Finding $nabla^2V $ if $V=frac2costheta+3sin^3theta cosphir^2$
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How do you find $nabla^2V $ if $V=frac2costheta+3sin^3theta cosphir^2$
The correct answer is supposedly $frac6sintheta cosphi(4-5sin^2theta)r^4$, but I can't seem to get the answer. Can anyone help?
multivariable-calculus spherical-coordinates
edited Aug 23 at 9:09
Bernard
111k635103
asked Aug 23 at 9:04
Yip Jung Hon
19911
add a comment |Â
up vote
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down vote
favorite
How do you find $nabla^2V $ if $V=frac2costheta+3sin^3theta cosphir^2$
The correct answer is supposedly $frac6sintheta cosphi(4-5sin^2theta)r^4$, but I can't seem to get the answer. Can anyone help?
multivariable-calculus spherical-coordinates
edited Aug 23 at 9:09
Bernard
111k635103
asked Aug 23 at 9:04
Yip Jung Hon
19911
Where exactly do you get stuck?
– Sobi
Aug 23 at 9:06
1
Del in spherical coordinates.
– Sobi
Aug 23 at 9:09
@Sobi Thanks for offering to help. This was where I got to: imgur.com/tDpPjxQ. Unfortunately, I can't get it to simplify. I'm not sure if I did a wrong step.
– Yip Jung Hon
Aug 23 at 10:16
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
How do you find $nabla^2V $ if $V=frac2costheta+3sin^3theta cosphir^2$
The correct answer is supposedly $frac6sintheta cosphi(4-5sin^2theta)r^4$, but I can't seem to get the answer. Can anyone help?
multivariable-calculus spherical-coordinates
edited Aug 23 at 9:09
Bernard
111k635103
asked Aug 23 at 9:04
Yip Jung Hon
19911
How do you find $nabla^2V $ if $V=frac2costheta+3sin^3theta cosphir^2$
The correct answer is supposedly $frac6sintheta cosphi(4-5sin^2theta)r^4$, but I can't seem to get the answer. Can anyone help?
multivariable-calculus spherical-coordinates
edited Aug 23 at 9:09
Bernard
111k635103
asked Aug 23 at 9:04
Yip Jung Hon
19911
edited Aug 23 at 9:09
Bernard
111k635103
edited Aug 23 at 9:09
Bernard
111k635103
edited Aug 23 at 9:09
Bernard
111k635103
111k635103
asked Aug 23 at 9:04
Yip Jung Hon
19911
asked Aug 23 at 9:04
Yip Jung Hon
19911
asked Aug 23 at 9:04
Yip Jung Hon
19911
19911
Where exactly do you get stuck?
– Sobi
Aug 23 at 9:06
1
Del in spherical coordinates.
– Sobi
Aug 23 at 9:09
@Sobi Thanks for offering to help. This was where I got to: imgur.com/tDpPjxQ. Unfortunately, I can't get it to simplify. I'm not sure if I did a wrong step.
– Yip Jung Hon
Aug 23 at 10:16
add a comment |Â
Where exactly do you get stuck?
– Sobi
Aug 23 at 9:06
1
Del in spherical coordinates.
– Sobi
Aug 23 at 9:09
@Sobi Thanks for offering to help. This was where I got to: imgur.com/tDpPjxQ. Unfortunately, I can't get it to simplify. I'm not sure if I did a wrong step.
– Yip Jung Hon
Aug 23 at 10:16
Where exactly do you get stuck?
– Sobi
Aug 23 at 9:06
Where exactly do you get stuck?
– Sobi
Aug 23 at 9:06
1
1
Del in spherical coordinates.
– Sobi
Aug 23 at 9:09
Del in spherical coordinates.
– Sobi
Aug 23 at 9:09
@Sobi Thanks for offering to help. This was where I got to: imgur.com/tDpPjxQ. Unfortunately, I can't get it to simplify. I'm not sure if I did a wrong step.
– Yip Jung Hon
Aug 23 at 10:16
@Sobi Thanks for offering to help. This was where I got to: imgur.com/tDpPjxQ. Unfortunately, I can't get it to simplify. I'm not sure if I did a wrong step.
