Cross product with surface in polar coordinates
$begingroup$
I'm struggling with a problem concerning cross products. I have a perturbation to a background magnetic field given by
$delta E times B$
where $B(r(phi),phi)= frac{B_0R_E^3}{r(phi)^3} + Delta B(r)cosphi$, so $r$ is a polar curve in terms of $phi$. I was wondering how to go about the cross product? I assume that
$delta E = delta E_r hat{r} + delta E_phi hat{phi}$
but how can I decompose the functional form of B into its vector form to complete the cross product?
EDIT
Am I correct in this approach?:
$vec{B} = (r(phi),phi,B(r,phi))$
begin{vmatrix}
hat{r} & hat{phi} &hat{z} \
delta E_r & delta E_phi & 0 \
r & phi & B
end{vmatrix}
where the zero is present because I am considering B which has no height component
differential-geometry polar-coordinates
asked Dec 6 '18 at 14:43
R ThompsonR Thompson
526
$endgroup$
add a comment |
$begingroup$
I'm struggling with a problem concerning cross products. I have a perturbation to a background magnetic field given by
$delta E times B$
where $B(r(phi),phi)= frac{B_0R_E^3}{r(phi)^3} + Delta B(r)cosphi$, so $r$ is a polar curve in terms of $phi$. I was wondering how to go about the cross product? I assume that
$delta E = delta E_r hat{r} + delta E_phi hat{phi}$
but how can I decompose the functional form of B into its vector form to complete the cross product?
EDIT
Am I correct in this approach?:
$vec{B} = (r(phi),phi,B(r,phi))$
begin{vmatrix}
hat{r} & hat{phi} &hat{z} \
delta E_r & delta E_phi & 0 \
r & phi & B
end{vmatrix}
where the zero is present because I am considering B which has no height component
differential-geometry polar-coordinates
asked Dec 6 '18 at 14:43
R ThompsonR Thompson
526
$endgroup$
add a comment |
$begingroup$
I'm struggling with a problem concerning cross products. I have a perturbation to a background magnetic field given by
$delta E times B$
where $B(r(phi),phi)= frac{B_0R_E^3}{r(phi)^3} + Delta B(r)cosphi$, so $r$ is a polar curve in terms of $phi$. I was wondering how to go about the cross product? I assume that
$delta E = delta E_r hat{r} + delta E_phi hat{phi}$
but how can I decompose the functional form of B into its vector form to complete the cross product?
EDIT
Am I correct in this approach?:
$vec{B} = (r(phi),phi,B(r,phi))$
begin{vmatrix}
hat{r} & hat{phi} &hat{z} \
delta E_r & delta E_phi & 0 \
r & phi & B
end{vmatrix}
where the zero is present because I am considering B which has no height component
differential-geometry polar-coordinates
asked Dec 6 '18 at 14:43
R ThompsonR Thompson
526
$endgroup$
I'm struggling with a problem concerning cross products. I have a perturbation to a background magnetic field given by
$delta E times B$
where $B(r(phi),phi)= frac{B_0R_E^3}{r(phi)^3} + Delta B(r)cosphi$, so $r$ is a polar curve in terms of $phi$. I was wondering how to go about the cross product? I assume that
$delta E = delta E_r hat{r} + delta E_phi hat{phi}$
but how can I decompose the functional form of B into its vector form to complete the cross product?
EDIT
Am I correct in this approach?:
$vec{B} = (r(phi),phi,B(r,phi))$
begin{vmatrix}
hat{r} & hat{phi} &hat{z} \
delta E_r & delta E_phi & 0 \
r & phi & B
end{vmatrix}
where the zero is present because I am considering B which has no height component
differential-geometry polar-coordinates
differential-geometry polar-coordinates
asked Dec 6 '18 at 14:43
R ThompsonR Thompson
526
asked Dec 6 '18 at 14:43
R ThompsonR Thompson
526
edited Jan 7 at 15:39
R Thompson
asked Dec 6 '18 at 14:43
R ThompsonR Thompson
526
asked Dec 6 '18 at 14:43
R ThompsonR Thompson
526
asked Dec 6 '18 at 14:43
R ThompsonR Thompson
526
526
add a comment |
add a comment |
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