Mixing
How to find limits of this volume integration?
$begingroup$
The question is
If $vec { F } = x hat { i } + y hat { j } + z hat { k }$ then find the value of $int int _ { S } vec { F } cdot hat { n } d s$ where $S$ is the sphere ${(x,y,z)inmathbb{R}^3 vert x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4}$.
Here if I'll calculate the answer using gauss divergence theorem, then I have to find the volume integral of the function, but how will I take the limits of the integration. Can someone help please?
Thank you.
definite-integrals
edited Jan 20 at 15:04
saz
81.3k861127
asked Jan 20 at 14:34
HawkingoHawkingo
84
$endgroup$
add a comment |
$begingroup$
The question is
If $vec { F } = x hat { i } + y hat { j } + z hat { k }$ then find the value of $int int _ { S } vec { F } cdot hat { n } d s$ where $S$ is the sphere ${(x,y,z)inmathbb{R}^3 vert x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4}$.
Here if I'll calculate the answer using gauss divergence theorem, then I have to find the volume integral of the function, but how will I take the limits of the integration. Can someone help please?
Thank you.
definite-integrals
edited Jan 20 at 15:04
saz
81.3k861127
asked Jan 20 at 14:34
HawkingoHawkingo
84
$endgroup$
1
$begingroup$
What about $$0le rhole2;,;;0lethetale2pi;,;;0lephilepi;?$$ These is the parametrization of the given sphere in spherical coordinates...
$endgroup$
– DonAntonio
Jan 20 at 15:30
add a comment |
$begingroup$
The question is
If $vec { F } = x hat { i } + y hat { j } + z hat { k }$ then find the value of $int int _ { S } vec { F } cdot hat { n } d s$ where $S$ is the sphere ${(x,y,z)inmathbb{R}^3 vert x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4}$.
Here if I'll calculate the answer using gauss divergence theorem, then I have to find the volume integral of the function, but how will I take the limits of the integration. Can someone help please?
Thank you.
definite-integrals
edited Jan 20 at 15:04
saz
81.3k861127
asked Jan 20 at 14:34
HawkingoHawkingo
84
$endgroup$
The question is
If $vec { F } = x hat { i } + y hat { j } + z hat { k }$ then find the value of $int int _ { S } vec { F } cdot hat { n } d s$ where $S$ is the sphere ${(x,y,z)inmathbb{R}^3 vert x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4}$.
Here if I'll calculate the answer using gauss divergence theorem, then I have to find the volume integral of the function, but how will I take the limits of the integration. Can someone help please?
Thank you.
definite-integrals
definite-integrals
edited Jan 20 at 15:04
saz
81.3k861127
asked Jan 20 at 14:34
HawkingoHawkingo
84
edited Jan 20 at 15:04
saz
81.3k861127
asked Jan 20 at 14:34
HawkingoHawkingo
84
edited Jan 20 at 15:04
saz
81.3k861127
edited Jan 20 at 15:04
saz
81.3k861127
edited Jan 20 at 15:04
saz
81.3k861127
81.3k861127
asked Jan 20 at 14:34
HawkingoHawkingo
84
asked Jan 20 at 14:34
HawkingoHawkingo
84
asked Jan 20 at 14:34
HawkingoHawkingo
84
84
1
$begingroup$
What about $$0le rhole2;,;;0lethetale2pi;,;;0lephilepi;?$$ These is the parametrization of the given sphere in spherical coordinates...
$endgroup$
– DonAntonio
Jan 20 at 15:30
add a comment |
1
$begingroup$
What about $$0le rhole2;,;;0lethetale2pi;,;;0lephilepi;?$$ These is the parametrization of the given sphere in spherical coordinates...
$endgroup$
– DonAntonio
Jan 20 at 15:30
1
1
$begingroup$
What about $$0le rhole2;,;;0lethetale2pi;,;;0lephilepi;?$$ These is the parametrization of the given sphere in spherical coordinates...
$endgroup$
– DonAntonio
Jan 20 at 15:30
$begingroup$
What about $$0le rhole2;,;;0lethetale2pi;,;;0lephilepi;?$$ These is the parametrization of the given sphere in spherical coordinates...
$endgroup$
– DonAntonio
Jan 20 at 15:30
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
You have ${bf F}({bf x}):={bf x}$, and on the sphere $S$ of radius $2$ you have ${bf n}({bf x})={1over2}{bf x}$. Furthermore $$int_S{rm d}omega=4cdot4pi .$$ It follows that
$$int_S{bf F}({bf x})cdot{bf n}({bf x})>{rm d}omega=int_S{bf x}cdot{1over2}{bf x}>{rm d}omega=2cdot16pi=32pi .$$
answered Jan 20 at 19:52
Christian BlatterChristian Blatter
174k8115327
$endgroup$
$begingroup$
sir,can you provide some method in which one can derive the limits directly from the given equation of the surface without knowing the geometry of the surface or plotting a graph?
