Integration by parts with vectors and matrices
$begingroup$
I need to integrate by parts the product of a vector (v) times the divergence of a matrix A. I think the solution is:
$$
DeclareMathOperator{div}{div}
DeclareMathOperator{grad}{grad}
int_Omega textbf{v} cdot div(A)dV=int_{partialOmega} (textbf{n}otimes textbf{v}): A dS-int_Omega grad(textbf{v}):AdV
$$
but I would like to be able to understand this and derive the equation myself. I would like to understand why the integration by parts leads to those double dot products, but also how to apply the divergence theorem with tensors.
Thanks for the help and sorry if the notation is not correct, it's been a while since I had to use tensors.
calculus
edited Aug 29 '18 at 13:03
mvw
31.5k22252
asked Mar 13 '15 at 22:28
JuanJuan
62
$endgroup$
add a comment |
$begingroup$
I need to integrate by parts the product of a vector (v) times the divergence of a matrix A. I think the solution is:
$$
DeclareMathOperator{div}{div}
DeclareMathOperator{grad}{grad}
int_Omega textbf{v} cdot div(A)dV=int_{partialOmega} (textbf{n}otimes textbf{v}): A dS-int_Omega grad(textbf{v}):AdV
$$
but I would like to be able to understand this and derive the equation myself. I would like to understand why the integration by parts leads to those double dot products, but also how to apply the divergence theorem with tensors.
Thanks for the help and sorry if the notation is not correct, it's been a while since I had to use tensors.
calculus
edited Aug 29 '18 at 13:03
mvw
31.5k22252
asked Mar 13 '15 at 22:28
JuanJuan
62
$endgroup$
$begingroup$
What does the colon mean? Is $otimes$ different from a vector product?
$endgroup$
– mvw
Aug 29 '18 at 13:05
add a comment |
$begingroup$
I need to integrate by parts the product of a vector (v) times the divergence of a matrix A. I think the solution is:
$$
DeclareMathOperator{div}{div}
DeclareMathOperator{grad}{grad}
int_Omega textbf{v} cdot div(A)dV=int_{partialOmega} (textbf{n}otimes textbf{v}): A dS-int_Omega grad(textbf{v}):AdV
$$
but I would like to be able to understand this and derive the equation myself. I would like to understand why the integration by parts leads to those double dot products, but also how to apply the divergence theorem with tensors.
Thanks for the help and sorry if the notation is not correct, it's been a while since I had to use tensors.
calculus
edited Aug 29 '18 at 13:03
mvw
31.5k22252
asked Mar 13 '15 at 22:28
JuanJuan
62
$endgroup$
I need to integrate by parts the product of a vector (v) times the divergence of a matrix A. I think the solution is:
$$
DeclareMathOperator{div}{div}
DeclareMathOperator{grad}{grad}
int_Omega textbf{v} cdot div(A)dV=int_{partialOmega} (textbf{n}otimes textbf{v}): A dS-int_Omega grad(textbf{v}):AdV
$$
but I would like to be able to understand this and derive the equation myself. I would like to understand why the integration by parts leads to those double dot products, but also how to apply the divergence theorem with tensors.
Thanks for the help and sorry if the notation is not correct, it's been a while since I had to use tensors.
calculus
calculus
edited Aug 29 '18 at 13:03
mvw
31.5k22252
asked Mar 13 '15 at 22:28
JuanJuan
62
edited Aug 29 '18 at 13:03
mvw
31.5k22252
asked Mar 13 '15 at 22:28
JuanJuan
62
edited Aug 29 '18 at 13:03
mvw
31.5k22252
edited Aug 29 '18 at 13:03
mvw
31.5k22252
edited Aug 29 '18 at 13:03
mvw
31.5k22252
31.5k22252
asked Mar 13 '15 at 22:28
JuanJuan
62
asked Mar 13 '15 at 22:28
JuanJuan
62
asked Mar 13 '15 at 22:28
JuanJuan
62
62
$begingroup$
What does the colon mean? Is $otimes$ different from a vector product?
$endgroup$
– mvw
Aug 29 '18 at 13:05
add a comment |
$begingroup$
What does the colon mean? Is $otimes$ different from a vector product?
$endgroup$
– mvw
Aug 29 '18 at 13:05
$begingroup$
What does the colon mean? Is $otimes$ different from a vector product?
$endgroup$
– mvw
Aug 29 '18 at 13:05
$begingroup$
What does the colon mean? Is $otimes$ different from a vector product?
