Mixing
Surface integrals, positive or negative normal?
$begingroup$
I'm unsure how to decide whether the normal should be positive or negative in $hat{n}dS=pm h_2 h_3 e_1 du_2 du_3$, where $h_i$ are the scale factors, $e_i $ are the base vectors, and $u_i$ are the curvilinear coordinates.
So for example, in spherical coordinate with $r, theta, phi$, and $S: 0<rleq2, theta = frac{pi}{4}, 0<phileq pi $. Orientation is given as $ hat{n} cdot e_z > 0 $. Here how do I know that $hat{n}dS = pm e_theta h_r h_phi dr d phi = -e_theta sin theta dr dphi $, it should be a negative?
calculus integration vectors vector-analysis orthonormal
asked Jan 15 at 1:19
user3221454user3221454
174
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add a comment |
$begingroup$
I'm unsure how to decide whether the normal should be positive or negative in $hat{n}dS=pm h_2 h_3 e_1 du_2 du_3$, where $h_i$ are the scale factors, $e_i $ are the base vectors, and $u_i$ are the curvilinear coordinates.
So for example, in spherical coordinate with $r, theta, phi$, and $S: 0<rleq2, theta = frac{pi}{4}, 0<phileq pi $. Orientation is given as $ hat{n} cdot e_z > 0 $. Here how do I know that $hat{n}dS = pm e_theta h_r h_phi dr d phi = -e_theta sin theta dr dphi $, it should be a negative?
calculus integration vectors vector-analysis orthonormal
asked Jan 15 at 1:19
user3221454user3221454
174
$endgroup$
add a comment |
$begingroup$
I'm unsure how to decide whether the normal should be positive or negative in $hat{n}dS=pm h_2 h_3 e_1 du_2 du_3$, where $h_i$ are the scale factors, $e_i $ are the base vectors, and $u_i$ are the curvilinear coordinates.
So for example, in spherical coordinate with $r, theta, phi$, and $S: 0<rleq2, theta = frac{pi}{4}, 0<phileq pi $. Orientation is given as $ hat{n} cdot e_z > 0 $. Here how do I know that $hat{n}dS = pm e_theta h_r h_phi dr d phi = -e_theta sin theta dr dphi $, it should be a negative?
calculus integration vectors vector-analysis orthonormal
asked Jan 15 at 1:19
user3221454user3221454
174
$endgroup$
I'm unsure how to decide whether the normal should be positive or negative in $hat{n}dS=pm h_2 h_3 e_1 du_2 du_3$, where $h_i$ are the scale factors, $e_i $ are the base vectors, and $u_i$ are the curvilinear coordinates.
So for example, in spherical coordinate with $r, theta, phi$, and $S: 0<rleq2, theta = frac{pi}{4}, 0<phileq pi $. Orientation is given as $ hat{n} cdot e_z > 0 $. Here how do I know that $hat{n}dS = pm e_theta h_r h_phi dr d phi = -e_theta sin theta dr dphi $, it should be a negative?
calculus integration vectors vector-analysis orthonormal
calculus integration vectors vector-analysis orthonormal
asked Jan 15 at 1:19
user3221454user3221454
174
asked Jan 15 at 1:19
user3221454user3221454
174
asked Jan 15 at 1:19
user3221454user3221454
174
asked Jan 15 at 1:19
user3221454user3221454
174
asked Jan 15 at 1:19
user3221454user3221454
174
174
add a comment |
add a comment |
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