円筒座標と球座標のデル – 日本語百科事典辞書

Del_in_cylindrical_and_spherical_coordinates
これは、一般的な曲線座標系を操作するためのベクトル計算式のリストです。

コンテンツ
1 ノート
2 座標変換
3 単位ベクトルの変換
4 デル式
4.1 重要な計算規則
5 デカルトの導出
6 円筒微分
7 球面微分
8 単位ベクトル換算式
9 こちらもご覧ください
10 参考文献
11 外部リンク

ノート
、球面座標に ISO 31-11 に取って代わる標準表記ISO 80000-2を使用します (他の情報源ではθとφの定義が逆になる場合があります)。
極角はθ ε
[ 0 π ] { theta in }
${$ : z軸と、原点を問題の点に接続する放射状ベクトルとの間の角度です。
方位角は、φ ε
[ 0 2π ]
{ varphi in }
: x軸と放射状ベクトルのxy平面への投影との間の角度です。
関数atan2 ( y ， x )は、その定義域とイメージにより、数学関数arctan ( y / x )の代わりに使用できます。従来の arctan 関数は(−π/2， +π/2)のイメージを持ちますが、atan2 は(−π， π]のイメージを持つように定義されています。

座標変換
デカルト座標、円柱座標、球座標の間の変換
から
デカルト
円筒形
球状デカルト— ρ コスφ y= ρ sin φ ぜ = ぜ { {begin{aligned}x&=rho cos varphi \y&=rho sin varphi \z&=zend{aligned}}}
r sin θ
コスφ y= r sin θ
sin φ ぜ = r コス θ { {begin{aligned}x&=rsin theta cos varphi \y&=rsin theta sin varphi \z&=rcos theta end{aligned}}}

円筒形 ρ =X2 2
アークタン ( yX ) ぜ { {begin{aligned}rho &={sqrt {x^{2}+y^{2}}}\varphi &=arctan left({frac {y}{x} }right)\z&=zend{整列}}}
— r sin θ φ =φ ぜ= r コス θ { {begin{aligned}rho &=rsin theta \varphi &=varphi \z&=rcos theta end{aligned}}}

球状 r =X2 2 2
アークタン (X 2+ y
2 ぜ )
アークタン yX )
{ {begin{aligned}r&={sqrt {x^{2}+y^{2}+z^{2}}}\theta &=arctan left({frac { sqrt {x^{2}+y^{2}}}{z}}right)\varphi &=arctan left({frac {y}{x}}right)end{整列}}}

ρ2 2
アークタン( ρ) φ= φ { {begin{aligned}r&={sqrt {rho ^{2}+z^{2}}}\theta &=arctan {left({frac {rho}{z }}right)}\varphi &=varphi end{整列}}}
—

単位ベクトルの変換
デカルト座標系、円柱座標系、球面座標系の単位ベクトル間の変換
先座標
デカルト
円筒形
球状
デカルト—X ^ = コスφ ρ ^ −
sin φφ ^ y ^ = sin φρ ^ +
コスφ φ ^ ぜ ^=ぜ
^ { {begin{aligned}{hat {mathbf {x} }}&=cos varphi {hat {boldsymbol {rho }}}-sin varphi {hat {boldsymbol { varphi }}}\{hat {mathbf {y} }}&=sin varphi {hat {boldsymbol {rho }}}+cos varphi {hat {boldsymbol {varphi }}}\{hat {mathbf {z} }}&={hat {mathbf {z} }}end{整列}}}
X ^ =
sin θ
コスφ r ^ +
コス θ コスφ θ ^ −
sin φφ ^ y ^ = sin θ
sin φr ^ +
コス θ sin φθ ^ +
コスφ φ ^ ぜ ^ = コスθ r ^ −
sin θθ
^ { {begin{aligned}{hat {mathbf {x} }}&=sin theta cos varphi {hat {mathbf {r} }}+cos theta cos varphi {hat {boldsymbol {theta }}}-sin varphi {hat {boldsymbol {varphi }}}\{hat {mathbf {y} }}&=sin theta sin varphi {hat {mathbf {r} }}+cos theta sin varphi {hat {boldsymbol {theta }}}+cos varphi {hat {boldsymbol {varphi }} }\{hat {mathbf {z} }}&=cos theta {hat {mathbf {r} }}-sin theta {hat {boldsymbol {theta }}}end {整列}}}

円筒形 ^ XX ^+ y y X 2 + y2 φ ^=− yX
^+X y X 2 + y 2ぜ ^=ぜ
^ { {begin{aligned}{hat {boldsymbol {rho }}}&={frac {x{hat {mathbf {x} }}+y{hat {mathbf {y} }}}{sqrt {x^{2}+y^{2}}}}\{hat {boldsymbol {varphi }}}&={frac {-y{hat {mathbf { x} }}+x{hat {mathbf {y} }}}{sqrt {x^{2}+y^{2}}}}\{hat {mathbf {z} }}& ={hat {mathbf {z} }}end{整列}}}
— ρ ^ =
sin θr ^ +
コスθ θ ^ φ ^=φ ^ ぜ ^ = コスθ r ^ −
sin θθ
^ { {begin{aligned}{hat {boldsymbol {rho }}}&=sin theta {hat {mathbf {r} }}+cos theta {hat {boldsymbol { theta }}}\{hat {boldsymbol {varphi }}}&={hat {boldsymbol {varphi }}}\{hat {mathbf {z} }}&=cos theta {hat {mathbf {r} }}-sin theta {hat {boldsymbol {theta }}}end{aligned}}}

