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Derivative of a double integral over a circular region
Surface integral of vector field over a unit ballComplicated surface integral/line integral.derivative of a surface integral with moving boundaryDivergence of the product of a scalar and a tensor fieldsDivergence of the product of a tensor and a vector fieldThe assumptions of the substitution theorem in double integralProving Some Relations for Gradient, Curl and Divergence of a Tensor FieldHow to calculate derivative under area integralVector Derivative With RotorMaterial Derivative in Cylindrical Coordinates
$begingroup$
Calculate the following derivative
$$fracddtiint_D_tF(x,y,t) , mathrm d x mathrm d y$$
where
$$D_t = lbrace (x,y) mid (x-t)^2+(y-t)^2leq r^2 rbrace$$
I've read here that the answer to the above question is in the form of:
beginalignfracddtiint_D_tF(x,y,t)dxdy &=int_partial D_t F(udy-vdx) + iint_D_tfracpartial Fpartial tdx dy \
&=iint_D_tleft[textdiv(Fmathbfv)+fracpartial Fpartial tright]dx dyendalign
which is a generalization of Leibniz integral rule.
I don't know how I can calculate $mathbfv$ and $mathbfn$ or $u$ and $v$ in my problem. The paper states that $mathbfv$ with components $u$ and $v$ is the velocity and $mathbfn$ is the normal vector.
Also, I would be grateful if someone could introduce some other references on the derivative of such double integrals to me.
calculus integration multivariable-calculus derivatives reference-request
edited 8 hours ago
Rodrigo de Azevedo
13.4k42165
asked 8 hours ago
SMA.DSMA.D
506421
$endgroup$
add a comment |
$begingroup$
Calculate the following derivative
$$fracddtiint_D_tF(x,y,t) , mathrm d x mathrm d y$$
where
$$D_t = lbrace (x,y) mid (x-t)^2+(y-t)^2leq r^2 rbrace$$
I've read here that the answer to the above question is in the form of:
beginalignfracddtiint_D_tF(x,y,t)dxdy &=int_partial D_t F(udy-vdx) + iint_D_tfracpartial Fpartial tdx dy \
&=iint_D_tleft[textdiv(Fmathbfv)+fracpartial Fpartial tright]dx dyendalign
which is a generalization of Leibniz integral rule.
I don't know how I can calculate $mathbfv$ and $mathbfn$ or $u$ and $v$ in my problem. The paper states that $mathbfv$ with components $u$ and $v$ is the velocity and $mathbfn$ is the normal vector.
Also, I would be grateful if someone could introduce some other references on the derivative of such double integrals to me.
calculus integration multivariable-calculus derivatives reference-request
edited 8 hours ago
Rodrigo de Azevedo
13.4k42165
asked 8 hours ago
SMA.DSMA.D
506421
$endgroup$
add a comment |
$begingroup$
Calculate the following derivative
$$fracddtiint_D_tF(x,y,t) , mathrm d x mathrm d y$$
where
$$D_t = lbrace (x,y) mid (x-t)^2+(y-t)^2leq r^2 rbrace$$
I've read here that the answer to the above question is in the form of:
beginalignfracddtiint_D_tF(x,y,t)dxdy &=int_partial D_t F(udy-vdx) + iint_D_tfracpartial Fpartial tdx dy \
&=iint_D_tleft[textdiv(Fmathbfv)+fracpartial Fpartial tright]dx dyendalign
which is a generalization of Leibniz integral rule.
I don't know how I can calculate $mathbfv$ and $mathbfn$ or $u$ and $v$ in my problem. The paper states that $mathbfv$ with components $u$ and $v$ is the velocity and $mathbfn$ is the normal vector.
Also, I would be grateful if someone could introduce some other references on the derivative of such double integrals to me.
calculus integration multivariable-calculus derivatives reference-request
edited 8 hours ago
Rodrigo de Azevedo
13.4k42165
asked 8 hours ago
SMA.DSMA.D
506421
$endgroup$
Calculate the following derivative
$$fracddtiint_D_tF(x,y,t) , mathrm d x mathrm d y$$
where
$$D_t = lbrace (x,y) mid (x-t)^2+(y-t)^2leq r^2 rbrace$$
I've read here that the answer to the above question is in the form of:
beginalignfracddtiint_D_tF(x,y,t)dxdy &=int_partial D_t F(udy-vdx) + iint_D_tfracpartial Fpartial tdx dy \
&=iint_D_tleft[textdiv(Fmathbfv)+fracpartial Fpartial tright]dx dyendalign
which is a generalization of Leibniz integral rule.
