Gauss's divergence theorem and change of coordination system
$begingroup$
Gauss's divergence theorem:
Let $V$ be a compact subset of $mathbb R^3$ with a piecewise smooth boundary $partial V in C^1_{pw}$.
Further, let $v$ be a continuous differentiable vector field on $V$. Then we have:
$$int_{partial V}vcdot n do=int_V div(v)dmu$$
wheras $n$ denotes the outwards pointing normal along $partial V$.
Question:
So I hope I can formulate my question correctly. When I first started calculating integrals using Gauss's convergence theorem, I did a lot of mistakes when changing the coordinate system. E.g. if I have to calculate the flow of a simple vector field over the unit sphere I'd use spherical coordinates and of course, since I just changed the coordinate system, I'd take the determinant of the jacobian matrix into account. But that's actually "wrong".
It is wrong because Gauss divergence theorem is tightly connected to the actually flow integral - we have an equation of integrals! So if we transform $int_V div(v)dmu$ to let's say spherical coordinates, we'd have to transform $int_{partial V}vcdot n do$ too.
Would we do that, we would see the determinant of the jacobian matrix of the coordination change show up on both sides: We can cancel it out.
Or simply: The flow does not depend on the chosen coordinate system.
Is that observation correct? Could I just cancel the determinant of the jacobian matrix of my coordination change?
vector-analysis
asked Dec 28 '18 at 15:47
xotixxotix
291411
$endgroup$
add a comment |
$begingroup$
Gauss's divergence theorem:
Let $V$ be a compact subset of $mathbb R^3$ with a piecewise smooth boundary $partial V in C^1_{pw}$.
Further, let $v$ be a continuous differentiable vector field on $V$. Then we have:
$$int_{partial V}vcdot n do=int_V div(v)dmu$$
wheras $n$ denotes the outwards pointing normal along $partial V$.
Question:
So I hope I can formulate my question correctly. When I first started calculating integrals using Gauss's convergence theorem, I did a lot of mistakes when changing the coordinate system. E.g. if I have to calculate the flow of a simple vector field over the unit sphere I'd use spherical coordinates and of course, since I just changed the coordinate system, I'd take the determinant of the jacobian matrix into account. But that's actually "wrong".
It is wrong because Gauss divergence theorem is tightly connected to the actually flow integral - we have an equation of integrals! So if we transform $int_V div(v)dmu$ to let's say spherical coordinates, we'd have to transform $int_{partial V}vcdot n do$ too.
Would we do that, we would see the determinant of the jacobian matrix of the coordination change show up on both sides: We can cancel it out.
Or simply: The flow does not depend on the chosen coordinate system.
Is that observation correct? Could I just cancel the determinant of the jacobian matrix of my coordination change?
vector-analysis
asked Dec 28 '18 at 15:47
xotixxotix
291411
$endgroup$
add a comment |
$begingroup$
Gauss's divergence theorem:
Let $V$ be a compact subset of $mathbb R^3$ with a piecewise smooth boundary $partial V in C^1_{pw}$.
Further, let $v$ be a continuous differentiable vector field on $V$. Then we have:
$$int_{partial V}vcdot n do=int_V div(v)dmu$$
wheras $n$ denotes the outwards pointing normal along $partial V$.
Question:
So I hope I can formulate my question correctly. When I first started calculating integrals using Gauss's convergence theorem, I did a lot of mistakes when changing the coordinate system. E.g. if I have to calculate the flow of a simple vector field over the unit sphere I'd use spherical coordinates and of course, since I just changed the coordinate system, I'd take the determinant of the jacobian matrix into account. But that's actually "wrong".
It is wrong because Gauss divergence theorem is tightly connected to the actually flow integral - we have an equation of integrals! So if we transform $int_V div(v)dmu$ to let's say spherical coordinates, we'd have to transform $int_{partial V}vcdot n do$ too.
Would we do that, we would see the determinant of the jacobian matrix of the coordination change show up on both sides: We can cancel it out.
Or simply: The flow does not depend on the chosen coordinate system.
Is that observation correct? Could I just cancel the determinant of the jacobian matrix of my coordination change?
vector-analysis
asked Dec 28 '18 at 15:47
xotixxotix
291411
$endgroup$
Gauss's divergence theorem:
Let $V$ be a compact subset of $mathbb R^3$ with a piecewise smooth boundary $partial V in C^1_{pw}$.
Further, let $v$ be a continuous differentiable vector field on $V$. Then we have:
$$int_{partial V}vcdot n do=int_V div(v)dmu$$
wheras $n$ denotes the outwards pointing normal along $partial V$.
Question:
So I hope I can formulate my question correctly. When I first started calculating integrals using Gauss's convergence theorem, I did a lot of mistakes when changing the coordinate system. E.g. if I have to calculate the flow of a simple vector field over the unit sphere I'd use spherical coordinates and of course, since I just changed the coordinate system, I'd take the determinant of the jacobian matrix into account. But that's actually "wrong".
It is wrong because Gauss divergence theorem is tightly connected to the actually flow integral - we have an equation of integrals! So if we transform $int_V div(v)dmu$ to let's say spherical coordinates, we'd have to transform $int_{partial V}vcdot n do$ too.
Would we do that, we would see the determinant of the jacobian matrix of the coordination change show up on both sides: We can cancel it out.
Or simply: The flow does not depend on the chosen coordinate system.
Is that observation correct? Could I just cancel the determinant of the jacobian matrix of my coordination change?
vector-analysis
vector-analysis
asked Dec 28 '18 at 15:47
xotixxotix
291411
asked Dec 28 '18 at 15:47
xotixxotix
291411
asked Dec 28 '18 at 15:47
xotixxotix
291411
asked Dec 28 '18 at 15:47
xotixxotix
291411
asked Dec 28 '18 at 15:47
xotixxotix
291411
291411
add a comment |
add a comment |
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