球状r ^ XX
^+ y y
^+ ぜぜ X 2 + y2 + ぜ 2 θ ^= (X X^ + y y ^)ぜ −(X2 2) ぜ
^X2 2
2X2 2φ ^=− yX
^+X y X 2 + y 2
{ {begin{aligned}{hat {mathbf {r} }}&={frac {x{hat {mathbf {x} }}+y{hat {mathbf {y} } }+z{hat {mathbf {z} }}}{sqrt {x^{2}+y^{2}+z^{2}}}}\{hat {boldsymbol {theta }}}&={frac {left(x{hat {mathbf {x} }}+y{hat {mathbf {y} }}right)z-left(x^{2} +y^{2}right){hat {mathbf {z} }}}{{sqrt {x^{2}+y^{2}+z^{2}}}{sqrt {x ^{2}+y^{2}}}}}\{hat {boldsymbol {varphi }}}&={frac {-y{hat {mathbf {x} }}+x{ hat {mathbf {y} }}}{sqrt {x^{2}+y^{2}}}}end{aligned}}}
r ^=ρ ρ ^ + ぜぜ
^ ρ 2+ ぜ 2 θ ^=ぜ ρ ^ − ρ ぜ
^ ρ 2+ ぜ 2 φ ^=φ
^ { {begin{aligned}{hat {mathbf {r} }}&={frac {rho {hat {boldsymbol {rho }}}+z{hat {mathbf {z } }}}{sqrt {rho ^{2}+z^{2}}}}\{hat {boldsymbol {theta }}}&={frac {z{hat {boldsymbol {rho }}}-rho {hat {mathbf {z} }}}{sqrt {rho ^{2}+z^{2}}}}\{hat {boldsymbol { varphi }}}&={hat {boldsymbol {varphi }}}end{整列}}}
— ソース座標
に関する直交座標系、円筒座標系、および球座標系の単位ベクトル間の変換デカルト円筒形
球状
デカルト—X ^ X ρ ^ − y
φ^X 2 + y 2 y ^=y ρ ^ +X φ ^X 2+ y 2 ぜ ^=ぜ
^ { {begin{aligned}{hat {mathbf {x} }}&={frac {x{hat {boldsymbol {rho }}}-y{hat {boldsymbol {varphi }}}}{sqrt {x^{2}+y^{2}}}}\{hat {mathbf {y} }}&={frac {y{hat {boldsymbol {ロー }}}+x{hat {boldsymbol {varphi }}}}{sqrt {x^{2}+y^{2}}}}\{hat {mathbf {z} }} &={hat {mathbf {z} }}end{整列}}}
X^ X (X 2 + y2 ^+ ぜ θ
^) − yX2 2 2 φ ^X2
2X2 2 2y ^=y (X 2 + y2 ^ +ぜ θ ^) +XX2 2 2 φ ^X2
2X2 2 2ぜ ^=ぜ r ^ −X2 2θ ^X 2 + y 2 +ぜ 2
{ {begin{aligned}{hat {mathbf {x} }}&={frac {xleft({sqrt {x^{2}+y^{2}}}{hat {mathbf {r} }}+z{hat {boldsymbol {theta }}}right)-y{sqrt {x^{2}+y^{2}+z^{2}}} {hat {boldsymbol {varphi }}}}{{sqrt {x^{2}+y^{2}}}{sqrt {x^{2}+y^{2}+z^{ 2}}}}}\{hat {mathbf {y} }}&={frac {yleft({sqrt {x^{2}+y^{2}}}{hat { mathbf {r} }}+z{hat {boldsymbol {theta }}}right)+x{sqrt {x^{2}+y^{2}+z^{2}}}{ hat {boldsymbol {varphi }}}}{{sqrt {x^{2}+y^{2}}}{sqrt {x^{2}+y^{2}+z^{2 }}}}}\{hat {mathbf {z} }}&={frac {z{hat {mathbf {r} }}-{sqrt {x^{2}+y^{ 2}}}{hat {boldsymbol {theta }}}}{sqrt {x^{2}+y^{2}+z^{2}}}}end{整列}}}

円筒形
ρ ^ =
コスφX ^ +
sin φy ^ φ ^= − sin φX^ +
コスφ y ^ ぜ ^=ぜ
^ { {begin{aligned}{hat {boldsymbol {rho }}}&=cos varphi {hat {mathbf {x} }}+sin varphi {hat {mathbf { y} }}\{hat {boldsymbol {varphi }}}&=-sin varphi {hat {mathbf {x} }}+cos varphi {hat {mathbf {y} }}\{hat {mathbf {z} }}&={hat {mathbf {z} }}end{整列}}}
— ρ ^ =ρ r
^+ ぜ θ ^ ρ
2+ ぜ 2 φ ^=φ ^ ぜ ^=ぜ r
^− ρ θ ^ ρ
2+ ぜ 2
{ {begin{aligned}{hat {boldsymbol {rho }}}&={frac {rho {hat {mathbf {r} }}+z{hat {boldsymbol {シータ }}}}{sqrt {rho ^{2}+z^{2}}}}\{hat {boldsymbol {varphi }}}&={hat {boldsymbol {varphi } }}\{hat {mathbf {z} }}&={frac {z{hat {mathbf {r} }}-rho {hat {boldsymbol {theta }}}}{ sqrt {rho ^{2}+z^{2}}}}end{整列}}}

球状r ^ = sin θ( コスX ^ + sinφ y ^ ) + コスθ ぜ ^ θ ^ = コス θ ( コスX ^ + sinφ y ^ ) − sin θぜ ^ φ ^= − sin φX^ +
コスφ y
^ { {begin{aligned}{hat {mathbf {r} }}&=sin theta left(cos varphi {hat {mathbf {x} }}+sin varphi { hat {mathbf {y} }}right)+cos theta {hat {mathbf {z} }}\{hat {boldsymbol {theta }}}&=cos theta left(cos varphi {hat {mathbf {x} }}+sin varphi {hat {mathbf {y} }}right)-sin theta {hat {mathbf {z} }}\{hat {boldsymbol {varphi }}}&=-sin varphi {hat {mathbf {x} }}+cos varphi {hat {mathbf {y} }} end{整列}}}
r ^ = sin θρ ^ +
コスθ ぜ ^ θ ^ = コスθ ρ ^ −
sin θぜ ^ φ ^=φ
^ { {begin{aligned}{hat {mathbf {r} }}&=sin theta {hat {boldsymbol {rho }}}+cos theta {hat {mathbf { z} }}\{hat {boldsymbol {theta }}}&=cos theta {hat {boldsymbol {rho }}}-sin theta {hat {mathbf {z} }}\{hat {boldsymbol {varphi }}}&={hat {boldsymbol {varphi }}}end{整列}}}
—

