veccalc

07f8d54f-b591-4fc0-ab84-bf7bb62aa822 196 4 5 false false 0 decimal

&[DATE] &[TIME] - &[FILENAME]

veccalc

Divergence F v h div n v length : k Clear H h k el k 1 n product : 13 mat 1 H / H F k el h k el / * v k el Jacob 1 el k 1 n sum * 21 line :

Rotational F v h curl m n c 1 h m el h n el * / h n el F n el * v m el Jacob 1 el h m el F m el * v n el Jacob 1 el - (* 11 line : v length 2 ≡ 0012 c 13 mat transpose 23 c 31 c 12 c 13 mat transpose if 21 line :

Gradient f v h grad n v length : G n 1 matrix : 12 mat k 1 n range G k el 1 h k el / f v k el Jacob 1 el * : for G 31 line :

Laplacian f v h lapl n v length : k Clear H h k el k 1 n product : 13 mat 1 H / H h k el (2^/ f v k el Jacob 1 el * v k el Jacob 1 el k 1 n sum * 21 line :

Change base matrix R v h CBMat n v length : M n n matrix : J R v Jacob : 13 mat r 1 n range c 1 n range M c r el J r c el h c el / : for for M 31 line :

Unit Vectors R v h UnitVectors M R v h CBMat : n M cols : C 1 n matrix : 13 mat k 1 n range C k el M k col : for C 31 line :

Scale factors R v ScaleFactors n v length : h n 1 matrix : J R v Jacob : j Clear 14 mat k 1 n range h k el J k col j el J k col j el Conjugate * j 1 n sum sqrt : for h 31 line :

F div F coords 1 el coords 2 el div 11 line :

f lapl f coords 1 el coords 2 el lapl 11 line :

F grad F coords 1 el coords 2 el grad 11 line :

F curl F coords 1 el coords 2 el curl 11 line :

F P div V coords 1 el : F V coords 2 el div V P ≡ vectorize at 21 line :

f P lapl V coords 1 el : f V coords 2 el lapl V P ≡ vectorize at 21 line :

F P grad V coords 1 el : F V coords 2 el grad V P ≡ vectorize at 21 line :

F P curl V coords 1 el : F V coords 2 el curl V P ≡ vectorize at 21 line :

R CBMat R coords 1 el coords 2 el CBMat 11 line :

R UnitVectors R coords 1 el coords 2 el UnitVectors 11 line :

Ê coords 3 el coords 1 el coords 2 el CBMat 11 line :

ê coords 3 el coords 1 el coords 2 el UnitVectors 11 line :

r coords 3 el :

r' coords 4 el :

v coords 1 el :

ê.x 10031 mat :

ê.y 01031 mat :

ê.z 00131 mat :

S V coords S V car coords 11 line :

S V W coords S V W 112 / 13 / 13 mat coords 11 line :

S V W p coords s S strtolower : V length 2 ≡ u v x y 14 mat V 1 el V 2 el W 1 el W 2 el 14 mat : u v w x y z 16 mat V 1 el V 2 el V 3 el W 1 el W 2 el W 3 el 16 mat : if see below s pol ≡ see below s cyl ≡ u v w 31 mat 1 u u v sin * 31 mat u v sin * w cos * u v sin * w sin * u v cos * 31 mat x 2^y 2^+ z 2^+ sqrt x 2^y 2^+ sqrt z atan y x atan 31 mat u 0 ∞ lelt v 0 π lele w 02 π * lelt 31 mat 15 mat s sph ≡ sph2 convention R r θ sin * φ sin * r θ sin * φ sin * r φ cos * 31 mat : 12 mat s sph2 ≡ Not implemented error cases 31 line :

car x y z 31 mat :

Default cartesian coordinate names.

Sph

Coords call

coords sph r θ ϕ 13 mat coords :

v v :

r r :

r' r' :

Needed for vectorizing under at's evaluations

If at can accept vector = vector, I guess this is not needed anymore.