$endgroup$
– Hawkingo
Jan 21 at 18:40
add a comment |
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1 Answer
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1 Answer
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$begingroup$
You have ${bf F}({bf x}):={bf x}$, and on the sphere $S$ of radius $2$ you have ${bf n}({bf x})={1over2}{bf x}$. Furthermore $$int_S{rm d}omega=4cdot4pi .$$ It follows that
$$int_S{bf F}({bf x})cdot{bf n}({bf x})>{rm d}omega=int_S{bf x}cdot{1over2}{bf x}>{rm d}omega=2cdot16pi=32pi .$$
answered Jan 20 at 19:52
Christian BlatterChristian Blatter
174k8115327
$endgroup$
$begingroup$
sir,can you provide some method in which one can derive the limits directly from the given equation of the surface without knowing the geometry of the surface or plotting a graph?
$endgroup$
– Hawkingo
Jan 21 at 18:40
add a comment |
$begingroup$
You have ${bf F}({bf x}):={bf x}$, and on the sphere $S$ of radius $2$ you have ${bf n}({bf x})={1over2}{bf x}$. Furthermore $$int_S{rm d}omega=4cdot4pi .$$ It follows that
$$int_S{bf F}({bf x})cdot{bf n}({bf x})>{rm d}omega=int_S{bf x}cdot{1over2}{bf x}>{rm d}omega=2cdot16pi=32pi .$$
answered Jan 20 at 19:52
Christian BlatterChristian Blatter
174k8115327
$endgroup$
$begingroup$
sir,can you provide some method in which one can derive the limits directly from the given equation of the surface without knowing the geometry of the surface or plotting a graph?
$endgroup$
– Hawkingo
Jan 21 at 18:40
add a comment |
$begingroup$
You have ${bf F}({bf x}):={bf x}$, and on the sphere $S$ of radius $2$ you have ${bf n}({bf x})={1over2}{bf x}$. Furthermore $$int_S{rm d}omega=4cdot4pi .$$ It follows that
$$int_S{bf F}({bf x})cdot{bf n}({bf x})>{rm d}omega=int_S{bf x}cdot{1over2}{bf x}>{rm d}omega=2cdot16pi=32pi .$$
answered Jan 20 at 19:52
Christian BlatterChristian Blatter
174k8115327
$endgroup$
You have ${bf F}({bf x}):={bf x}$, and on the sphere $S$ of radius $2$ you have ${bf n}({bf x})={1over2}{bf x}$. Furthermore $$int_S{rm d}omega=4cdot4pi .$$ It follows that
$$int_S{bf F}({bf x})cdot{bf n}({bf x})>{rm d}omega=int_S{bf x}cdot{1over2}{bf x}>{rm d}omega=2cdot16pi=32pi .$$
answered Jan 20 at 19:52
Christian BlatterChristian Blatter
174k8115327
answered Jan 20 at 19:52
Christian BlatterChristian Blatter
174k8115327
answered Jan 20 at 19:52
Christian BlatterChristian Blatter
174k8115327
answered Jan 20 at 19:52
Christian BlatterChristian Blatter
174k8115327
174k8115327
$begingroup$
sir,can you provide some method in which one can derive the limits directly from the given equation of the surface without knowing the geometry of the surface or plotting a graph?
$endgroup$
– Hawkingo
Jan 21 at 18:40
add a comment |
$begingroup$
sir,can you provide some method in which one can derive the limits directly from the given equation of the surface without knowing the geometry of the surface or plotting a graph?
$endgroup$
– Hawkingo
Jan 21 at 18:40
$begingroup$
sir,can you provide some method in which one can derive the limits directly from the given equation of the surface without knowing the geometry of the surface or plotting a graph?
$endgroup$
– Hawkingo
Jan 21 at 18:40
$begingroup$
sir,can you provide some method in which one can derive the limits directly from the given equation of the surface without knowing the geometry of the surface or plotting a graph?
$endgroup$
– Hawkingo
Jan 21 at 18:40
add a comment |
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$begingroup$
What about $$0le rhole2;,;;0lethetale2pi;,;;0lephilepi;?$$ These is the parametrization of the given sphere in spherical coordinates...
$endgroup$
– DonAntonio
Jan 20 at 15:30