handout_af1_Differential_Operators_in_Curvilinear_Coordinates
PDF · 6 pages · 232.0 KB
Open PDF file
Reference handout in the style of the NRL Plasma Formulary pages, listing vector identities, integral theorems (divergence, Stokes and Green type), and the divergence, gradient, curl and Laplacian in cylindrical and spherical coordinates. It also gives the Laplacian of a vector, components of (A·∇)B, and divergence of a tensor in each system. The text is garbled by symbol encoding, but the formulas are recognizable.
AI-written summary; may contain errors. This description is approximate.
Extracted text (machine-read; may contain errors)
VECTOR IDENTITIES
/4Notation/: f/; g /; are scalars/; A /, B /, etc/./, are v ectors/; T is a tensor/; I is the unitdy ad/./#28/1/#29 A /#01 B /#02 C /= A /#02 B /#01 C /= B /#01 C /#02 A /= B /#02 C /#01 A /= C /#01 A /#02 B /= C /#02 A /#01 B/#28/2/#29 A /#02 /#28 B /#02 C /#29/= /#28 C /#02 B /#29 /#02 A /=/#28 A /#01 C /#29 B /, /#28 A /#01 B /#29 C/#28/3/#29 A /#02 /#28 B /#02 C /#29/+ B /#02 /#28 C /#02 A /#29/+ C /#02 /#28 A /#02 B /#29/= /0/#28/4/#29 /#28 A /#02 B /#29 /#01 /#28 C /#02 D /#29/=/#28 A /#01 C /#29/#28 B /#01 D /#29 /, /#28 A /#01 D /#29/#28 B /#01 C /#29/#28/5/#29 /#28 A /#02 B /#29 /#02 /#28 C /#02 D /#29/= /#28 A /#02 B /#01 D /#29 C /, /#28 A /#02 B /#01 C /#29 D/#28/6/#29 r /#28 fg /#29/= r /#28 gf /#29/= f r g /+ g r f/#28/7/#29 r/#01 /#28 f A /#29/= f r/#01 A /+ A /#01r f/#28/8/#29 r/#02 /#28 f A /#29/= f r/#02 A /+ r f /#02 A/#28/9/#29 r/#01 /#28 A /#02 B /#29/= B /#01r/#02 A /, A /#01r/#02 B/#28/1/0/#29 r/#02 /#28 A /#02 B /#29/= A /#28 r/#01 B /#29 /, B /#28 r/#01 A /#29/+/#28 B /#01r /#29 A /, /#28 A /#01r /#29 B/#28/1/1/#29 A /#02 /#28 r/#02 B /#29/= /#28 r B /#29 /#01 A /, /#28 A /#01r /#29 B/#28/1/2/#29 r /#28 A /#01 B /#29/= A /#02 /#28 r/#02 B /#29/+ B /#02 /#28 r/#02 A /#29/+/#28 A /#01r /#29 B /+/#28 B /#01r /#29 A/#28/1/3/#29 r
/2f /= r/#01r f/#28/1/4/#29 r
/2A /= r /#28 r/#01 A /#29 /,r/#02r/#02 A/#28/1/5/#29 r/#02 r f /=/0/#28/1/6/#29 r/#01r/#02 A /=/0If e/1
/,e/2
/,e/3
are orthonormal unit v ectors/, a second/-order tensor T can b ewritten in the dy adic form/#28/1/7/#29 T /=
Pi/;j
Tij
ei
ejIn cartesian co ordinates the div ergence of a tensor is a v ector with comp onen ts/#28/1/8/#29 /#28 r/#01 T /#29i