Unit vectors and Change base matrix

ê θ sin ϕ cos * θ cos ϕ cos * ϕ sin - 31 mat θ sin ϕ sin * θ cos ϕ sin * ϕ cos 31 mat θ cos θ sin - 031 mat 13 mat

Ê θ sin ϕ cos * θ sin ϕ sin * θ cos θ cos ϕ cos * θ cos ϕ sin * θ sin - ϕ sin - ϕ cos 033 mat

coords v h r r' rng 15 mat ≡

where: v coordinates names h scale factors r formulas for [u v w] -> [x y z] r' formulas for [x y z] -> [u v w] rng coordinates ranges

Given a point in cartesians, their spherical coordinates and back to cartesians, are

P' 12331 mat :

P r' car P' ≡ vectorize at : 3.7417 0.6405 1.1071 31 mat

r θ ϕ

r v P ≡ vectorize at 12331 mat

x y z

coords r θ ϕ 31 mat 1 r r θ sin * 31 mat r θ sin * ϕ cos * r θ sin * ϕ sin * r θ cos * 31 mat x 2^y 2^+ z 2^+ sqrt x 2^y 2^+ sqrt z atan y x atan 31 mat 0 r ≤ (r ∞ < (& 0 θ ≤ (θ π ≤ (& 0 ϕ ≤ (ϕ 2 π * < (& 31 mat 15 mat

Given a scalar function in cartesians, it's expression in sphericals is

f' z x * :

f f' car r ≡ vectorize at : r 2^θ cos * θ sin * ϕ cos *

and can be evaluated at P as

f' car P' ≡ vectorize at 3

f v P ≡ vectorize at 3

Given a vectorial function in the cartesians, we can get it expression in sphericals

F' 1 x 2^y 2^+ z 31 mat :

F F' car r ≡ vectorize at : 1 r 2^θ sin 2^* r θ cos * 31 mat

F' 1 el ê.x * F' 2 el ê.y * + F' 3 el ê.z * + 10031 mat y 2^x 2^+ (01031 mat * + z 00131 mat * +

and evaluate them at P as

F'.P' F' car P' ≡ vectorize at : 15331 mat

x component y component z component

F v P ≡ vectorize at 15331 mat

It's components in sperical coordinates are expressed, using the change base matrix or unit vectors, by

F.P Ê F * v P ≡ vectorize at : 5.3452 2.1514 1.3416 31 mat

F'.P' 1 el ê 1 el * F'.P' 2 el ê 2 el * + F'.P' 3 el ê 3 el * + v P ≡ vectorize at 5.3452 2.1514 1.3416 31 mat

We can find the gradient and divergence of the f and the curl and divergence of F

f grad 2 r * θ cos * θ sin * ϕ cos * r ϕ cos * θ cos θ sin - (* θ cos θ sin + (* r ϕ sin * θ cos * - 31 mat

f P grad 1.6036 0.4781 2.6833 - 31 mat

F curl θ cos θ sin - (θ cos θ sin + (* θ sin / 2 θ cos * - 3 r * θ sin 2^* 31 mat

F P curl 0.4781 1.6036 - 4.0089 31 mat

F P div 5.9131

F div 23 θ cos * θ sin * r 2^* + r /

f P lapl 1.0616 1015 -^*

f lapl ϕ cos θ cos θ sin + (θ cos θ cos θ sin - (* θ sin θ sin θ cos + (* - (* θ sin θ cos θ sin - (2^6 θ sin * θ cos * + (* + θ cos - (* θ sin /

Cyl

ρ φ cos * ρ φ sin * z 31 mat ρ φ z 31 mat ScaleFactors 1 ρ 2^sqrt 131 mat

then

coords ρ φ z 31 mat 1 ρ 131 mat 12 mat :

f ρ φ sin * z 2^* :

F ρ z * φ sin * φ cos z 2^31 mat :

f P grad 0.7325 0.9828 4.9513 31 mat

F P curl 0 2.2361 0.6734 - 31 mat

f grad φ sin z 2^* φ cos z 2^* 2 ρ * φ sin * z * 31 mat

F curl 0 ρ φ sin * φ cos 1 ρ z * - (* ρ / 31 mat

F P div 3.3779

f lapl 2 ρ * φ sin *

F div 2 ρ * z * 1 φ sin + (* φ sin - ρ /

f P lapl 4.4721

Pol

ρ φ cos * ρ φ sin * 21 mat ρ φ 21 mat ScaleFactors 1 ρ 2^sqrt 21 mat

then

coords ρ φ 21 mat 1 ρ 21 mat 12 mat :

P P 12 range el :

car x y 21 mat :

f ρ 2^φ sin * :

F ρ φ sin * φ cos 21 mat :

f P grad 4.4721 321 mat

this curl isn't "correct", but just useful.

f grad 2 ρ * φ sin * ρ φ cos * 21 mat

F curl 00 φ cos 1 ρ - (* ρ / 31 mat

F P curl 00 0.5875 - 31 mat

F P div 1.0355

f lapl 3 φ sin *

F div φ sin 1 - 2 ρ * + (* ρ /

f P lapl 1.7928

Alvaro

From Maple's 11 VectorCalculus[Coordinates] help (guess the same in Maple's 5 linalg[coords]

References Moon, P., and Spencer, D.E. Field Theory Handbook. 2d ed. Berlin: Springer-Verlag, 1971. Spiegel, Murray R. Mathematical Handbook of Formulas and Tables. New York: McGraw Hill Book Company, 1968, pp. 126-130.