デル式
直交座標、円筒座標、および球座標のdel演算子を
含むテーブル
手術
デカルト座標 ( x、y、z ) 円筒座標 ( ρ ， φ ， z ) 球面座標 ( r、θ、φ )。ここで、θは極角、φは方位角αです。
ベクトル場 AあXX ^ +
あy y ^ +
あぜぜ
^ { A_{x}{hat {mathbf {x} }}+A_{y}{hat {mathbf {y} }}+A_{z}{hat {mathbf {z} }} }
hat{mathbf x} + A_y hat{mathbf y}+ A_z hat{mathbf z}””>あ ρ ρ ^ +
あφ φ ^ +
あぜぜ
^ { A_{rho }{hat {boldsymbol {rho }}}+A_{varphi }{hat {boldsymbol {varphi }}}+A_{z}{hat {mathbf { z} }}}
あ r r ^ +
あθ θ ^ +
あφ φ
^ { A_{r}{hat {mathbf {r} }}+A_{theta }{hat {boldsymbol {theta }}}+A_{varphi }{hat {boldsymbol {バルフィ}}}}
hat{mathbf r} + A_theta hat{boldsymbol theta} + A_varphi hat{boldsymbol varphi}””>
勾配 ∇ f ∂ へ ∂XX ^ + ∂ へ
∂y y ^ + ∂ へ ∂ぜぜ
^ { {partial f over partial x}{hat {mathbf {x} }}+{partial f over partial y}{hat {mathbf {y} }}+{partial f over partial z}{hat {mathbf {z} }}}
∂ へ ∂ ρ ρ ^ +
1ρ ∂ へ ∂ φ φ ^+ ∂ へ ∂ ぜぜ ^ { {partial f over partial rho}{hat {boldsymbol {rho }}}+{1 over rho}{partial f over partial varphi }{hat { boldsymbol {varphi }}}+{partial f over partial z}{hat {mathbf {z} }}}
∂ へ ∂ r r ^ +1 r
∂へ ∂ θ θ ^ + 1 r sin θ∂ へ ∂ φ φ
^ { {partial f over partial r}{hat {mathbf {r} }}+{1 over r}{partial f over partial theta }{hat {boldsymbol { theta }}}+{1 over rsin theta }{partial f over partial varphi }{hat {boldsymbol {varphi }}}}

発散 ∇ ⋅ A ∂ あX ∂X + ∂ あ y
∂y + ∂ あぜ ∂ ぜ
{ {partial A_{x} over partial x}+{partial A_{y} over partial y}+{partial A_{z} over partial z}}
1 ρ ∂( ρ ) ∂ ρ + 1 ρ ∂あ φ
∂φ + ∂ あぜ ∂ ぜ
{ {1 over rho }{partial left(rho A_{rho }right) over partial rho }+{1 over rho }{partial A_{varphi } オーバー partial varphi }+{partial A_{z} over partial z}}
1 r 2 ∂ ( r 2 あ r) ∂ r + 1 r
sin θ ∂ ∂ θ ( あ θ sin θ) + 1 r
sin θ∂ あ φ
∂ φ { {1 over r^{2}}{partial left(r^{2}A_{r}right) over partial r}+{1 over rsin theta }{ partial over partial theta }left(A_{theta }sin theta right)+{1 over rsin theta }{partial A_{varphi } over partial varphi }}

カール ∇× A ( ∂あぜ ∂ y − ∂ あy ∂ ぜ )
X^ +( ∂あX ∂ ぜ − ∂ あぜ∂X )
y^ +( ∂あ y ∂X − ∂ あX ∂y )
ぜ ^ { {begin{aligned}left({frac {partial A_{z}}{partial y}}-{frac {partial A_{y}}{partial z}}right) &{hat {mathbf {x} }}\+left({frac {partial A_{x}}{partial z}}-{frac {partial A_{z}}{partial x}}right)&{hat {mathbf {y} }}\+left({frac {partial A_{y}}{partial x}}-{frac {partial A_{ x}}{partial y}}right)&{hat {mathbf {z} }}end{整列}}}
( 1ρ ∂ あぜ ∂ φ −
∂あ φ ∂ ぜ ) ρ ^ + ( ∂あ ρ ∂ ぜ − ∂ あ
ぜ∂ ρ ) φ ^ + 1 ρ ( ∂( ρ
あφ ) ∂ ρ − ∂ あρ ∂ φ )
ぜ ^ { {begin{aligned}left({frac {1}{rho }}{frac {partial A_{z}}{partial varphi }}-{frac {partial A_{ varphi }}{partial z}}right)&{hat {boldsymbol {rho }}}\+left({frac {partial A_{rho }}{partial z}} -{frac {partial A_{z}}{partial rho }}right)&{hat {boldsymbol {varphi }}}\+{frac {1}{rho }} left({frac {partial left(rho A_{varphi }right)}{partial rho }}-{frac {partial A_{rho }}{partial varphi }}右)&{hat {mathbf {z} }}end{整列}}}
1
sin θ( ∂
∂ θ ( あ φ sin θ) − ∂ あ θ ∂ φ ) r^ + 1 r( 1sinθ ∂ あ r ∂ φ − ∂ ∂ r ( r あ φ) ) θ ^ + 1 r( ∂
∂ r ( r あ θ) − ∂ あ r ∂ θ) φ
^ { {begin{aligned}{frac {1}{rsin theta }}left({frac {partial}{partial theta }}left(A_{varphi }sin theta right)-{frac {partial A_{theta }}{partial varphi }}right)&{hat {mathbf {r} }}\{}+{frac {1 }{r}}left({frac {1}{sin theta }}{frac {partial A_{r}}{partial varphi }}-{frac {partial }{partial r}}left(rA_{varphi }right)right)&{hat {boldsymbol {theta }}}\{}+{frac {1}{r}}left({ frac {partial }{partial r}}left(rA_{theta}right)-{frac {partial A_{r}}{partial theta }}right)&{hat { boldsymbol {varphi }}}end{aligned}}}

ラプラス演算子 ∇ 2 f ≡ ∆ f 2 ∂X 2 +
∂2 ∂ y 2 +
∂2 ∂ ぜ 2
{ {partial ^{2}f over partial x^{2}}+{partial ^{2}f over partial y^{2}}+{partial ^{2}f partial z^{2}}}以上
1 ρ ∂
∂ ρ ( ρ
∂へ ∂ ρ ) + 1 ρ 2 ∂2 ∂ φ 2 +
∂2 ∂ ぜ 2
{ {1 over rho }{partial over partial rho }left(rho {partial f over partial rho }right)+{1 over rho ^{2} }{partial ^{2}f over partial varphi ^{2}}+{partial ^{2}f over partial z^{2}}}
1 r 2 ∂
∂ r ( r2 ∂ へ ∂ r) + 1r 2
sin θ ∂ ∂ θ ( sin θ
∂へ ∂ θ ) + 1 r2 in 2 θ
∂2 ∂ φ 2
{ {1 over r^{2}}{partial over partial r}!left(r^{2}{partial f over partial r}right)!+! {1 over r^{2}!sin theta }{partial over partial theta }!left(sin theta {partial f over partial theta }right) !+!{1 over r^{2}!sin ^{2}theta }{partial ^{2}f over partial varphi ^{2}}}

