veccalc

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veccalc

M col n M cols : C 1 n matrix : 12 mat k 1 n range C k el M k col : for C 31 line :

M row n M rows : R 1 n matrix : 12 mat k 1 n range R k el M k row : for R 31 line :

Derivative for f:R->R f x ®j f x Jacob 1 el 11 line :

Scalar by matrix for symbolics s A ®s m A rows : n A cols : M m n matrix : 13 mat r 1 m range c 1 n range M r c el s A r c el * : for for M 31 line :

Norme for symbolics A ®n k Clear A k el A k el Conjugate * k 1 A length sum sqrt 21 line :

Matrix product for symbolics A B ®p m A rows : n B cols : M m n matrix : 13 mat r 1 m range c 1 n range M r c el A r row B c col ®d : for for M 31 line :

Dot product for symbolics A B ®d k Clear A k el B k el * k 1 A length sum 21 line :

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 ®j 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 ®j h m el F m el * v n el ®j - (* 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 ®j * : 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 ®j * v k el ®j 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 :

Scale factors R v ScaleFactors n v length : h n 1 matrix : J R v Jacob : 13 mat k 1 n range h k el J k col ®n : 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 :

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

Sph

Coordinate names and position vector, formed whith the transform formulas

v r θ ϕ 31 mat :

R r θ sin * ϕ cos * r θ sin * ϕ sin * r θ cos * 31 mat :

R r θ sin * φ sin * r θ sin * φ sin * r φ cos * 31 mat :

radial distance polar angle azimut angle

0 r ≤ θ 0 π lele φ 02 π * lelt 31 sys

Given the Jacobian of the position vector

J R v Jacob : θ sin ϕ cos * r θ cos * ϕ cos * r θ sin * ϕ sin * - θ sin ϕ sin * r θ cos * ϕ sin * r θ sin * ϕ cos * θ cos r θ sin * - 033 mat

the scale factors are the norme of the columns of J, and we simplify it by hand

h R v ScaleFactors : 1 r 2^sqrt r 2^θ sin 2^* sqrt 31 mat

h 1 r r θ sin * 31 mat :

We introduce here the definition for the coordinate system as

coords v h 12 mat :

The change base matrix is formed with each Jacobian's column divide by each scale factor

Ê R CBMat : 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

Columns of the change base matrix are the unit vectors

ê Ê col : 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

Given a point in cartesians, their spherical coordinates are

P' 12331 mat :

x component y component z component

v' x y z 31 mat :

R' x 2^y 2^+ z 2^+ sqrt x 2^y 2^+ sqrt z atan y x atan 31 mat :

P R' v' P' ≡ vectorize at : 3.7417 0.6405 1.1071 31 mat

r component θ component ϕ component

Given a scalar function in cartesians

f' z x * :

it's expression in sphericals is

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

and can be evaluated at P as

f' v' 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 y 2^x 2^+ z 31 mat :

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

and evaluate them at P as

F'.P' F' v' 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, by

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

or, using the unit vectors in shpericals, by

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

F div 23 θ cos * θ sin * r 2^* + r /

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

Short forms

v ρ φ z 31 mat :

R ρ φ cos * ρ φ sin * z 31 mat :

h R v ScaleFactors : 1 ρ 2^sqrt 131 mat

h 1 ρ 131 mat :

coords v h 12 mat :

f ρ φ sin * z 2^* :

F ρ z * φ sin * φ cos z 2^31 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 lapl 2 ρ * φ sin *

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

Pol

Short forms

v ρ φ 21 mat :

R ρ φ cos * ρ φ sin * 21 mat :

h R v ScaleFactors : 1 ρ 2^sqrt 21 mat

h 1 ρ 21 mat :

coords v h 12 mat :

f ρ 2^φ sin * :

F ρ z * φ sin * φ cos 21 mat :

f grad 2 ρ * φ sin * ρ φ cos * 21 mat

F curl 00 φ cos 1 ρ z * - (* ρ / 31 mat

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

f lapl 3 φ sin *

F div φ sin 1 - 2 ρ * z * + (* ρ /

Alvaro