Description The VectorCalculus package supports the following coordinate systems: In two dimensions - bipolar, cardioid, cassinian, cartesian, elliptic, hyperbolic, invcassinian, logarithmic, logcosh, parabolic, polar, rose, and tangent. In three dimensions - bipolarcylindrical, bispherical, cardioidal, cardioidcylindrical, cartesian, casscylindrical, conical, cylindrical, ellcylindrical, hypercylindrical, invcasscylindrical, invprospheroidal, logcoshcylindrical, logcylindrical, oblatespheroidal, paraboloidal, paracylindrical, prolatespheroidal, rosecylindrical, sixsphere, spherical, tangentcylindrical, tangentsphere, and toroidal. Note: Only the positive roots have been used for the following transformations: In two dimensions - cassinian, hyperbolic, invcassinian, and rose In three dimensions - casscylindrical, conical, hypercylindrical, invcasscylindrical, and rosecylindrical The conversions from the various coordinate systems to cartesian (rectangular) coordinates in 2-space (u, v) --> (x, y) are given by: bipolar: (Spiegel) x = sinh(v)/(cosh(v)-cos(u)) y = sin(u)/(cosh(v)-cos(u)) cardioid: x = 1/2*(u^2-v^2)/(u^2+v^2)^2 y = u*v/(u^2+v^2)^2 cartesian: x = u y = v cassinian: (Cassinian-oval) x = a*2^(1/2)/2*((exp(2*u)+2*exp(u)*cos(v)+1)^(1/2) + exp(u)*cos(v)+1)^(1/2) y = a*2^(1/2)/2*((exp(2*u)+2*exp(u)*cos(v)+1)^(1/2) - exp(u)*cos(v)-1)^(1/2)] elliptic: x = cosh(u)*cos(v) y = sinh(u)*sin(v) hyperbolic: x = ((u^2+v^2)^(1/2)+u)^(1/2) y = ((u^2+v^2)^(1/2)-u)^(1/2) invcassinian: (inverse Cassinian-oval) x = a*2^(1/2)/2*((exp(2*u)+2*exp(u)*cos(v)+1)^(1/2) + exp(u)*cos(v)+1)^(1/2)/(exp(2*u)+2*exp(u)*cos(v)+1)^(1/2) y = a*2^(1/2)/2*((exp(2*u)+2*exp(u)*cos(v)+1)^(1/2) - exp(u)*cos(v)-1)^(1/2)/(exp(2*u)+2*exp(u)*cos(v)+1)^(1/2) logarithmic: x = a/Pi*ln(u^2+v^2) y = 2*a/Pi*arctan(v/u) logcosh: (ln cosh) x = a/Pi*ln(cosh(u)^2-sin(v)^2) y = 2*a/Pi*arctan(tanh(u)*tan(v)) parabolic: x = (u^2-v^2)/2 y = u*v polar: x = u*cos(v) y = u*sin(v) rose: x = ((u^2+v^2)^(1/2)+u)^(1/2)/(u^2+v^2)^(1/2) y = ((u^2+v^2)^(1/2)-u)^(1/2)/(u^2+v^2)^(1/2) tangent: x = u/(u^2+v^2) y = v/(u^2+v^2) The conversions from the various coordinate systems to cartesian coordinates in 3-space (u, v, w) --> (x, y, z) are given as follows (the author is indicated where applicable): bipolarcylindrical: (Spiegel) x = a*sinh(v)/(cosh(v)-cos(u)) y = a*sin(u)/(cosh(v)-cos(u)) z = w bispherical: x = sin(u)*cos(w)/d y = sin(u)*sin(w)/d z = sinh(v)/d where d = cosh(v) - cos(u) cardioidal: x = u*v*cos(w)/(u^2+v^2)^2 y = u*v*sin(w)/(u^2+v^2)^2 z = (u^2-v^2)/2/(u^2+v^2)^2 cardioidcylindrical: x = (u^2-v^2)/2/(u^2+v^2)^2 y = u*v/(u^2+v^2)^2 z = w cartesian: x = u y = v z = w casscylindrical: (Cassinian-oval cylinder) x = a*2^(1/2)/2*((exp(2*u)+2*exp(u)*cos(v)+1)^(1/2)+exp(u)*cos(v)+1)^(1/2) y = a*2^(1/2)/2*((exp(2*u)+2*exp(u)*cos(v)+1)^(1/2)-exp(u)*cos(v)-1)^(1/2) z = w conical: x = u*v*w/(a*b) y = u/b*((v^2 - b^2)*(b^2-w^2)/(a^2-b^2))^(1/2) z = u/a*((a^2 - v^2)*(a^2 - w^2)/(a^2-b^2))^(1/2) cylindrical: x = u*cos(v) y = u*sin(v) z = w ellcylindrical: (elliptic cylindrical) x = a*cosh(u)*cos(v) y = a*sinh(u)*sin(v) z = w hypercylindrical: (hyperbolic cylinder) x = ((u^2+v^2)^(1/2)+u)^(1/2) y = ((u^2+v^2)^(1/2)-u)^(1/2) z = w invcasscylindrical: (inverse Cassinian-oval cylinder) x = a*2^(1/2)/2*((exp(2*u)+2*exp(u)*cos(v)+1)^(1/2) + exp(u)*cos(v)+1)^(1/2)/(exp(2*u)+2*exp(u)*cos(v)+1)^(1/2) y = a*2^(1/2)/2*((exp(2*u)+2*exp(u)*cos(v)+1)^(1/2) - exp(u)*cos(v)-1)^(1/2)/(exp(2*u)+2*exp(u)*cos(v)+1)^(1/2) z = w invprospheroidal: (inverse prolate spheroidal) x = a*sinh(u)*sin(v)*cos(w)/(cosh(u)^2-sin(v)^2) y = a*sinh(u)*sin(v)*sin(w)/(cosh(u)^2-sin(v)^2) z = a*cosh(u)*cos(v)/(cosh(u)^2-sin(v)^2) logcylindrical: (logarithmic cylinder) x = a/Pi*ln(u^2+v^2) y = 2*a/Pi*arctan(v/u) z = w logcoshcylindrical: (ln cosh cylinder) x = a/Pi*ln(cosh(u)^2-sin(v)^2) y = 2*a/Pi*arctan(tanh(u)*tan(v)) z = w oblatespheroidal: x = a*cosh(u)*sin(v)*cos(w) y = a*cosh(u)*sin(v)*sin(w) z = a*sinh(u)*cos(v) paraboloidal: (Spiegel) x = u*v*cos(w) y = u*v*sin(w) z = (u^2 - v^2)/2 paracylindrical: x = (u^2 - v^2)/2 y = u*v z = w prolatespheroidal: x = a*sinh(u)*sin(v)*cos(w) y=a*sinh(u)*sin(v)*sin(w) z=a*cosh(u)*cos(v) rosecylindrical: x = ((u^2+v^2)^(1/2)+u)^(1/2)/(u^2+v^2)^(1/2) y = ((u^2+v^2)^(1/2)-u)^(1/2)/(u^2+v^2)^(1/2) z = w sixsphere: (6-sphere) x = u/(u^2+v^2+w^2) y = v/(u^2+v^2+w^2) z = w/(u^2+v^2+w^2) spherical: x = u*cos(w)*sin(v) y = u*sin(w)*sin(v) z = u*cos(v) tangentcylindrical: x = u/(u^2+v^2) y = v/(u^2+v^2) z = w tangentsphere: x = u*cos(w)/(u^2+v^2) y = u*sin(w)/(u^2+v^2) z = v/(u^2+v^2) toroidal: x = a*sinh(v)*cos(w)/d y = a*sinh(v)*sin(w)/d z = a*sin(u)/d where d = cosh(v) - cos(u) The a, b, and c values in the above coordinate transformations can be queried and set by using the GetCoordinateParameters and SetCoordinateParameters commands from the VectorCalculus package. The default values are a = 1, b = 1/2, and c = 1/3. The GetCoordinateParameters command returns an expression sequence containing the current values of a, b, and c. The SetCoordinateParameters command takes either 1, 2, or 3 arguments, and sets the values of a, a and b, or a, b, and c respectively.