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MATH2023 Chapter 15 study guide New Update …//
Vector Fields
Contents
15.1 Vector fields
15.3 Line integrals in space
15.4 Line integrals of vector fields
15.2 Conservative vector fields
15.5 Introduction to surface integrals
15.6 Surface integrals of vector fields, flux
Review
r( t ) = x ( t ) i + y ( t ) j + z ( t ) k
— vector-valued functions of a single (scalar) variable, that is, curves.
z = f ( x 1
, x 2
,... , x n
) = f (r)
— scalar valued functions of a vector variable r, (that is, functions of
several real variables). This is a scalar field.
In the next two chapters, we will look at vector-valued function F of a vector
variable r, i.e.
F(r).
15.1 Vector Fields
MATH2023 Chapter 15 study guide New Update …//
Definition: A vector field is a function that associates a unique vector F( P ) with
each point P
in a region of 2D or 3D, i.e.
F( x, y ) = F 1 ( x, y ) i + F 2 ( x, y ) j (2D)
or F(r) = F 1 (r) i + F 2 (r) j , where the position vector r
= ( x, y ).
F( x, y, z ) = F 1 ( x, y, z ) i + F 2 ( x, y, z ) j + F 3 ( x, y, z ) k
(3D) or F(r) = F 1 (r) i + F 2 (r) j + F 3 (r) k ,
where the position vector r = ( x, y, z ).
Note that the components of a vector field are scalar fields.
A vector field is smooth when its component scalar fields have
continuous partial derivatives of all orders. (For most purposes, however,
second order would be sufficient.)
3
^
Ex. 1.3 The gravitational field of a point mass at
the origin.
m
F(r) = − k
r
r ,
where k is a constant and m is the mass.
y
y
z
x
1
-1 -0.5 0.5 1
-0.
3
x
2D field 3D field
^
3
grad f = ∇ f = f x i + f y j + f z k_._
Therefore ∇ f is called a gradient vector field.
Gradient of a scalar field f
Let f = f ( x, y, z ), then
df =
∂f
dx
∂x
∂f ∂f
dy + dz
∂y ∂z
= ∇ f · d r where r = ( x, y, z )
d r
= ∇ f · n^ ds where
ds
= n^ ,
n^ is the unit normal to the level surface and s is a distance measured along the
normal.
df
ds
= ∇ f · n^ =
ǁ∇ f ǁ
(∵ ∇ f ǁ n).
Hence the magnitude of ∇ f is the rate of change of f with position along the
normal, and points in the direction of the maximum upward gradient.
∇
3
Ex. 1.5If f ( x, y ) = x
2
2
, then f = 2 x i + 2 y j.
y
x
Ex. 1.6 If f ( x, y, z ) = xyz , then ∇ f = yz i + xz j + xy k.
1
-1-0.5 0.
-0.
y
1
0
-0.
1
0
-0.
-0.
0
x0.
1
^
δV → 0
δV
3
Divergence of a vector field
The divergence at any point P is defined as the limit (as the size of the
region tends to zero) of the flux of F out of some small volume δV (has surface
δS and outward normal n) surrounding P , divided by δV. Thus
∇ · F = lim
F· n^ dS
Hence the integration extends over closed surface surrounding the small
volume. This can be written in terms of the differential operator ∇ · F.
Curl of a vector field
The curl of a vector field F is a vector field. Its component in the direction of the
unit vector n is
n^ · ∇ × F = lim 1
I
n
S
P
V
δ
S
δS → 0
δS
3
F · d r
where δS is a small surface element perpendicular to n, δS is the closed curve
forming the boundary of δS and δC and n are oriented in a right-handed
sense.
The small surface δS is enclosed by the curve δC and has unit normal vector n ^
. n
small surface
S
C
δ
C
3
appearing to “diverge” at all, and it is possible for field to have a nontrivial
curl and yet have flow lines that do not bend at all.
Ex. 1.7 F(r) = y i − x j y
Note ∇ · ( x j) = 0 and ∇ × ( x j) = − 2 k.
x
Ex. 1.8F(r) = x j y
Note ∇ · ( x j) = 0 and ∇ × ( x j) = k.
x
1
-1 -0.5 0.5 1
-0.
1
-1 -0.5 0.5 1
-0.
..
3
Ex. 1.9 Let r = x i + y j + z k and u = a i + b j + c k, where a , b and c are constants,
show that
(a) ∇· r = 3 ,
(b) ∇ × r = 0,
(c) ∇· (u × r) = 0 ,
(d) ∇ × (u × r) = 2u.
(a) ∇· r = i
· ( x i + y j + z k) =
∂x
∂y
∂z
∂x ∂y ∂z ∂x ∂y ∂z
(b) ∇ × r = = 0
i j k
(c) u × r = a b c = ( bz − cy ) i − ( az − cx ) j + ( ay − bx ) k.
x y z
∴ ∇· (u × r) =
( bz − cy ) −
( az − cx ) +
( ay − bx ) = 0.
∂x ∂y ∂z
i j k
i
∂
j
∂
k
∂
∂
x
x
∂
y
y
∂
z
z
..
3
Some identities involving Grad, Div and Curl
Let f be a scalar field and F(r) = F 1 (r) i + F 2 (r) j + F 3 (r) k be a vector field, then
∂f ∂f ∂f
∇ f =
∂x
i +
∂y
j +
∂z
k (vector field)
∇ · F =
∂F
1
∂x
∂F
2
∂x
∂F
3
(scalar field)
∂z
i j k
∇ × F =
(vector field)
∂x ∂y ∂z
Definition: Laplacian Operator
= ∇ · ∇ = i
· i
∂x ∂y ∂z
∂x ∂y ∂z
∂x
∂y
∂z
∂
F 1 F 2
F 3
2
2 2 2
2
2
3
∇ is a scalar differential operator. Note that
2
2
f ∂
2
f ∂
2
f
∇ f =
∂x
∂y
∂z
∇ F = ∇ F
1
i + ∇ F 2
j + ∇ F 3
k_._
Vector differential identities
Let φ , ψ are scalar fields and F and G are vector fields, then
(a) ∇( φψ ) = φ ∇ ψ + ψ ∇ φ
(b) ∇ · ( φ F) = ∇ φ · F + φ (∇ · F)
(c) ∇ × ( φ F) = ∇ φ × F + φ (∇ × F)
(d) ∇ · (F × G) = (∇ × F) · G − F · (∇ × G)
(e) ∇ · (∇ × F) =
0 (f) ∇ × (∇ φ ) =
(g) ∇ × (∇ × F) = ∇(∇ · F) − ∇ F
2
3
i
(∇ · F) − ∇ F
i
∇ × (∇ × F) = ∇(∇ · F) − ∇ F
Ex. 1.10 Verify the
identity ∇ · ( f (∇ g × ∇ h )) = ∇ f · (∇ g × ∇ h )
for smooth scalar fields f , g and h.
3
Field lines
If the velocity of the particle (with position vector: r( t )) is given by the field, then
d r
= F(r).
dt
The path of the particle will be a curve to which the field is tangent at
every point. Such curves are called field lines. If we break the equation into
components, then
dx
F
1 (r) ,
dt
dy
F
2 (r) ,
dt
dz
= F
3 (r).
dt
∴ The differential equation for the field lines is
dx
F
1
r)
dy
F
2
r)
dz
F
3 (r)
Note that the field lines of F do not depend on the magnitude of F at any
point, but only on the direction of the field.