– Yip Jung Hon
Aug 23 at 10:16
add a comment |Â
1 Answer
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The Laplacian in spherical coordinates is given by:
$$
nabla^2V=frac1r^2fracpartialpartialr left(r^2 fracpartial Vpartial r right) + frac1r^2sinthetafracpartialpartialtheta left(sintheta fracpartial Vpartial theta right) + frac1r^2sin^2thetafracpartial^2Vpartialphi^2 = 0$$
Proceeding carefully, the first term gives:
$$ begin align
fracpartial Vpartialr &=frac-4costheta-6sin^3thetacosphir^3 \
r^2fracpartial Vpartialr &= frac-4costheta-6sin^3thetacosphir \
fracpartialpartialr left(r^2fracpartial Vpartialrright) &= frac4costheta + 6sin^3thetacosphir \
Rightarrow frac1r^2fracpartialpartialr left(r^2 fracpartial Vpartial r right) &= frac4costheta + 6sin^3thetacosphir^4
endalign nonumber $$
The second term gives:
$$ begin align
fracpartial Vpartialtheta &=frac-2sintheta + 9sin^2thetacosthetacosphir^2 \
sinthetafracpartial Vpartialtheta &=frac-2sin^2theta + 9sin^3thetacosthetacosphir^2 \
fracpartialpartialthetaleft( sinthetafracpartial Vpartialtheta right) &= frac-4sinthetacostheta + 9cosphi left(3sin^2theta - 4sin^4thetaright)r^2 \
Rightarrow frac1r^2 sinthetafracpartialpartialthetaleft( sinthetafracpartial Vpartialtheta right) &= frac-4costheta + 9cosphi left(3sintheta - 4sin^3thetaright)r^4
endalign nonumber $$
The third term gives:
$$ begin align
fracpartial Vpartialphi &= -frac3sin^3thetar^2sinphi \
fracpartial^2 Vpartialphi^2 &= -frac3sin^3thetar^2cosphi \
Rightarrow frac1r^2 sin^2theta left(fracpartial^2 Vpartialphi^2 right) &= -frac3sinthetacosphir^4 \
endalign nonumber $$
Adding up all three terms gives:
$$ begin align
nabla^2V &= frac4costheta + 6sin^3thetacosphi - 4costheta + 9cosphi left(3sintheta - 4sin^3theta right) -3sinthetacosphi r^4 \
&= frac6sin^3thetacosphi + 27sinthetacosphi -36sin^3thetacosphi -3sinthetacosphi r^4 \
&= frac24sinthetacosphi -30sin^3thetacosphi r^4 \
&= frac6sinthetacosphi left(4 -5sin^2theta right)r^4 \
Rightarrow nabla^2V &= frac6sinthetacosphi left(4 -5sin^2thetaright)r^4
endalign nonumber $$
answered Aug 23 at 22:35
Winter Soldier
350211
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
The Laplacian in spherical coordinates is given by:
$$
nabla^2V=frac1r^2fracpartialpartialr left(r^2 fracpartial Vpartial r right) + frac1r^2sinthetafracpartialpartialtheta left(sintheta fracpartial Vpartial theta right) + frac1r^2sin^2thetafracpartial^2Vpartialphi^2 = 0$$
Proceeding carefully, the first term gives:
$$ begin align
fracpartial Vpartialr &=frac-4costheta-6sin^3thetacosphir^3 \
r^2fracpartial Vpartialr &= frac-4costheta-6sin^3thetacosphir \
fracpartialpartialr left(r^2fracpartial Vpartialrright) &= frac4costheta + 6sin^3thetacosphir \
Rightarrow frac1r^2fracpartialpartialr left(r^2 fracpartial Vpartial r right) &= frac4costheta + 6sin^3thetacosphir^4
endalign nonumber $$
The second term gives:
$$ begin align
fracpartial Vpartialtheta &=frac-2sintheta + 9sin^2thetacosthetacosphir^2 \
sinthetafracpartial Vpartialtheta &=frac-2sin^2theta + 9sin^3thetacosthetacosphir^2 \
fracpartialpartialthetaleft( sinthetafracpartial Vpartialtheta right) &= frac-4sinthetacostheta + 9cosphi left(3sin^2theta - 4sin^4thetaright)r^2 \
Rightarrow frac1r^2 sinthetafracpartialpartialthetaleft( sinthetafracpartial Vpartialtheta right) &= frac-4costheta + 9cosphi left(3sintheta - 4sin^3thetaright)r^4
endalign nonumber $$
The third term gives:
$$ begin align
fracpartial Vpartialphi &= -frac3sin^3thetar^2sinphi \
fracpartial^2 Vpartialphi^2 &= -frac3sin^3thetar^2cosphi \
Rightarrow frac1r^2 sin^2theta left(fracpartial^2 Vpartialphi^2 right) &= -frac3sinthetacosphir^4 \
