How can we show that the limit of the following surface integral is finite?











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I have an arbitrary parameterized surface $S(u,v)$ in Cartesian coordinates (with finite area) and the surface passes through origin. There is a very small circular hole of radius $epsilon$ at the origin.



Is there a way to show that the limit of the following surface integral is finite?
$$ displaystyle vec{V} = lim limits_{epsilon to 0} iint_S dfrac{dS}{r^2} hat{r}$$
Here $r$ is the distance between origin and points on our parameterized surface $S(u,v)$. $hat {r}$ is a unit vector from our surface to origin.










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  • What is $hat{r}$?
    – AlgebraicallyClosed
    Nov 15 at 12:07










  • $hat {r}$ is a unit vector from our surface to origin.
    – Joe
    Nov 15 at 12:16















up vote
1
down vote

favorite












I have an arbitrary parameterized surface $S(u,v)$ in Cartesian coordinates (with finite area) and the surface passes through origin. There is a very small circular hole of radius $epsilon$ at the origin.



Is there a way to show that the limit of the following surface integral is finite?
$$ displaystyle vec{V} = lim limits_{epsilon to 0} iint_S dfrac{dS}{r^2} hat{r}$$
Here $r$ is the distance between origin and points on our parameterized surface $S(u,v)$. $hat {r}$ is a unit vector from our surface to origin.










share|cite|improve this question
























  • What is $hat{r}$?
    – AlgebraicallyClosed
    Nov 15 at 12:07










  • $hat {r}$ is a unit vector from our surface to origin.
    – Joe
    Nov 15 at 12:16













up vote
1
down vote

favorite









up vote
1
down vote

favorite











I have an arbitrary parameterized surface $S(u,v)$ in Cartesian coordinates (with finite area) and the surface passes through origin. There is a very small circular hole of radius $epsilon$ at the origin.



Is there a way to show that the limit of the following surface integral is finite?
$$ displaystyle vec{V} = lim limits_{epsilon to 0} iint_S dfrac{dS}{r^2} hat{r}$$
Here $r$ is the distance between origin and points on our parameterized surface $S(u,v)$. $hat {r}$ is a unit vector from our surface to origin.










share|cite|improve this question















I have an arbitrary parameterized surface $S(u,v)$ in Cartesian coordinates (with finite area) and the surface passes through origin. There is a very small circular hole of radius $epsilon$ at the origin.



Is there a way to show that the limit of the following surface integral is finite?
$$ displaystyle vec{V} = lim limits_{epsilon to 0} iint_S dfrac{dS}{r^2} hat{r}$$
Here $r$ is the distance between origin and points on our parameterized surface $S(u,v)$. $hat {r}$ is a unit vector from our surface to origin.







calculus limits multivariable-calculus vectors surface-integrals






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edited Nov 15 at 12:16

























asked Nov 15 at 11:36









Joe

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395113












  • What is $hat{r}$?
    – AlgebraicallyClosed
    Nov 15 at 12:07










  • $hat {r}$ is a unit vector from our surface to origin.
    – Joe
    Nov 15 at 12:16


















  • What is $hat{r}$?
    – AlgebraicallyClosed
    Nov 15 at 12:07










  • $hat {r}$ is a unit vector from our surface to origin.
    – Joe
    Nov 15 at 12:16
















What is $hat{r}$?
– AlgebraicallyClosed
Nov 15 at 12:07




What is $hat{r}$?
– AlgebraicallyClosed
Nov 15 at 12:07












$hat {r}$ is a unit vector from our surface to origin.
– Joe
Nov 15 at 12:16




$hat {r}$ is a unit vector from our surface to origin.
– Joe
Nov 15 at 12:16















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