I don't know how I can calculate $mathbfv$ and $mathbfn$ or $u$ and $v$ in my problem. The paper states that $mathbfv$ with components $u$ and $v$ is the velocity and $mathbfn$ is the normal vector.
Also, I would be grateful if someone could introduce some other references on the derivative of such double integrals to me.
calculus integration multivariable-calculus derivatives reference-request
calculus integration multivariable-calculus derivatives reference-request
edited 8 hours ago
Rodrigo de Azevedo
13.4k42165
asked 8 hours ago
SMA.DSMA.D
506421
edited 8 hours ago
Rodrigo de Azevedo
13.4k42165
asked 8 hours ago
SMA.DSMA.D
506421
edited 8 hours ago
Rodrigo de Azevedo
13.4k42165
edited 8 hours ago
Rodrigo de Azevedo
13.4k42165
edited 8 hours ago
Rodrigo de Azevedo
13.4k42165
13.4k42165
asked 8 hours ago
SMA.DSMA.D
506421
asked 8 hours ago
SMA.DSMA.D
506421
asked 8 hours ago
SMA.DSMA.D
506421
506421
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Your domain $D_t$ is quite simple, you should be able to compute whatever you need. But also because of how simple it is you can just use a simple change of variables
$$ fracdd t iint_D_t F(x, y, t) d x d y= fracdd tiint_D_0 F(x+t,y+t,t)dxdy = iint_D_0 fracdd t(F(x+t,y+t,t))dxdy= iint_D_0 (partial_xF+partial_yF+partial_t F)(x+t,y+t,t)dxdy = iint_D_tpartial_xF+partial_yF+partial_t Fdxdy $$
answered 8 hours ago
Calvin KhorCalvin Khor
13.4k21540
$endgroup$
add a comment |
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1 Answer
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1 Answer
1
active
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active
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active
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votes
$begingroup$
Your domain $D_t$ is quite simple, you should be able to compute whatever you need. But also because of how simple it is you can just use a simple change of variables
$$ fracdd t iint_D_t F(x, y, t) d x d y= fracdd tiint_D_0 F(x+t,y+t,t)dxdy = iint_D_0 fracdd t(F(x+t,y+t,t))dxdy= iint_D_0 (partial_xF+partial_yF+partial_t F)(x+t,y+t,t)dxdy = iint_D_tpartial_xF+partial_yF+partial_t Fdxdy $$
answered 8 hours ago
Calvin KhorCalvin Khor
13.4k21540
$endgroup$
add a comment |
$begingroup$
Your domain $D_t$ is quite simple, you should be able to compute whatever you need. But also because of how simple it is you can just use a simple change of variables
$$ fracdd t iint_D_t F(x, y, t) d x d y= fracdd tiint_D_0 F(x+t,y+t,t)dxdy = iint_D_0 fracdd t(F(x+t,y+t,t))dxdy= iint_D_0 (partial_xF+partial_yF+partial_t F)(x+t,y+t,t)dxdy = iint_D_tpartial_xF+partial_yF+partial_t Fdxdy $$
answered 8 hours ago
Calvin KhorCalvin Khor
13.4k21540
$endgroup$
add a comment |
$begingroup$
Your domain $D_t$ is quite simple, you should be able to compute whatever you need. But also because of how simple it is you can just use a simple change of variables
$$ fracdd t iint_D_t F(x, y, t) d x d y= fracdd tiint_D_0 F(x+t,y+t,t)dxdy = iint_D_0 fracdd t(F(x+t,y+t,t))dxdy= iint_D_0 (partial_xF+partial_yF+partial_t F)(x+t,y+t,t)dxdy = iint_D_tpartial_xF+partial_yF+partial_t Fdxdy $$
answered 8 hours ago
Calvin KhorCalvin Khor
13.4k21540
$endgroup$
Your domain $D_t$ is quite simple, you should be able to compute whatever you need. But also because of how simple it is you can just use a simple change of variables
$$ fracdd t iint_D_t F(x, y, t) d x d y= fracdd tiint_D_0 F(x+t,y+t,t)dxdy = iint_D_0 fracdd t(F(x+t,y+t,t))dxdy= iint_D_0 (partial_xF+partial_yF+partial_t F)(x+t,y+t,t)dxdy = iint_D_tpartial_xF+partial_yF+partial_t Fdxdy $$
answered 8 hours ago
Calvin KhorCalvin Khor
13.4k21540
answered 8 hours ago
Calvin KhorCalvin Khor
13.4k21540
answered 8 hours ago
Calvin KhorCalvin Khor
13.4k21540
answered 8 hours ago
Calvin KhorCalvin Khor
13.4k21540
13.4k21540
add a comment |
add a comment |
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