endalign nonumber $$
Adding up all three terms gives:
$$ begin align
nabla^2V &= frac4costheta + 6sin^3thetacosphi - 4costheta + 9cosphi left(3sintheta - 4sin^3theta right) -3sinthetacosphi r^4 \
&= frac6sin^3thetacosphi + 27sinthetacosphi -36sin^3thetacosphi -3sinthetacosphi r^4 \
&= frac24sinthetacosphi -30sin^3thetacosphi r^4 \
&= frac6sinthetacosphi left(4 -5sin^2theta right)r^4 \
Rightarrow nabla^2V &= frac6sinthetacosphi left(4 -5sin^2thetaright)r^4
endalign nonumber $$
answered Aug 23 at 22:35
Winter Soldier
350211
add a comment |Â
up vote
1
down vote
The Laplacian in spherical coordinates is given by:
$$
nabla^2V=frac1r^2fracpartialpartialr left(r^2 fracpartial Vpartial r right) + frac1r^2sinthetafracpartialpartialtheta left(sintheta fracpartial Vpartial theta right) + frac1r^2sin^2thetafracpartial^2Vpartialphi^2 = 0$$
Proceeding carefully, the first term gives:
$$ begin align
fracpartial Vpartialr &=frac-4costheta-6sin^3thetacosphir^3 \
r^2fracpartial Vpartialr &= frac-4costheta-6sin^3thetacosphir \
fracpartialpartialr left(r^2fracpartial Vpartialrright) &= frac4costheta + 6sin^3thetacosphir \
Rightarrow frac1r^2fracpartialpartialr left(r^2 fracpartial Vpartial r right) &= frac4costheta + 6sin^3thetacosphir^4
endalign nonumber $$
The second term gives:
$$ begin align
fracpartial Vpartialtheta &=frac-2sintheta + 9sin^2thetacosthetacosphir^2 \
sinthetafracpartial Vpartialtheta &=frac-2sin^2theta + 9sin^3thetacosthetacosphir^2 \
fracpartialpartialthetaleft( sinthetafracpartial Vpartialtheta right) &= frac-4sinthetacostheta + 9cosphi left(3sin^2theta - 4sin^4thetaright)r^2 \
Rightarrow frac1r^2 sinthetafracpartialpartialthetaleft( sinthetafracpartial Vpartialtheta right) &= frac-4costheta + 9cosphi left(3sintheta - 4sin^3thetaright)r^4
endalign nonumber $$
The third term gives:
$$ begin align
fracpartial Vpartialphi &= -frac3sin^3thetar^2sinphi \
fracpartial^2 Vpartialphi^2 &= -frac3sin^3thetar^2cosphi \
Rightarrow frac1r^2 sin^2theta left(fracpartial^2 Vpartialphi^2 right) &= -frac3sinthetacosphir^4 \
endalign nonumber $$
Adding up all three terms gives:
$$ begin align
nabla^2V &= frac4costheta + 6sin^3thetacosphi - 4costheta + 9cosphi left(3sintheta - 4sin^3theta right) -3sinthetacosphi r^4 \
&= frac6sin^3thetacosphi + 27sinthetacosphi -36sin^3thetacosphi -3sinthetacosphi r^4 \
&= frac24sinthetacosphi -30sin^3thetacosphi r^4 \
&= frac6sinthetacosphi left(4 -5sin^2theta right)r^4 \
Rightarrow nabla^2V &= frac6sinthetacosphi left(4 -5sin^2thetaright)r^4
endalign nonumber $$
answered Aug 23 at 22:35
Winter Soldier
350211
add a comment |Â
up vote
1
down vote
up vote
1
down vote
The Laplacian in spherical coordinates is given by:
$$
nabla^2V=frac1r^2fracpartialpartialr left(r^2 fracpartial Vpartial r right) + frac1r^2sinthetafracpartialpartialtheta left(sintheta fracpartial Vpartial theta right) + frac1r^2sin^2thetafracpartial^2Vpartialphi^2 = 0$$
Proceeding carefully, the first term gives:
$$ begin align
fracpartial Vpartialr &=frac-4costheta-6sin^3thetacosphir^3 \
r^2fracpartial Vpartialr &= frac-4costheta-6sin^3thetacosphir \
fracpartialpartialr left(r^2fracpartial Vpartialrright) &= frac4costheta + 6sin^3thetacosphir \
Rightarrow frac1r^2fracpartialpartialr left(r^2 fracpartial Vpartial r right) &= frac4costheta + 6sin^3thetacosphir^4
endalign nonumber $$
The second term gives:
$$ begin align
fracpartial Vpartialtheta &=frac-2sintheta + 9sin^2thetacosthetacosphir^2 \
sinthetafracpartial Vpartialtheta &=frac-2sin^2theta + 9sin^3thetacosthetacosphir^2 \
