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 =
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) |
|
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.