$endgroup$
– mvw
Aug 29 '18 at 13:05
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Usually, when you have something like that hard to understand, return to the formula component by component.
$$v cdot div(A) = sum_{i=1}^n v_i times div(A)_i = sum_{i=1}^nleft( v_i times sum_{j=1}^n partial_j a_{ij} right) = sum_{i=1}^n sum_{j=1}^n v_ipartial_j a_{ij} $$
You apply the integration by part, and you get, on the border
$$sum_{i=1}^n sum_{j=1}^n n_jv_i a_{ij} = sum_{i=1}^n sum_{j=1}^n (notimes v)_{ji} a_{ij} = (notimes v): A$$
And on the domain
$$- sum_{i=1}^n sum_{j=1}^n a_{ij} partial_jv_i = grad(v) : A$$
answered Mar 13 '15 at 22:42
TryssTryss
13.1k1229
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add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Usually, when you have something like that hard to understand, return to the formula component by component.
$$v cdot div(A) = sum_{i=1}^n v_i times div(A)_i = sum_{i=1}^nleft( v_i times sum_{j=1}^n partial_j a_{ij} right) = sum_{i=1}^n sum_{j=1}^n v_ipartial_j a_{ij} $$
You apply the integration by part, and you get, on the border
$$sum_{i=1}^n sum_{j=1}^n n_jv_i a_{ij} = sum_{i=1}^n sum_{j=1}^n (notimes v)_{ji} a_{ij} = (notimes v): A$$
And on the domain
$$- sum_{i=1}^n sum_{j=1}^n a_{ij} partial_jv_i = grad(v) : A$$
answered Mar 13 '15 at 22:42
TryssTryss
13.1k1229
$endgroup$
add a comment |
$begingroup$
Usually, when you have something like that hard to understand, return to the formula component by component.
$$v cdot div(A) = sum_{i=1}^n v_i times div(A)_i = sum_{i=1}^nleft( v_i times sum_{j=1}^n partial_j a_{ij} right) = sum_{i=1}^n sum_{j=1}^n v_ipartial_j a_{ij} $$
You apply the integration by part, and you get, on the border
$$sum_{i=1}^n sum_{j=1}^n n_jv_i a_{ij} = sum_{i=1}^n sum_{j=1}^n (notimes v)_{ji} a_{ij} = (notimes v): A$$
And on the domain
$$- sum_{i=1}^n sum_{j=1}^n a_{ij} partial_jv_i = grad(v) : A$$
answered Mar 13 '15 at 22:42
TryssTryss
13.1k1229
$endgroup$
add a comment |
$begingroup$
Usually, when you have something like that hard to understand, return to the formula component by component.
$$v cdot div(A) = sum_{i=1}^n v_i times div(A)_i = sum_{i=1}^nleft( v_i times sum_{j=1}^n partial_j a_{ij} right) = sum_{i=1}^n sum_{j=1}^n v_ipartial_j a_{ij} $$
You apply the integration by part, and you get, on the border
$$sum_{i=1}^n sum_{j=1}^n n_jv_i a_{ij} = sum_{i=1}^n sum_{j=1}^n (notimes v)_{ji} a_{ij} = (notimes v): A$$
And on the domain
$$- sum_{i=1}^n sum_{j=1}^n a_{ij} partial_jv_i = grad(v) : A$$
answered Mar 13 '15 at 22:42
TryssTryss
13.1k1229
$endgroup$
Usually, when you have something like that hard to understand, return to the formula component by component.
$$v cdot div(A) = sum_{i=1}^n v_i times div(A)_i = sum_{i=1}^nleft( v_i times sum_{j=1}^n partial_j a_{ij} right) = sum_{i=1}^n sum_{j=1}^n v_ipartial_j a_{ij} $$
You apply the integration by part, and you get, on the border
$$sum_{i=1}^n sum_{j=1}^n n_jv_i a_{ij} = sum_{i=1}^n sum_{j=1}^n (notimes v)_{ji} a_{ij} = (notimes v): A$$
And on the domain
$$- sum_{i=1}^n sum_{j=1}^n a_{ij} partial_jv_i = grad(v) : A$$
answered Mar 13 '15 at 22:42
TryssTryss
13.1k1229
edited Mar 14 '15 at 0:13
answered Mar 13 '15 at 22:42
TryssTryss
13.1k1229
answered Mar 13 '15 at 22:42
TryssTryss
13.1k1229
answered Mar 13 '15 at 22:42
TryssTryss
13.1k1229
13.1k1229
add a comment |
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$begingroup$
What does the colon mean? Is $otimes$ different from a vector product?
$endgroup$
– mvw
Aug 29 '18 at 13:05