_y$$
– user8469759
Aug 1 at 9:35
@Rahul, No, I'm looking for a shorthand for $$A = fracpartialpartial x fracpartialpartial v_x + fracpartialpartial y fracpartialpartial v_y$$
– user8469759
Aug 1 at 9:35
add a comment |Â
1 Answer
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1
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It is fairly common to use the notation $nabla_mathbfu$ for gradient with respect to the velocity field. An alternative notation is $partial/partialmathbfu$ so that the usual gradient is written as $partial/partialmathbfx$.
answered Aug 1 at 9:40
Amey Joshi
579210
If I applied $nabla_bfu f$ wouldn't I get $langle nabla f, bfu rangle $?
– user8469759
Aug 1 at 9:41
No, you should get $hate_xpartial f/partial u_x + hate_ypartial f/partial u_y$ in two dimensions.
– Amey Joshi
Aug 1 at 9:43
But couldn't $nabla_bfu$ be confused with the directional derivative?
– user8469759
Aug 1 at 9:45
I did mean the directional derivative. I gave an analog of the nabla operator. If it is not a directional derivative with respect to a velocity field, why would want to write it as a vector. Can you possible write an expression in a long form that you want to write in a compact notation?
– Amey Joshi
Aug 1 at 9:49
Euler Lagrange equations are an example. I want ti write to gradient wrt a different basis I think (instead of the canonical basis I'd use the velocity field I think).
– user8469759
Aug 1 at 9:52
 |Â
show 4 more comments
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
It is fairly common to use the notation $nabla_mathbfu$ for gradient with respect to the velocity field. An alternative notation is $partial/partialmathbfu$ so that the usual gradient is written as $partial/partialmathbfx$.
answered Aug 1 at 9:40
Amey Joshi
579210
If I applied $nabla_bfu f$ wouldn't I get $langle nabla f, bfu rangle $?
– user8469759
Aug 1 at 9:41
No, you should get $hate_xpartial f/partial u_x + hate_ypartial f/partial u_y$ in two dimensions.
– Amey Joshi
Aug 1 at 9:43
But couldn't $nabla_bfu$ be confused with the directional derivative?
– user8469759
Aug 1 at 9:45
I did mean the directional derivative. I gave an analog of the nabla operator. If it is not a directional derivative with respect to a velocity field, why would want to write it as a vector. Can you possible write an expression in a long form that you want to write in a compact notation?
– Amey Joshi
Aug 1 at 9:49
Euler Lagrange equations are an example. I want ti write to gradient wrt a different basis I think (instead of the canonical basis I'd use the velocity field I think).
– user8469759
Aug 1 at 9:52
 |Â
show 4 more comments
up vote
1
down vote
It is fairly common to use the notation $nabla_mathbfu$ for gradient with respect to the velocity field. An alternative notation is $partial/partialmathbfu$ so that the usual gradient is written as $partial/partialmathbfx$.
answered Aug 1 at 9:40
Amey Joshi
579210
If I applied $nabla_bfu f$ wouldn't I get $langle nabla f, bfu rangle $?
– user8469759
Aug 1 at 9:41
No, you should get $hate_xpartial f/partial u_x + hate_ypartial f/partial u_y$ in two dimensions.
– Amey Joshi
Aug 1 at 9:43
But couldn't $nabla_bfu$ be confused with the directional derivative?
– user8469759
Aug 1 at 9:45
I did mean the directional derivative. I gave an analog of the nabla operator. If it is not a directional derivative with respect to a velocity field, why would want to write it as a vector. Can you possible write an expression in a long form that you want to write in a compact notation?
– Amey Joshi
Aug 1 at 9:49
Euler Lagrange equations are an example. I want ti write to gradient wrt a different basis I think (instead of the canonical basis I'd use the velocity field I think).
– user8469759
Aug 1 at 9:52
 |Â
show 4 more comments
up vote
1
down vote
up vote
1
down vote
It is fairly common to use the notation $nabla_mathbfu$ for gradient with respect to the velocity field. An alternative notation is $partial/partialmathbfu$ so that the usual gradient is written as $partial/partialmathbfx$.
answered Aug 1 at 9:40
Amey Joshi
579210
It is fairly common to use the notation $nabla_mathbfu$ for gradient with respect to the velocity field. An alternative notation is $partial/partialmathbfu$ so that the usual gradient is written as $partial/partialmathbfx$.
answered Aug 1 at 9:40
Amey Joshi
579210
answered Aug 1 at 9:40
Amey Joshi
579210
answered Aug 1 at 9:40
Amey Joshi
579210
answered Aug 1 at 9:40
Amey Joshi
579210
579210
If I applied $nabla_bfu f$ wouldn't I get $langle nabla f, bfu rangle $?