/=
Pj
/#28 /@Tji
/=/@ xj
/#29/#5BThis de/#0Cnition is required for consistency with Eq/. /#28/2/9/#29/#5D/. In general/#28/1/9/#29 r/#01 /#28 AB /#29/= /#28 r/#01 A /#29 B /+/#28 A /#01r /#29 B/#28/2/0/#29 r/#01 /#28 f T /#29/= r f /#01 T /+ f r/#01 T/4
Let r /= i x /+ j y /+ k z b e the radius v ector of magnitude r /, from the origin tothe p oin t x/; y /; z /. Then/#28/2/1/#29 r/#01 r /=/3/#28/2/2/#29 r/#02 r /=/0/#28/2/3/#29 r r /= r /=r/#28/2/4/#29 r /#28/1 /=r /#29/= /, r /=r
/3/#28/2/5/#29 r/#01 /#28 r /=r
/3/#29/=/4 /#19/#0E /#28 r /#29/#28/2/6/#29 r r /= IIf V is a v olume enclosed b y a surface S and d S /= n dS /, where n is the unitnormal out w ard from V/;/#28/2/7/#29
ZV
dV r f /=
ZS
d S f/#28/2/8/#29
ZV
dV r/#01 A /=
ZS
d S /#01 A/#28/2/9/#29
ZV
dV r/#01 T /=
ZS
d S /#01 T/#28/3/0/#29
ZV
dV r/#02 A /=
ZS
d S /#02 A/#28/3/1/#29
ZV
dV /#28 f r
/2g /, g r
/2f /#29/=
ZS
d S /#01 /#28 f r g /, g r f /#29/#28/3/2/#29
ZV
dV /#28 A /#01r/#02r/#02 B /, B /#01r/#02r/#02 A /#29/=
ZS
d S /#01 /#28 B /#02r/#02 A /, A /#02r/#02 B /#29If S is an op en surface b ounded b y the con tour C /, of whic h the line elemen ti sd l /,/#28/3/3/#29
ZS
d S /#02r f /=
IC
d l f/5
/#28/3/4/#29
ZS
d S /#01r/#02 A /=
IC
d l /#01 A/#28/3/5/#29
ZS
/#28 d S /#02r /#29 /#02 A /=
IC
d l /#02 A/#28/3/6/#29
ZS
d S /#01 /#28 r f /#02r g /#29/=
IC
fd g /= /,
IC
gd fDIFFERENTIAL OPERA TORS INCUR VILINEAR COORDINA TES
/5Cylindrical Co ordinatesDiv ergencer/#01 A /=
/1r
/@/@r
/#28 rAr
/#29/+
/1r
/@A/#1E/@/#1E
/+
/@Az/@zGradien t/#28 r f /#29r
/=
/@f/@r
/; /#28 r f /#29/#1E
/=
/1r
/@f/@/#1E
/; /#28 r f /#29z
/=
/@f/@zCurl/#28 r/#02 A /#29r
/=
/1r
/@Az/@/#1E
/,
/@A/#1E/@z/#28 r/#02 A /#29/#1E
/=
/@Ar/@z
/,
/@Az/@r/#28 r/#02 A /#29z
/=
/1r
/@/@r
/#28 rA/#1E
/#29 /,
/1r
/@Ar/@/#1ELaplacianr
/2f /=
/1r
/@/@r
/#10r
/@f/@r
/#11/+
/1r
/2
/@
/2f/@/#1E
/2
/+
/@
/2f/@z
/2/6
Laplacian of a v ector/#28 r
/2A /#29r
/= r
/2Ar
/,
/2r
/2
/@A/#1E/@/#1E
/,
Arr
/2/#28 r
/2A /#29/#1E
/= r
/2A/#1E
/+
/2r
/2
/@Ar/@/#1E
/,
A/#1Er
/2/#28 r
/2A /#29z
/= r
/2AzComp onen ts of /#28 A /#01r /#29 B/#28 A /#01r B /#29r
/= Ar
/@Br/@r
/+
A/#1Er
/@Br/@/#1E
/+ Az
/@Br/@z
/,
A/#1E
B/#1Er/#28 A /#01r B /#29/#1E
/= Ar
/@B/#1E/@r
/+
A/#1Er
/@B/#1E/@/#1E
/+ Az
/@B/#1E/@z
/+
A/#1E
Brr/#28 A /#01r B /#29z
/= Ar
/@Bz/@r
/+
A/#1Er
/@Bz/@/#1E