fracpartialpartialthetaleft( sinthetafracpartial Vpartialtheta right) &= frac-4sinthetacostheta + 9cosphi left(3sin^2theta - 4sin^4thetaright)r^2 \
Rightarrow frac1r^2 sinthetafracpartialpartialthetaleft( sinthetafracpartial Vpartialtheta right) &= frac-4costheta + 9cosphi left(3sintheta - 4sin^3thetaright)r^4
endalign nonumber $$
The third term gives:
$$ begin align
fracpartial Vpartialphi &= -frac3sin^3thetar^2sinphi \
fracpartial^2 Vpartialphi^2 &= -frac3sin^3thetar^2cosphi \
Rightarrow frac1r^2 sin^2theta left(fracpartial^2 Vpartialphi^2 right) &= -frac3sinthetacosphir^4 \
endalign nonumber $$
Adding up all three terms gives:
$$ begin align
nabla^2V &= frac4costheta + 6sin^3thetacosphi - 4costheta + 9cosphi left(3sintheta - 4sin^3theta right) -3sinthetacosphi r^4 \
&= frac6sin^3thetacosphi + 27sinthetacosphi -36sin^3thetacosphi -3sinthetacosphi r^4 \
&= frac24sinthetacosphi -30sin^3thetacosphi r^4 \
&= frac6sinthetacosphi left(4 -5sin^2theta right)r^4 \
Rightarrow nabla^2V &= frac6sinthetacosphi left(4 -5sin^2thetaright)r^4
endalign nonumber $$
answered Aug 23 at 22:35
Winter Soldier
350211
The Laplacian in spherical coordinates is given by:
$$
nabla^2V=frac1r^2fracpartialpartialr left(r^2 fracpartial Vpartial r right) + frac1r^2sinthetafracpartialpartialtheta left(sintheta fracpartial Vpartial theta right) + frac1r^2sin^2thetafracpartial^2Vpartialphi^2 = 0$$
Proceeding carefully, the first term gives:
$$ begin align
fracpartial Vpartialr &=frac-4costheta-6sin^3thetacosphir^3 \
r^2fracpartial Vpartialr &= frac-4costheta-6sin^3thetacosphir \
fracpartialpartialr left(r^2fracpartial Vpartialrright) &= frac4costheta + 6sin^3thetacosphir \
Rightarrow frac1r^2fracpartialpartialr left(r^2 fracpartial Vpartial r right) &= frac4costheta + 6sin^3thetacosphir^4
endalign nonumber $$
The second term gives:
$$ begin align
fracpartial Vpartialtheta &=frac-2sintheta + 9sin^2thetacosthetacosphir^2 \
sinthetafracpartial Vpartialtheta &=frac-2sin^2theta + 9sin^3thetacosthetacosphir^2 \
fracpartialpartialthetaleft( sinthetafracpartial Vpartialtheta right) &= frac-4sinthetacostheta + 9cosphi left(3sin^2theta - 4sin^4thetaright)r^2 \
Rightarrow frac1r^2 sinthetafracpartialpartialthetaleft( sinthetafracpartial Vpartialtheta right) &= frac-4costheta + 9cosphi left(3sintheta - 4sin^3thetaright)r^4
endalign nonumber $$
The third term gives:
$$ begin align
fracpartial Vpartialphi &= -frac3sin^3thetar^2sinphi \
fracpartial^2 Vpartialphi^2 &= -frac3sin^3thetar^2cosphi \
Rightarrow frac1r^2 sin^2theta left(fracpartial^2 Vpartialphi^2 right) &= -frac3sinthetacosphir^4 \
endalign nonumber $$
Adding up all three terms gives:
$$ begin align
nabla^2V &= frac4costheta + 6sin^3thetacosphi - 4costheta + 9cosphi left(3sintheta - 4sin^3theta right) -3sinthetacosphi r^4 \
&= frac6sin^3thetacosphi + 27sinthetacosphi -36sin^3thetacosphi -3sinthetacosphi r^4 \
&= frac24sinthetacosphi -30sin^3thetacosphi r^4 \
&= frac6sinthetacosphi left(4 -5sin^2theta right)r^4 \
Rightarrow nabla^2V &= frac6sinthetacosphi left(4 -5sin^2thetaright)r^4
endalign nonumber $$
answered Aug 23 at 22:35
Winter Soldier
350211
answered Aug 23 at 22:35
Winter Soldier
350211
answered Aug 23 at 22:35
Winter Soldier
350211
answered Aug 23 at 22:35
Winter Soldier
350211
350211
add a comment |Â
add a comment |Â
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Where exactly do you get stuck?
– Sobi
Aug 23 at 9:06
1
Del in spherical coordinates.
– Sobi
Aug 23 at 9:09
@Sobi Thanks for offering to help. This was where I got to: imgur.com/tDpPjxQ. Unfortunately, I can't get it to simplify. I'm not sure if I did a wrong step.
– Yip Jung Hon
Aug 23 at 10:16