What is the meaning of notation $ nabla u + nabla u^T$?
$begingroup$
In my exercises it appears ($u:mathbb{R}^3rightarrowmathbb{R}^3$):
$$ nabla u + nabla u^T$$
What is the meaning of this notation when writing it in terms of partials of $u_i$?
notation vector-analysis
edited Dec 19 '18 at 20:51
Jean Marie
30.9k42155
asked Dec 19 '18 at 19:24
davidivadfuldavidivadful
13610
$endgroup$
add a comment |
$begingroup$
In my exercises it appears ($u:mathbb{R}^3rightarrowmathbb{R}^3$):
$$ nabla u + nabla u^T$$
What is the meaning of this notation when writing it in terms of partials of $u_i$?
notation vector-analysis
edited Dec 19 '18 at 20:51
Jean Marie
30.9k42155
asked Dec 19 '18 at 19:24
davidivadfuldavidivadful
13610
$endgroup$
add a comment |
$begingroup$
In my exercises it appears ($u:mathbb{R}^3rightarrowmathbb{R}^3$):
$$ nabla u + nabla u^T$$
What is the meaning of this notation when writing it in terms of partials of $u_i$?
notation vector-analysis
edited Dec 19 '18 at 20:51
Jean Marie
30.9k42155
asked Dec 19 '18 at 19:24
davidivadfuldavidivadful
13610
$endgroup$
In my exercises it appears ($u:mathbb{R}^3rightarrowmathbb{R}^3$):
$$ nabla u + nabla u^T$$
What is the meaning of this notation when writing it in terms of partials of $u_i$?
notation vector-analysis
notation vector-analysis
edited Dec 19 '18 at 20:51
Jean Marie
30.9k42155
asked Dec 19 '18 at 19:24
davidivadfuldavidivadful
13610
edited Dec 19 '18 at 20:51
Jean Marie
30.9k42155
asked Dec 19 '18 at 19:24
davidivadfuldavidivadful
13610
edited Dec 19 '18 at 20:51
Jean Marie
30.9k42155
edited Dec 19 '18 at 20:51
Jean Marie
30.9k42155
edited Dec 19 '18 at 20:51
Jean Marie
30.9k42155
30.9k42155
asked Dec 19 '18 at 19:24
davidivadfuldavidivadful
13610
asked Dec 19 '18 at 19:24
davidivadfuldavidivadful
13610
asked Dec 19 '18 at 19:24
davidivadfuldavidivadful
13610
13610
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Typically, when $u:Bbb R^3 to Bbb R^3$, we define
$$
newcommand{pwrt}[2]{frac{partial #1}{partial #2}}\
nabla u = pmatrix{pwrt {u_1}{x_1} & pwrt {u_1}{x_2} & pwrt {u_1}{x_3}\
pwrt {u_2}{x_1} & pwrt {u_2}{x_2} & pwrt {u_2}{x_3}\
pwrt {u_3}{x_1} & pwrt {u_3}{x_2} & pwrt {u_3}{x_3}\}
$$
In some contexts, this is referred to instead as the Jacobian of $u$.
$M^T$ refers to the transpose of the matrix $M$.
answered Dec 19 '18 at 19:30
OmnomnomnomOmnomnomnom
129k792185
$endgroup$
add a comment |
$begingroup$
$M=nabla u ; ;$ is a $(3times 3)$ matrix
with
$$M_{i,j}=frac{partial u_i}{partial x_j}$$
and
$$nabla u^T=M^T=N$$
with
$$N_{i,j}=M_{j,i}=frac{partial u_j}{partial x_i}$$ is the transpose matrix of $M$.
answered Dec 19 '18 at 19:31
hamam_Abdallahhamam_Abdallah
38.2k21634
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Typically, when $u:Bbb R^3 to Bbb R^3$, we define
$$
newcommand{pwrt}[2]{frac{partial #1}{partial #2}}\
nabla u = pmatrix{pwrt {u_1}{x_1} & pwrt {u_1}{x_2} & pwrt {u_1}{x_3}\
pwrt {u_2}{x_1} & pwrt {u_2}{x_2} & pwrt {u_2}{x_3}\
pwrt {u_3}{x_1} & pwrt {u_3}{x_2} & pwrt {u_3}{x_3}\}
$$
In some contexts, this is referred to instead as the Jacobian of $u$.
$M^T$ refers to the transpose of the matrix $M$.
answered Dec 19 '18 at 19:30
OmnomnomnomOmnomnomnom
129k792185
$endgroup$
add a comment |
$begingroup$
Typically, when $u:Bbb R^3 to Bbb R^3$, we define
$$
newcommand{pwrt}[2]{frac{partial #1}{partial #2}}\
nabla u = pmatrix{pwrt {u_1}{x_1} & pwrt {u_1}{x_2} & pwrt {u_1}{x_3}\
pwrt {u_2}{x_1} & pwrt {u_2}{x_2} & pwrt {u_2}{x_3}\
pwrt {u_3}{x_1} & pwrt {u_3}{x_2} & pwrt {u_3}{x_3}\}
$$
In some contexts, this is referred to instead as the Jacobian of $u$.