– user8469759
Aug 1 at 9:41
No, you should get $hate_xpartial f/partial u_x + hate_ypartial f/partial u_y$ in two dimensions.
– Amey Joshi
Aug 1 at 9:43
But couldn't $nabla_bfu$ be confused with the directional derivative?
– user8469759
Aug 1 at 9:45
I did mean the directional derivative. I gave an analog of the nabla operator. If it is not a directional derivative with respect to a velocity field, why would want to write it as a vector. Can you possible write an expression in a long form that you want to write in a compact notation?
– Amey Joshi
Aug 1 at 9:49
Euler Lagrange equations are an example. I want ti write to gradient wrt a different basis I think (instead of the canonical basis I'd use the velocity field I think).
– user8469759
Aug 1 at 9:52
 |Â
show 4 more comments
If I applied $nabla_bfu f$ wouldn't I get $langle nabla f, bfu rangle $?
– user8469759
Aug 1 at 9:41
No, you should get $hate_xpartial f/partial u_x + hate_ypartial f/partial u_y$ in two dimensions.
– Amey Joshi
Aug 1 at 9:43
But couldn't $nabla_bfu$ be confused with the directional derivative?
– user8469759
Aug 1 at 9:45
I did mean the directional derivative. I gave an analog of the nabla operator. If it is not a directional derivative with respect to a velocity field, why would want to write it as a vector. Can you possible write an expression in a long form that you want to write in a compact notation?
– Amey Joshi
Aug 1 at 9:49
Euler Lagrange equations are an example. I want ti write to gradient wrt a different basis I think (instead of the canonical basis I'd use the velocity field I think).
– user8469759
Aug 1 at 9:52
If I applied $nabla_bfu f$ wouldn't I get $langle nabla f, bfu rangle $?
– user8469759
Aug 1 at 9:41
If I applied $nabla_bfu f$ wouldn't I get $langle nabla f, bfu rangle $?
– user8469759
Aug 1 at 9:41
No, you should get $hate_xpartial f/partial u_x + hate_ypartial f/partial u_y$ in two dimensions.
– Amey Joshi
Aug 1 at 9:43
No, you should get $hate_xpartial f/partial u_x + hate_ypartial f/partial u_y$ in two dimensions.
– Amey Joshi
Aug 1 at 9:43
But couldn't $nabla_bfu$ be confused with the directional derivative?
– user8469759
Aug 1 at 9:45
But couldn't $nabla_bfu$ be confused with the directional derivative?
– user8469759
Aug 1 at 9:45
I did mean the directional derivative. I gave an analog of the nabla operator. If it is not a directional derivative with respect to a velocity field, why would want to write it as a vector. Can you possible write an expression in a long form that you want to write in a compact notation?
– Amey Joshi
Aug 1 at 9:49
I did mean the directional derivative. I gave an analog of the nabla operator. If it is not a directional derivative with respect to a velocity field, why would want to write it as a vector. Can you possible write an expression in a long form that you want to write in a compact notation?
– Amey Joshi
Aug 1 at 9:49
Euler Lagrange equations are an example. I want ti write to gradient wrt a different basis I think (instead of the canonical basis I'd use the velocity field I think).
– user8469759
Aug 1 at 9:52
Euler Lagrange equations are an example. I want ti write to gradient wrt a different basis I think (instead of the canonical basis I'd use the velocity field I think).
– user8469759
Aug 1 at 9:52
 |Â
show 4 more comments
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Like the gradient on the fibre of the tangent bundle?
– Troy Woo
Aug 1 at 9:28
1
I'm not exactly sure what you're asking, but the directional derivative along a vector field $mathbf v = (v_x, v_y)$ is usually denoted $(mathbf v cdot nabla)$, which means $v_x fracpartialpartial x + v_y fracpartialpartial y$. This is used to define the material derivative in continuum mechanics.
– Rahul
Aug 1 at 9:30
@Rahul, No, I'm looking for a shorthand for $$A = fracpartialpartial x fracpartialpartial v_x + fracpartialpartial y fracpartialpartial v_y$$
– user8469759
Aug 1 at 9:35