ベクトルラプラシアン ∇ 2 A ≡ Δ A ∇ 2 あXX ^+ 2あ y y ^+ 2あぜぜ
^ { nabla ^{2}A_{x}{hat {mathbf {x} }}+nabla ^{2}A_{y}{hat {mathbf {y} }}+nabla ^ {2}A_{z}{hat {mathbf {z} }}}
( ∇
2あ ρ − あ ρ ρ 2
−2 2∂ あ φ ∂ φ ) ρ
^ + ( ∇
2あ φ − あ φ ρ 2+2 2∂ あ ρ ∂ φ )
φ ∇ 2 あ
ぜぜ
^ { {begin{aligned}{mathopen {}}left(nabla ^{2}A_{rho }-{frac {A_{rho }}{rho ^{2}}}- {frac {2}{rho ^{2}}}{frac {partial A_{varphi }}{partial varphi }}right){mathclose {}}&{hat {boldsymbol {rho }}}\+{mathopen {}}left(nabla ^{2}A_{varphi }-{frac {A_{varphi }}{rho ^{2}}}+ {frac {2}{rho ^{2}}}{frac {partial A_{rho }}{partial varphi }}right){mathclose {}}&{hat {boldsymbol {varphi }}}\{}+nabla ^{2}A_{z}&{hat {mathbf {z} }}end{aligned}}}
( ∇
2あ r − 2 あ r 2 −2 r 2
sin θ ∂ ( あ θ sinθ)∂ θ − 2 r 2
sin θ∂ あ φ ∂ φ ) r^ +( ∇
2あ θ − あ θ r 2sin 2 θ+2 2∂ あ r
∂θ − 2
コスθ r 2 sin 2 θ ∂ あφ ∂ φ ) θ
^ + ( ∇
2あ φ − あ φ r 2sin 2 θ + 2 r 2
sin θ∂ あ r
∂φ + 2
コスθ r 2 sin 2 θ ∂ あθ ∂ φ ) φ ^ { {begin{aligned}left(nabla ^{2}A_{r}-{frac {2A_{r}}{r^{2}}}-{frac {2}{r^ {2}sin theta }}{frac {partial left(A_{theta }sin theta right)}{partial theta }}-{frac {2}{r^{2 }sin theta }}{frac {partial A_{varphi }}{partial varphi }}right)&{hat {mathbf {r} }}\+left(nabla ^ {2}A_{theta}-{frac {A_{theta}}{r^{2}sin^{2}theta}}+{frac{2}{r^{2}}} {frac {partial A_{r}}{partial theta }}-{frac {2cos theta}{r^{2}sin ^{2}theta }}{frac {部分 A_{varphi }}{部分 varphi }}right)&{hat {boldsymbol {theta }}}\+left(nabla ^{2}A_{varphi }-{ frac {A_{varphi }}{r^{2}sin ^{2}theta }}+{frac {2}{r^{2}sin theta }}{frac {partial A_ {r}}{partial varphi }}+{frac {2cos theta}{r^{2}sin ^{2}theta }}{frac {partial A_{theta }} {partial varphi }}right)&{hat {boldsymbol {varphi }}}end{aligned}}}

