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6.641 Electromagnetic Fields, Forces, and Motion, Spring 2005
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Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.641 FORMULA SHEET
1. DIFFERENTIAL OPERATORS IN CYLINDRICAL AND SPHERICAL COORDINATES
If r, *, and z are circular cylindrical coordinates and ı̂r , ı̂* , and ı̂z are unit vectors in the directions
of increasing values of the corresponding coordinates,
nU = grad U = ı̂r
(U
1 (U
(U
+ ı̂*
+ ı̂z
(r
r (*
(z
n = div A
n = 1 ( (rAr ) + 1 (A* + (Az
n·A
r (*
(z
r (r
w
n = curlA
n = ı̂r
n×A
(A*
1 (Az
(z
r (*
w
W
+ ı̂*
1 (
n U = div grad U =
r (r
2
(Az
(Ar
(z
(r
W
w
+ ı̂z
1 ( (rA* ) 1 (Ar
r (r
r (*
W
w
W
(U
1 (2U
(2U
r
+ 2
+
(r
r (*2
(z 2
If r, , and * are spherical coordinates and ı̂r , ı̂ , and ı̂* are unit vectors in the directions of
increasing values of the corresponding coordinates,
nU = grad U = ı̂r
(U
1 (U
1 (U
+ ı̂
+ ı̂*
(r
r (
r sin (*
D
i
1 ( (A sin )
1 (A*
1 ( r 2 Ar
n
n
+
+
n · A = div A = 2
r
(r
r sin (
r sin (*
w
n = curlA
n = ı̂r
n×A
1 (A
1 ( (A* sin )
r sin (
r sin (*
1 (
n U = div grad U = 2
r (r
2
W
w
+ ı̂
1 ( (rA* )
1 (Ar
r sin (*
r (r
w
W
+ ı̂*
w
W
w
W
1
(U
1
(2U
(
2 (U
r
+ 2
sin + 2 2
(r
r sin (
(
r sin (*2
2. SOLUTIONS OF LAPLACE’S EQUATIONS
A. Rectangular coordinates, two dimensions (independent of z):
= ekx (A1 sin ky + A2 cos ky) + ekx (B1 sin ky + B2 cos ky)
(or replace ekx and ekx by sinh kx and cosh kx).
= Axy + Bx + Cy + D; (k = 0) B. Cylindrical coordinates, two dimensions (independent of z):
= r n (A1 sin n* + A2 cos n*) + r n (B1 sin n* + B2 cos n*)
= ln
1 ( (rA ) 1 (Ar
r (r
r (
R
(A1 * + A2 ) + B1 * + B2 ; (n = 0)
r
C. Spherical coordinates, two dimensions (independent of *):
= Ar cos +
1
B
C
cos + + D
2
r
r
W