Trouble calculating the Laplace-Beltrami operator through this formula
Let $U$ be an open, bounded and connected subset of $mathbb R^3$ with a $C^2−$regular boundary $partial U$. For an arbitary $x_0 in partial U$ define the function $f:B(x_0,r) cap partial U to mathbb R$ as follows: $f(x)=C|x−x_0|^2$. I'm interested in computing $Δf$ where $Δ$ denotes the Laplace-Beltrami operator.
After some research (since I'm not familiar to the Laplace-Beltrami operator) I found this formula:
$Delta f= {Delta}_{mathbb R^2} f-{nabla}_{mathbb
R^2}^2f(eta,eta)+H{nabla}_{mathbb R^2}f$
where $Δ_{mathbb R^2}$ denotes the usual Laplace operator, $H$ is the mean curvature vector over $partial U$ and $eta$ is a unit normal vector field.
How do I compute the Hessian term ${nabla}_{mathbb R^2}^2f(eta,eta)$?
DISCLAIMER: I've only seen an introduction to differential geometry on manifolds in the past so maybe my question is too elementary
I would appreciate any help.
Thanks in advance!
analysis multivariable-calculus differential-geometry laplacian hessian-matrix
asked Dec 13 '18 at 10:05
kaithkolesidoukaithkolesidou
959511
add a comment |
Let $U$ be an open, bounded and connected subset of $mathbb R^3$ with a $C^2−$regular boundary $partial U$. For an arbitary $x_0 in partial U$ define the function $f:B(x_0,r) cap partial U to mathbb R$ as follows: $f(x)=C|x−x_0|^2$. I'm interested in computing $Δf$ where $Δ$ denotes the Laplace-Beltrami operator.
After some research (since I'm not familiar to the Laplace-Beltrami operator) I found this formula:
$Delta f= {Delta}_{mathbb R^2} f-{nabla}_{mathbb
R^2}^2f(eta,eta)+H{nabla}_{mathbb R^2}f$
where $Δ_{mathbb R^2}$ denotes the usual Laplace operator, $H$ is the mean curvature vector over $partial U$ and $eta$ is a unit normal vector field.
How do I compute the Hessian term ${nabla}_{mathbb R^2}^2f(eta,eta)$?
DISCLAIMER: I've only seen an introduction to differential geometry on manifolds in the past so maybe my question is too elementary
I would appreciate any help.
Thanks in advance!
analysis multivariable-calculus differential-geometry laplacian hessian-matrix
asked Dec 13 '18 at 10:05
kaithkolesidoukaithkolesidou
959511
add a comment |
Let $U$ be an open, bounded and connected subset of $mathbb R^3$ with a $C^2−$regular boundary $partial U$. For an arbitary $x_0 in partial U$ define the function $f:B(x_0,r) cap partial U to mathbb R$ as follows: $f(x)=C|x−x_0|^2$. I'm interested in computing $Δf$ where $Δ$ denotes the Laplace-Beltrami operator.
After some research (since I'm not familiar to the Laplace-Beltrami operator) I found this formula:
$Delta f= {Delta}_{mathbb R^2} f-{nabla}_{mathbb
R^2}^2f(eta,eta)+H{nabla}_{mathbb R^2}f$
where $Δ_{mathbb R^2}$ denotes the usual Laplace operator, $H$ is the mean curvature vector over $partial U$ and $eta$ is a unit normal vector field.
How do I compute the Hessian term ${nabla}_{mathbb R^2}^2f(eta,eta)$?
DISCLAIMER: I've only seen an introduction to differential geometry on manifolds in the past so maybe my question is too elementary
I would appreciate any help.
Thanks in advance!
analysis multivariable-calculus differential-geometry laplacian hessian-matrix
asked Dec 13 '18 at 10:05
kaithkolesidoukaithkolesidou
959511
Let $U$ be an open, bounded and connected subset of $mathbb R^3$ with a $C^2−$regular boundary $partial U$. For an arbitary $x_0 in partial U$ define the function $f:B(x_0,r) cap partial U to mathbb R$ as follows: $f(x)=C|x−x_0|^2$. I'm interested in computing $Δf$ where $Δ$ denotes the Laplace-Beltrami operator.
After some research (since I'm not familiar to the Laplace-Beltrami operator) I found this formula:
$Delta f= {Delta}_{mathbb R^2} f-{nabla}_{mathbb
R^2}^2f(eta,eta)+H{nabla}_{mathbb R^2}f$
where $Δ_{mathbb R^2}$ denotes the usual Laplace operator, $H$ is the mean curvature vector over $partial U$ and $eta$ is a unit normal vector field.
How do I compute the Hessian term ${nabla}_{mathbb R^2}^2f(eta,eta)$?
DISCLAIMER: I've only seen an introduction to differential geometry on manifolds in the past so maybe my question is too elementary
I would appreciate any help.
Thanks in advance!
analysis multivariable-calculus differential-geometry laplacian hessian-matrix
analysis multivariable-calculus differential-geometry laplacian hessian-matrix
asked Dec 13 '18 at 10:05
kaithkolesidoukaithkolesidou
959511
asked Dec 13 '18 at 10:05
kaithkolesidoukaithkolesidou
959511
edited Dec 13 '18 at 11:22
kaithkolesidou
asked Dec 13 '18 at 10:05
kaithkolesidoukaithkolesidou
959511
asked Dec 13 '18 at 10:05
kaithkolesidoukaithkolesidou
959511
asked Dec 13 '18 at 10:05
kaithkolesidoukaithkolesidou
959511
959511
add a comment |
add a comment |
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