物質微分α ( A ⋅ ∇) Bあ ⋅ ∇ BXX ^ +
あ⋅ ∇ B y y ^ +
あ⋅ ∇ B ぜぜ
^ { mathbf {A} cdot nabla B_{x}{hat {mathbf {x} }}+mathbf {A} cdot nabla B_{y}{hat {mathbf {y} }}+mathbf {A} cdot nabla B_{z}{hat {mathbf {z} }}}
( あρ ∂ B ρ ∂ ρ + あ φρ ∂ B ρ ∂ φ +
あぜ ∂ B ρ ∂ ぜ −
あφ B φ ρ ) ρ ^ + ( あρ ∂ B φ ∂ ρ + あ φρ ∂ B φ ∂ φ +
あぜ ∂ B φ ∂ ぜ +
あφ B ρ ρ ) φ ^ + ( あρ ∂ B ぜ ∂ ρ + あ φρ ∂ B ぜ ∂ φ +
あぜ ∂ B ぜ ∂ ぜ )
ぜ ^ { {begin{aligned}left(A_{rho }{frac {partial B_{rho }}{partial rho }}+{frac {A_{varphi }}{rho }}{frac {partial B_{rho }}{partial varphi }}+A_{z}{frac {partial B_{rho }}{partial z}}-{frac {A_ {varphi }B_{varphi }}{rho }}right)&{hat {boldsymbol {rho }}}\+left(A_{rho }{frac {partial B_{ varphi }}{partial rho }}+{frac {A_{varphi }}{rho }}{frac {partial B_{varphi }}{partial varphi }}+A_{z }{frac {partial B_{varphi }}{partial z}}+{frac {A_{varphi }B_{rho }}{rho }}right)&{hat {boldsymbol {varphi }}}\+left(A_{rho }{frac {partial B_{z}}{partial rho }}+{frac {A_{varphi }}{rho } }{frac {partial B_{z}}{partial varphi }}+A_{z}{frac {partial B_{z}}{partial z}}right)&{hat { mathbf {z} }}end{aligned}}}
( あr ∂ B r
∂r + あ θ r ∂ B r ∂θ + あ φ r
sin θ∂ B r
∂φ − あ θ
Bθ + あ φ
Bφ r ) r ^ +( あr ∂ B θ ∂ r +
あθ r ∂ B θ ∂ θ+ あ φ r
sin θ∂ B θ ∂ φ + あ θ Br r − あ φ
B φ ベビーベッドθ r ) θ
^ + ( あr ∂ B φ ∂ r +
あθ r ∂ B φ ∂ θ+あ φ r sin θ ∂ Bφ ∂ φ + A φ
Br r + A φ
Bθ cot θ r ) φ ^ { {begin{aligned}left(A_{r}{frac {partial B_{r}}{partial r}}+{frac {A_{theta }}{r}}{frac {partial B_{r}}{partial theta }}+{frac {A_{varphi }}{rsin theta }}{frac {partial B_{r}}{partial varphi }}-{frac {A_{theta }B_{theta }+A_{varphi }B_{varphi }}{r}}right)&{hat {mathbf {r} }}\+left(A_{r}{frac {partial B_{theta }}{partial r}}+{frac {A_{theta }}{r}}{frac {partial B_{theta }}{partial theta }}+{frac {A_{varphi }}{rsin theta }}{frac {partial B_{theta }}{partial varphi }}+{frac {A_{theta }B_{r}}{r}}-{frac {A_{varphi }B_{varphi }cot theta }{r}}right)&{hat {boldsymbol {theta }}}\+left(A_{r}{frac {partial B_{varphi }}{partial r}}+{frac {A_{theta }}{r}}{frac {partial B_{varphi }}{partial theta }}+{frac {A_{varphi }}{rsin theta }}{frac {partial B_{varphi }}{partial varphi }}+{frac {A_{varphi }B_{r}}{r}}+{frac {A_{varphi }B_{theta }cot theta }{r}}right)&{hat {boldsymbol {varphi }}}end{aligned}}}
Tensor ∇ ⋅ T (not to be confused with 2nd order tensor divergence)∂ T x x
∂x + ∂ T y x
∂y + ∂ T z x
∂z ) x ^ +(∂T x y
∂x + ∂ T y y
∂y + ∂ T z y
∂z ) y ^ +(∂T x z
∂x + ∂ T y z
∂y + ∂ T z z
∂z ) z
^ { {begin{aligned}left({frac {partial T_{xx}}{partial x}}+{frac {partial T_{yx}}{partial y}}+{frac {partial T_{zx}}{partial z}}right)&{hat {mathbf {x} }}\+left({frac {partial T_{xy}}{partial x}}+{frac {partial T_{yy}}{partial y}}+{frac {partial T_{zy}}{partial z}}right)&{hat {mathbf {y} }}\+left({frac {partial T_{xz}}{partial x}}+{frac {partial T_{yz}}{partial y}}+{frac {partial T_{zz}}{partial z}}right)&{hat {mathbf {z} }}end{aligned}}}
[∂T ρ ρ ∂ ρ + 1
ρ∂ T φ ρ ∂ φ +
∂T z ρ ∂ z + 1 ρ ( Tρ ρ − T φ φ )] ρ
^ + [∂T ρ φ ∂ ρ + 1
ρ∂ T φ φ ∂ φ +
∂T z φ ∂ z + 1 ρ ( Tρ φ + T φ ρ )] φ
^ + [∂T ρ z
∂ρ + 1 ρ ∂ T φ z ∂φ + ∂ T z z
∂z + T ρ z
ρ] z
^ { {begin{aligned}left[{frac {partial T_{rho rho }}{partial rho }}+{frac {1}{rho }}{frac {partial T_{varphi rho }}{partial varphi }}+{frac {partial T_{zrho }}{partial z}}+{frac {1}{rho }}(T_{rho rho }-T_{varphi varphi })right]&{hat {boldsymbol {rho }}}\+left[{frac {partial T_{rho varphi }}{partial rho }}+{frac {1}{rho }}{frac {partial T_{varphi varphi }}{partial varphi }}+{frac {partial T_{zvarphi }}{partial z}}+{frac {1}{rho }}(T_{rho varphi }+T_{varphi rho })right]&{hat {boldsymbol {varphi }}}\+left[{frac {partial T_{rho z}}{partial rho }}+{frac {1}{rho }}{frac {partial T_{varphi z}}{partial varphi }}+{frac {partial T_{zz}}{partial z}}+{frac {T_{rho z}}{rho }}right]&{hat {mathbf {z} }}end{aligned}}}
[∂T r r
∂r + 2 T r r
r+ 1 r ∂ T θ r
∂θ + cot θ r T θ r +1 sin θ ∂ T φ r
∂φ − 1 r( Tθ θ + T φ φ )] r ^ +
[∂T r θ ∂ r + 2
Tr θ r + 1 r ∂
Tθ θ ∂ θ + cot θr T θ θ+1 sin θ ∂ T φ θ ∂φ + T θ r
r− cot θ r T φ φ] θ
^ + [∂T r φ ∂ r + 2
Tr φ r + 1 r ∂T θ φ ∂ θ+1 sin θ ∂ T φ φ ∂φ + T φ r
r+ cot θ r( Tθ φ + T φ θ )] φ ^ { {begin{aligned}left[{frac {partial T_{rr}}{partial r}}+2{frac {T_{rr}}{r}}+{frac {1}{r}}{frac {partial T_{theta r}}{partial theta }}+{frac {cot theta }{r}}T_{theta r}+{frac {1}{rsin theta }}{frac {partial T_{varphi r}}{partial varphi }}-{frac {1}{r}}(T_{theta theta }+T_{varphi varphi })right]&{hat {mathbf {r} }}\+left[{frac {partial T_{rtheta }}{partial r}}+2{frac {T_{rtheta }}{r}}+{frac {1}{r}}{frac {partial T_{theta theta }}{partial theta }}+{frac {cot theta }{r}}T_{theta theta }+{frac {1}{rsin theta }}{frac {partial T_{varphi theta }}{partial varphi }}+{frac {T_{theta r}}{r}}-{frac {cot theta }{r}}T_{varphi varphi }right]&{hat {boldsymbol {theta }}}\+left[{frac {partial T_{rvarphi }}{partial r}}+2{frac {T_{rvarphi }}{r}}+{frac {1}{r}}{frac {partial T_{theta varphi }}{partial theta }}+{frac {1}{rsin theta }}{frac {partial T_{varphi varphi }}{partial varphi }}+{frac {T_{varphi r}}{r}}+{frac {cot theta }{r}}(T_{theta varphi }+T_{varphi theta })right]&{hat {boldsymbol {varphi }}}end{aligned}}}
Differential displacement dℓd x x ^ + d yy ^ + d z z ^ { dx，{hat {mathbf {x} }}+dy，{hat {mathbf {y} }}+dz，{hat {mathbf {z} }}}
d ρ ρ ^ + ρ d
φφ ^ + d z z
^ { drho ，{hat {boldsymbol {rho }}}+rho ，dvarphi ，{hat {boldsymbol {varphi }}}+dz，{hat {mathbf {z} }}}
d r r ^ + r d
θθ ^ + r sin θ dφ φ
^ { dr，{hat {mathbf {r} }}+r，dtheta ，{hat {boldsymbol {theta }}}+r，sin theta ，dvarphi ，{hat {boldsymbol {varphi }}}}

Differential normal area dSd y d z x ^ +
dx d z y ^ + dx d y z
^ { {begin{aligned}dy，dz&，{hat {mathbf {x} }}\{}+dx，dz&，{hat {mathbf {y} }}\{}+dx，dy&，{hat {mathbf {z} }}end{aligned}}}
ρ d φ d z
ρ^ + d ρ d z
φ^ + ρ d ρ d φz
^ { {begin{aligned}rho ，dvarphi ，dz&，{hat {boldsymbol {rho }}}\{}+drho ，dz&，{hat {boldsymbol {varphi }}}\{}+rho ，drho ，dvarphi &，{hat {mathbf {z} }}end{aligned}}}
r 2 sin θ d θ d
φr ^ + r sin θ dr d φ
θ^ + r d r d θ
φ ^ { {begin{aligned}r^{2}sin theta ，dtheta ，dvarphi &，{hat {mathbf {r} }}\{}+rsin theta ，dr，dvarphi &，{hat {boldsymbol {theta }}}\{}+r，dr，dtheta &，{hat {boldsymbol {varphi }}}end{aligned}}}

