BASIC ELEMENTS OF VECTOR CALCULUS>
In the following, S is a scalar function of x,y,z:
S(x,y,z)
In the following, V and W are vector functions of x,y,z:
V = Vx(x,y,z) x + Vy(x,y,z)
y + Vz(x,y,z) z
where x, y, and z are unit vectors in the x, y, and z directions respectively [rectangular], and r, q , and f are unit vectors in the r, q, and f directions [polar, cylindrical, and spherical].
Operator
Ñ = del operator: in
rectangular form it is: Ñ = (¶[ ]/¶x)
x + (¶[ ]/¶y) y + (¶[ ]/¶z)
z
Gradient
gradient: ÑS = (¶S/¶x) x + (¶S/¶y) y + (¶S/¶z) z (a vector!)
gradient shows the change in S over space and is in the direction of that greatest change, and hence is a vector.
Divergence
divergence: Ñ · V = (¶Vx/¶x) + (¶Vy/¶y) + (¶Vz/¶z) (a scalar!)
idea of divergence is best indicated through divergence theorem:
òòò Ñ · V dVolume = ò òclosed V· n dSurface surface enclosed volume
Curl
curl: Ñ ´ V = [(¶Vz/¶y)-(¶Vy/¶z)] x + [(¶Vx/¶z)-(¶Vz/¶x)]
y + [(¶Vy/¶x)-(¶Vx/¶y)] z
idea of curl is best indicated through Stokes theorem:
òò (Ñ ´ V)· n dArea = òclosed V· dlength length encircles area
IDENTITIES
|
Ñ · (V+W) = Ñ · V + Ñ · W |
(scalar) |
|
Ñ · (SV) = (ÑS)· V + S(Ñ · V) |
(scalar) |
|
Ñ · (ÑS) = ѲS = (¶²S/¶x² + ¶²S/¶y² + ¶²S/¶z²) |
(scalar) |
|
Ñ ´ (V+W) = (Ñ ´ V) + (Ñ ´ W) |
(vector) |
|
Ñ ´ (SV) = (ÑS)´V + S(Ñ ´ V) |
(vector) |
|
Ñ · (V´ W) = W· (Ñ ´ V) - V· (Ñ ´ W) |
(scalar) |
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Ñ ´ (V´ W) = (W· Ñ )V - (V· Ñ )W + (Ñ · W)V - (Ñ · V)W |
(vector) |
|
Ñ (V· W) = (W· Ñ )V + (V· Ñ )W + W´ (Ñ ´ V) + V´ (Ñ ´ W) |
(vector) |
|
Ñ ´ (Ñ ´ V) = Ñ (Ñ · V) - Ñ ²V |
(vector) |
|
Ñ ´ (ÑS) = 0 |
(vector) |
|
Ñ · (Ñ ´ V) = 0 |
(scalar) |
In spherical coordinates:
Ñ = r (¶ /¶r) + q (1/r) (¶ /¶q) + f (1/[r sinq ]) (¶ /¶f)
ѲS = (1/r²) (¶/¶r)(r² ¶S/¶r) + (1/[r²sinq]) (¶/¶q)(sinq ¶S/¶q) + (1/[r²sin²q]) (¶²S/¶f²)
In cylindrical coordinates:
Ñ = r (¶ /¶r) + f (1/r) (¶ /¶f) + z (¶ /¶z)
ѲS = (1/r) (¶ /¶r)(r ¶S/¶r) + (1/r²) (¶²S/¶f²) + (¶²S/¶z²)
Homework Problem #17: Calculate curl of A, Ñ ´ A, in cylindrical coordinates.