$M^T$ refers to the transpose of the matrix $M$.
answered Dec 19 '18 at 19:30
OmnomnomnomOmnomnomnom
129k792185
$endgroup$
add a comment |
$begingroup$
Typically, when $u:Bbb R^3 to Bbb R^3$, we define
$$
newcommand{pwrt}[2]{frac{partial #1}{partial #2}}\
nabla u = pmatrix{pwrt {u_1}{x_1} & pwrt {u_1}{x_2} & pwrt {u_1}{x_3}\
pwrt {u_2}{x_1} & pwrt {u_2}{x_2} & pwrt {u_2}{x_3}\
pwrt {u_3}{x_1} & pwrt {u_3}{x_2} & pwrt {u_3}{x_3}\}
$$
In some contexts, this is referred to instead as the Jacobian of $u$.
$M^T$ refers to the transpose of the matrix $M$.
answered Dec 19 '18 at 19:30
OmnomnomnomOmnomnomnom
129k792185
$endgroup$
Typically, when $u:Bbb R^3 to Bbb R^3$, we define
$$
newcommand{pwrt}[2]{frac{partial #1}{partial #2}}\
nabla u = pmatrix{pwrt {u_1}{x_1} & pwrt {u_1}{x_2} & pwrt {u_1}{x_3}\
pwrt {u_2}{x_1} & pwrt {u_2}{x_2} & pwrt {u_2}{x_3}\
pwrt {u_3}{x_1} & pwrt {u_3}{x_2} & pwrt {u_3}{x_3}\}
$$
In some contexts, this is referred to instead as the Jacobian of $u$.
$M^T$ refers to the transpose of the matrix $M$.
answered Dec 19 '18 at 19:30
OmnomnomnomOmnomnomnom
129k792185
answered Dec 19 '18 at 19:30
OmnomnomnomOmnomnomnom
129k792185
answered Dec 19 '18 at 19:30
OmnomnomnomOmnomnomnom
129k792185
answered Dec 19 '18 at 19:30
OmnomnomnomOmnomnomnom
129k792185
129k792185
add a comment |
add a comment |
$begingroup$
$M=nabla u ; ;$ is a $(3times 3)$ matrix
with
$$M_{i,j}=frac{partial u_i}{partial x_j}$$
and
$$nabla u^T=M^T=N$$
with
$$N_{i,j}=M_{j,i}=frac{partial u_j}{partial x_i}$$ is the transpose matrix of $M$.
answered Dec 19 '18 at 19:31
hamam_Abdallahhamam_Abdallah
38.2k21634
$endgroup$
add a comment |
$begingroup$
$M=nabla u ; ;$ is a $(3times 3)$ matrix
with
$$M_{i,j}=frac{partial u_i}{partial x_j}$$
and
$$nabla u^T=M^T=N$$
with
$$N_{i,j}=M_{j,i}=frac{partial u_j}{partial x_i}$$ is the transpose matrix of $M$.
answered Dec 19 '18 at 19:31
hamam_Abdallahhamam_Abdallah
38.2k21634
$endgroup$
add a comment |
$begingroup$
$M=nabla u ; ;$ is a $(3times 3)$ matrix
with
$$M_{i,j}=frac{partial u_i}{partial x_j}$$
and
$$nabla u^T=M^T=N$$
with
$$N_{i,j}=M_{j,i}=frac{partial u_j}{partial x_i}$$ is the transpose matrix of $M$.
answered Dec 19 '18 at 19:31
hamam_Abdallahhamam_Abdallah
38.2k21634
$endgroup$
$M=nabla u ; ;$ is a $(3times 3)$ matrix
with
$$M_{i,j}=frac{partial u_i}{partial x_j}$$
and
$$nabla u^T=M^T=N$$
with
$$N_{i,j}=M_{j,i}=frac{partial u_j}{partial x_i}$$ is the transpose matrix of $M$.
answered Dec 19 '18 at 19:31
hamam_Abdallahhamam_Abdallah
38.2k21634
edited Dec 19 '18 at 20:05
answered Dec 19 '18 at 19:31
hamam_Abdallahhamam_Abdallah
38.2k21634
answered Dec 19 '18 at 19:31
hamam_Abdallahhamam_Abdallah
38.2k21634
answered Dec 19 '18 at 19:31
hamam_Abdallahhamam_Abdallah
38.2k21634
38.2k21634
add a comment |
add a comment |
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