Differential volume dVd x d y d z
{ dx，dy，dz}
ρ d ρ d φ d z
{ rho ，drho ，dvarphi ，dz}
r 2 in θ d r dθ d φ
{ r^{2}sin theta ，dr，dtheta ，dvarphi }

^α This page uses θ { theta }

for the polar angle and φ { varphi }

for the azimuthal angle， which is common notation in physics. The source that is used for these formulae uses θ { theta }

for the azimuthal angle and φ { varphi }

for the polar angle， which is common mathematical notation. In order to get the mathematics formulae， switch θ { theta }
and φ
{ varphi }

in the formulae shown in the table above.
Non-trivial calculation rulesEditiv grad f ≡ ∇ ⋅ ∇f ≡ ∇
2 f { operatorname {div} ，operatorname {grad} fequiv nabla cdot nabla fequiv nabla ^{2}f}

curl grad f ≡ ∇ × ∇f = 0
{ operatorname {curl} ，operatorname {grad} fequiv nabla times nabla f=mathbf {0} }
equiv nabla times nabla f = mathbf 0″”>
div curl A
≡∇ ⋅( ∇× A) = 0
{ operatorname {div} ，operatorname {curl} mathbf {A} equiv nabla cdot (nabla times mathbf {A} )=0}

curl curl A
≡∇ ×( ∇× A) = ∇( ∇⋅ A) − ∇
2 A { operatorname {curl} ，operatorname {curl} mathbf {A} equiv nabla times (nabla times mathbf {A} )=nabla (nabla cdot mathbf {A} )-nabla ^{2}mathbf {A} }

(Lagrange’s formula for del) ∇ 2( fg ) = f ∇
2g + 2 ∇ f ⋅ ∇g + g ∇
2 f { nabla ^{2}(fg)=fnabla ^{2}g+2nabla fcdot nabla g+gnabla ^{2}f}

デカルトの導出

iv A = im V
0∬ ∂ V A ⋅ d S∭ V
dV = A x x + d
x) d y d z − A
x x ) d y d z +
Ay y + d y ) d
xd z − A y y )
dx d z + A z z+d z ) d x d y
−A z z ) d x d
yd x d y d z =
∂A x ∂ x + ∂ A
y∂ y + ∂ A z ∂ z { {begin{aligned}operatorname {div} mathbf {A} =lim _{Vto 0}{frac {iint _{partial V}mathbf {A} cdot dmathbf {S} }{iiint _{V}dV}}&={frac {A_{x}(x+dx)，dy，dz-A_{x}(x)，dy，dz+A_{y}(y+dy)，dx，dz-A_{y}(y)，dx，dz+A_{z}(z+dz)，dx，dy-A_{z}(z)，dx，dy}{dx，dy，dz}}\&={frac {partial A_{x}}{partial x}}+{frac {partial A_{y}}{partial y}}+{frac {partial A_{z}}{partial z}}end{aligned}}}
curl A) x=im S ⊥ x
^ 0∫ ∂ S A ⋅ d ℓ∬ S
dS = A z y + d
y) d z − A z y
)d z + A y z )
dy − A y z + d
z) d y d y d z=∂ A z ∂ y − ∂A y ∂ z
{ {begin{aligned}(operatorname {curl} mathbf {A} )_{x}=lim _{S^{perp mathbf {hat {x}} }to 0}{frac {int _{partial S}mathbf {A} cdot dmathbf {ell } }{iint _{S}dS}}&={frac {A_{z}(y+dy)，dz-A_{z}(y)，dz+A_{y}(z)，dy-A_{y}(z+dz)，dy}{dy，dz}}\&={frac {partial A_{z}}{partial y}}-{frac {partial A_{y}}{partial z}}end{aligned}}}

The expressions for( curlA) y
{ (operatorname {curl} mathbf {A} )_{y}}

and( curlA) z
{ (operatorname {curl} mathbf {A} )_{z}}

are found in the same way.