/+ Az
/@Bz/@zDiv ergence of a tensor/#28 r/#01 T /#29r
/=
/1r
/@/@r
/#28 rTrr
/#29/+
/1r
/@T/#1Er/@/#1E
/+
/@Tzr/@z
/,
T/#1E/#1Er/#28 r/#01 T /#29/#1E
/=
/1r
/@/@r
/#28 rTr/#1E
/#29/+
/1r
/@T/#1E/#1E/@/#1E
/+
/@Tz/#1E/@z
/+
T/#1Err/#28 r/#01 T /#29z
/=
/1r
/@/@r
/#28 rTrz
/#29/+
/1r
/@T/#1Ez/@/#1E
/+
/@Tzz/@z/7
Spherical Co ordinatesDiv ergencer/#01 A /=
/1r
/2
/@/@r
/#28 r
/2Ar
/#29/+
/1r sin /#12
/@/@/#12
/#28sin /#12A/#12
/#29/+
/1r sin /#12
/@A/#1E/@/#1EGradien t/#28 r f /#29r
/=
/@f/@r
/; /#28 r f /#29/#12
/=
/1r
/@f/@/#12
/; /#28 r f /#29/#1E
/=
/1r sin /#12
/@f/@/#1ECurl/#28 r/#02 A /#29r
/=
/1r sin /#12
/@/@/#12
/#28sin /#12A/#1E
/#29 /,
/1r sin /#12
/@A/#12/@/#1E/#28 r/#02 A /#29/#12
/=
/1r sin /#12
/@Ar/@/#1E
/,
/1r
/@/@r
/#28 rA/#1E
/#29/#28 r/#02 A /#29/#1E
/=
/1r
/@/@r
/#28 rA/#12
/#29 /,
/1r
/@Ar/@/#12Laplacianr
/2f /=
/1r
/2
/@/@r
/#10r
/2
/@f/@r
/#11/+
/1r
/2sin /#12
/@/@/#12
/#10sin /#12
/@f/@/#12
/#11/+
/1r
/2sin
/2/#12
/@
/2f/@/#1E
/2Laplacian of a v ector/#28 r
/2A /#29r
/= r
/2Ar
/,
/2 Arr
/2
/,
/2r
/2
/@A/#12/@/#12
/,
/2 cot /#12A/#12r
/2
/,
/2r
/2sin /#12
/@A/#1E/@/#1E/#28 r
/2A /#29/#12
/= r
/2A/#12
/+
/2r
/2
/@Ar/@/#12
/,
A/#12r
/2sin
/2/#12
/,
/2 cos /#12r
/2sin
/2/#12
/@A/#1E/@/#1E/#28 r
/2A /#29/#1E
/= r
/2A/#1E
/,
A/#1Er
/2sin
/2/#12
/+
/2r
/2sin /#12
/@Ar/@/#1E
/+
/2 cos /#12r
/2sin
/2/#12
/@A/#12/@/#1E/8
Comp onen ts of /#28 A /#01r /#29 B/#28 A /#01r B /#29r
/= Ar
/@Br/@r
/+
A/#12r
/@Br/@/#12
/+
A/#1Er sin /#12
/@Br/@/#1E
/,
A/#12
B/#12
/+ A/#1E
B/#1Er/#28 A /#01r B /#29/#12
/= Ar
/@B/#12/@r
/+
A/#12r
/@B/#12/@/#12
/+
A/#1Er sin /#12
/@B/#12/@/#1E
/+
A/#12
Brr
/,
cot /#12A/#1E
B/#1Er/#28 A /#01r B /#29/#1E
/= Ar
/@B/#1E/@r
/+
A/#12r
/@B/#1E/@/#12
/+
A/#1Er sin /#12
/@B/#1E/@/#1E
/+
A/#1E
Brr
/+
cot /#12A/#1E
B/#12rDiv ergence of a tensor/#28 r/#01 T /#29r
/=
/1r
/2
/@/@r
/#28 r
/2Trr
/#29/+
/1r sin /#12
/@/@/#12
/#28sin /#12T/#12r
/#29/+
/1r sin /#12
/@T/#1Er/@/#1E
/,
T/#12/#12
/+ T/#1E/#1Er/#28 r/#01 T /#29/#12
/=
/1r
/2
/@/@r
/#28 r
/2Tr/#12
/#29/+
/1r sin /#12
/@/@/#12
/#28sin /#12T/#12/#12
/#29/+
/1r sin /#12
/@T/#1E/#12/@/#1E
/+
T/#12rr
/,
cot /#12T/#1E/#1Er/#28 r/#01 T /#29/#1E
/=
/1r
/2
/@/@r
/#28 r
/2Tr/#1E
/#29/+
/1r sin /#12
/@/@/#12
/#28sin /#12T/#12/#1E
/#29/+
/1r sin /#12
/@T/#1E/#1E/@/#1E
/+
T/#1Err
/+
cot /#12T/#1E/#12r/9