円筒微分

iv A = im V
0∬ ∂ V A ⋅ d S∭ V
dV = A ρ( ρ+ d ρ )( ρ+ d ρ ) d ϕ dz − A ρ( ρ) ρ d ϕ d z +A ϕ( ϕ+ d ϕ ) d ρ dz − A ϕ( ϕ) d ρ d z + A
z z + d z ) d ρ( ρ+ d ρ / 2 ) d
ϕ− A z z ) d ρ( ρ+ d ρ / 2 ) d ϕ ρd ϕ d ρ d z =1 ρ ∂( ρA ρ ) ∂ ρ + 1ρ ∂ A ϕ
∂ϕ + ∂ A z ∂ z
{ {begin{aligned}operatorname {div} mathbf {A} &=lim _{Vto 0}{frac {iint _{partial V}mathbf {A} cdot dmathbf {S} }{iiint _{V}dV}}\&={frac {A_{rho }(rho +drho )(rho +drho )，dphi ，dz-A_{rho }(rho )rho ，dphi ，dz+A_{phi }(phi +dphi )，drho ，dz-A_{phi }(phi )，drho ，dz+A_{z}(z+dz)，drho ，(rho +drho /2)，dphi -A_{z}(z)，drho (rho +drho /2)，dphi }{rho ，dphi ，drho ，dz}}\&={frac {1}{rho }}{frac {partial (rho A_{rho })}{partial rho }}+{frac {1}{rho }}{frac {partial A_{phi }}{partial phi }}+{frac {partial A_{z}}{partial z}}end{aligned}}}
curl A) ρ = im S ⊥ρ
^ 0∫ ∂ S A ⋅ d ℓ∬ S
dS = A ϕ( z ) ( ρ+ d ρ ) d ϕ −A ϕ( z+ d z )( ρ+ d ρ ) d ϕ +A z( ϕ+ d ϕ ) d z −A z( ϕ) d z( ρ+ d ρ ) d ϕ dz = − ∂ A ϕ
∂z + 1 ρ ∂ A z∂ ϕ
{ {begin{aligned}(operatorname {curl} mathbf {A} )_{rho }&=lim _{S^{perp {boldsymbol {hat {rho }}}}to 0}{frac {int _{partial S}mathbf {A} cdot dmathbf {ell } }{iint _{S}dS}}\&={frac {A_{phi }(z)(rho +drho )，dphi -A_{phi }(z+dz)(rho +drho )，dphi +A_{z}(phi +dphi )，dz-A_{z}(phi )，dz}{(rho +drho )，dphi ，dz}}\&=-{frac {partial A_{phi }}{partial z}}+{frac {1}{rho }}{frac {partial A_{z}}{partial phi }}end{aligned}}}
curl A) ϕ = im S ⊥ϕ
^ 0∫ ∂ S A ⋅ d ℓ∬ S
dS = A z( ρ) d z − A z( ρ+ d ρ ) d z +A ρ( z+ d z ) d ρ −A ρ( z) d ρ
dρ d z = − ∂ Az ∂ ρ + ∂ A ρ ∂ z { {begin{aligned}(operatorname {curl} mathbf {A} )_{phi }&=lim _{S^{perp {boldsymbol {hat {phi }}}}to 0}{frac {int _{partial S}mathbf {A} cdot dmathbf {ell } }{iint _{S}dS}}\&={frac {A_{z}(rho )，dz-A_{z}(rho +drho )，dz+A_{rho }(z+dz)，drho -A_{rho }(z)，drho }{drho ，dz}}\&=-{frac {partial A_{z}}{partial rho }}+{frac {partial A_{rho }}{partial z}}end{aligned}}}
curl A) z = im S ⊥z
^ 0∫ ∂ S A ⋅ d ℓ∬ S
dS = A ρ( ϕ) d ρ − A ρ( ϕ+ d ϕ ) d ρ +A ϕ( ρ+ d ρ )( ρ+ d ρ ) d ϕ −A ϕ( ρ) ρ d ϕ
ρd ρ d ϕ=− 1 ρ ∂ A ρ
∂ϕ + 1 ρ ∂( ρA ϕ ) ∂ ρ
{ {begin{aligned}(operatorname {curl} mathbf {A} )_{z}&=lim _{S^{perp {boldsymbol {hat {z}}}}to 0}{frac {int _{partial S}mathbf {A} cdot dmathbf {ell } }{iint _{S}dS}}\&={frac {A_{rho }(phi )，drho -A_{rho }(phi +dphi )，drho +A_{phi }(rho +drho )(rho +drho )，dphi -A_{phi }(rho )rho ，dphi }{rho ，drho ，dphi }}\&=-{frac {1}{rho }}{frac {partial A_{rho }}{partial phi }}+{frac {1}{rho }}{frac {partial (rho A_{phi })}{partial rho }}end{aligned}}}
url A =( curlA) ρ ρ ^ +( curlA) ϕ ϕ ^ +( curlA) z
z ^ = 1 ρ
∂ ∂ ϕ ∂ z) ρ ^ +( ∂ ∂ z
− ∂ ρ ) ϕ ^ + 1 ρ ( ∂( ρ ) ∂
ρ ∂ ϕ ) z
^ { {begin{aligned}operatorname {curl} mathbf {A} &=(operatorname {curl} mathbf {A} )_{rho }{hat {boldsymbol {rho }}}+(operatorname {curl} mathbf {A} )_{phi }{hat {boldsymbol {phi }}}+(operatorname {curl} mathbf {A} )_{z}{hat {boldsymbol {z}}}\&=left({frac {1}{rho }}{frac {partial A_{z}}{partial phi }}-{frac {partial A_{phi }}{partial z}}right){hat {boldsymbol {rho }}}+left({frac {partial A_{rho }}{partial z}}-{frac {partial A_{z}}{partial rho }}right){hat {boldsymbol {phi }}}+{frac {1}{rho }}left({frac {partial (rho A_{phi })}{partial rho }}-{frac {partial A_{rho }}{partial phi }}right){hat {boldsymbol {z}}}end{aligned}}}

球面微分

div A = im V
0∬ ∂ V A ⋅ d S∭ V
dV = A r r + dr )( r+ d r ) d θ( r+ d r ) sin θ d
ϕ− A r r ) r d
θr sin θ d ϕ + A θ ( θ+ d θ ) sin( θ+ d θ ) r d rd ϕ − A θ( θ) sin( θ) r d r d ϕ +A ϕ( ϕ+ d ϕ ) r d rd θ − A ϕ( ϕ) r d r d θ
dr r d θ r sin θ d ϕ 1r 2 ∂( r2 A r) ∂ r + 1r sin θ ∂( Aθ sin θ ) ∂ θ +1 r sin θ ∂ A ϕ ∂ ϕ { {begin{aligned}operatorname {div} mathbf {A} &=lim _{Vto 0}{frac {iint _{partial V}mathbf {A} cdot dmathbf {S} }{iiint _{V}dV}}\&={frac {A_{r}(r+dr)(r+dr)，dtheta ，(r+dr)sin theta ，dphi -A_{r}(r)r，dtheta ，rsin theta ，dphi +A_{theta }(theta +dtheta )sin(theta +dtheta )r，dr，dphi -A_{theta }(theta )sin(theta )r，dr，dphi +A_{phi }(phi +dphi )r，dr，dtheta -A_{phi }(phi )r，dr，dtheta }{dr，r，dtheta ，rsin theta ，dphi }}\&={frac {1}{r^{2}}}{frac {partial (r^{2}A_{r})}{partial r}}+{frac {1}{rsin theta }}{frac {partial (A_{theta }sin theta )}{partial theta }}+{frac {1}{rsin theta }}{frac {partial A_{phi }}{partial phi }}end{aligned}}}
curl A) r=im S ⊥ r
^ 0∫ ∂ S A ⋅ d ℓ∬ S
dS = A θ( ϕ) r d θ + A ϕ( θ+ d θ ) r sin( θ+ d θ ) d ϕ −A θ( ϕ+ d ϕ ) r d θ− A ϕ( θ) r sin( θ) d ϕ
rd θ r sin θ d
ϕ 1r sin θ ∂( Aϕ sin θ ) ∂ θ −1 r sin θ ∂ A θ ∂ ϕ { {begin{aligned}(operatorname {curl} mathbf {A} )_{r}=lim _{S^{perp {boldsymbol {hat {r}}}}to 0}{frac {int _{partial S}mathbf {A} cdot dmathbf {ell } }{iint _{S}dS}}&={frac {A_{theta }(phi )r，dtheta +A_{phi }(theta +dtheta )rsin(theta +dtheta )，dphi -A_{theta }(phi +dphi )r，dtheta -A_{phi }(theta )rsin(theta )，dphi }{r，dtheta ，rsin theta ，dphi }}\&={frac {1}{rsin theta }}{frac {partial (A_{phi }sin theta )}{partial theta }}-{frac {1}{rsin theta }}{frac {partial A_{theta }}{partial phi }}end{aligned}}}
curl A) θ = im S ⊥θ
^ 0∫ ∂ S A ⋅ d ℓ∬ S
dS = A ϕ( r) r sin θ d ϕ +A r( ϕ+ d ϕ ) d r −A ϕ( r+ d r )( r+ d r ) sin θ dϕ − A r( ϕ) d r d r r sinθ d
ϕ 1r sin θ ∂ A r ∂ϕ − 1 r ∂ ( rA ϕ ) ∂ r
{ {begin{aligned}(operatorname {curl} mathbf {A} )_{theta }=lim _{S^{perp {boldsymbol {hat {theta }}}}to 0}{frac {int _{partial S}mathbf {A} cdot dmathbf {ell } }{iint _{S}dS}}&={frac {A_{phi }(r)rsin theta ，dphi +A_{r}(phi +dphi )，dr-A_{phi }(r+dr)(r+dr)sin theta ，dphi -A_{r}(phi )，dr}{dr，rsin theta ，dphi }}\&={frac {1}{rsin theta }}{frac {partial A_{r}}{partial phi }}-{frac {1}{r}}{frac {partial (rA_{phi })}{partial r}}end{aligned}}}
curl A) ϕ = im S ⊥ϕ
^ 0∫ ∂ S A ⋅ d ℓ∬ S
dS = A r( θ) d r + A θ( r+ d r )( r+ d r ) d θ −A r( θ+ d θ ) d r −A θ( r) r d θ
rd r d
θ 1 r ∂( rA θ ) ∂ r − 1 r ∂A r ∂ θ
{ {begin{aligned}(operatorname {curl} mathbf {A} )_{phi }=lim _{S^{perp {boldsymbol {hat {phi }}}}to 0}{frac {int _{partial S}mathbf {A} cdot dmathbf {ell } }{iint _{S}dS}}&={frac {A_{r}(theta )，dr+A_{theta }(r+dr)(r+dr)，dtheta -A_{r}(theta +dtheta )，dr-A_{theta }(r)r，dtheta }{r，dr，dtheta }}\&={frac {1}{r}}{frac {partial (rA_{theta })}{partial r}}-{frac {1}{r}}{frac {partial A_{r}}{partial theta }}end{aligned}}}
curl A =( curlA ) r r ^ + ( curlA ) θ
θ ( curlA ) ϕ
ϕ 1r sin θ (∂ ( Aϕ sin θ )
∂θ − ∂ A θ ∂ ϕ) r
^ 1 r ( 1in θ ∂ A r
∂ϕ − ∂( rA ϕ )
∂r )
θ 1 r (∂( rA θ )
∂r − ∂ A r
∂θ ) ϕ ^ { operatorname {curl} mathbf {A} =(operatorname {curl} mathbf {A} )_{r}，{hat {boldsymbol {r}}}+(operatorname {curl} mathbf {A} )_{theta }，{hat {boldsymbol {theta }}}+(operatorname {curl} mathbf {A} )_{phi }，{hat {boldsymbol {phi }}}={frac {1}{rsin theta }}left({frac {partial (A_{phi }sin theta )}{partial theta }}-{frac {partial A_{theta }}{partial phi }}right){hat {boldsymbol {r}}}+{frac {1}{r}}left({frac {1}{sin theta }}{frac {partial A_{r}}{partial phi }}-{frac {partial (rA_{phi })}{partial r}}right){hat {boldsymbol {theta }}}+{frac {1}{r}}left({frac {partial (rA_{theta })}{partial r}}-{frac {partial A_{r}}{partial theta }}right){hat {boldsymbol {phi }}}}

単位ベクトル換算式
The unit vector of a coordinate parameter u is defined in such a way that a small positive change in u causes the position vector r { mathbf {r} }

to change in u { mathbf {u} }

direction. Therefore， ∂r ∂
u= ∂
s ∂ u u { {frac {partial {mathbf {r} }}{partial u}}={frac {partial {s}}{partial u}}mathbf {u} }
where s is the arc length parameter.
For two sets of coordinate systems
u i { u_{i}}

and
v j { v_{j}}

， according to chain rule，
dr = ∑ i ∂ r ∂ d u i = ∑ i ∂ s ∂ u ^ i d u i =∑ j ∂ s ∂ v
^ d v j =∑ j ∂ s ∂ v
^ ∑ i
∂
∂ d u i = ∑ i ∑j ∂ s ∂
∂
∂ v
^ d u i .
{ dmathbf {r} =sum _{i}{frac {partial mathbf {r} }{partial u_{i}}}，du_{i}=sum _{i}{frac {partial s}{partial u_{i}}}{hat {mathbf {u} }}_{i}du_{i}=sum _{j}{frac {partial s}{partial v_{j}}}{hat {mathbf {v} }}_{j}，dv_{j}=sum _{j}{frac {partial s}{partial v_{j}}}{hat {mathbf {v} }}_{j}sum _{i}{frac {partial v_{j}}{partial u_{i}}}，du_{i}=sum _{i}sum _{j}{frac {partial s}{partial v_{j}}}{frac {partial v_{j}}{partial u_{i}}}{hat {mathbf {v} }}_{j}，du_{i}.}

Now， we isolate the i { i}

th component. Fori ≠ k
{ i{neq }k}

， let k 0 { mathrm {d} u_{k}=0}

. Then divide on both sides by i { mathrm {d} u_{i}}

to get:
∂ u ^ i = ∑ j ∂ s ∂
∂
∂ v
^ .
{ {frac {partial s}{partial u_{i}}}{hat {mathbf {u} }}_{i}=sum _{j}{frac {partial s}{partial v_{j}}}{frac {partial v_{j}}{partial u_{i}}}{hat {mathbf {v} }}_{j}.}

こちらもご覧くださいDel Orthogonal coordinates Curvilinear coordinates
Vector fields in cylindrical and spherical coordinates

参考文献
^ Griffiths， David J. （2012). Introduction to Electrodynamics. Pearson. ISBN 978-0-321-85656-2.
^ Arfken， George; Weber， Hans; Harris， Frank （2012). Mathematical Methods for Physicists (Seventh ed.). Academic Press. p. 192. ISBN 9789381269558.
^ Weisstein， Eric W. “Convective Operator”. Mathworld. Retrieved 23 March 2011.

外部リンク
Maxima Computer Algebra system scripts to generate some of these operators in cylindrical and spherical coordinates.”