Section 9.7 Divergence and Curl
"Del", ∇ - A defined operator
∇=
∂ ∂ ∂
, ,
∂x ∂y ∂z
The gradient of a function (at a point) is a vector that points in the direction
in which the function increases most rapidly.
A vector field is a vector function that can be thought of as a velocity field
of a fluid. At each point it assigns a vector that represents the velocity of
a particle at that point.
9.7
The flux of a vector field is the volume of fluid flowing through an element
of surface area per unit time.
The divergence of a vector field is the flux per unit volume.
The divergence of a vector field is a number
that can be thought of as a measure of the
rate of change of the density of the fluid at a point.
The curl of a vector field measures
the tendency of the vector field
to rotate about a point.
The curl of a vector field at a point is a vector
that points in the direction of the axis of
rotation and has magnitude represents
the speed of the rotation.
1
V ector Field
Scalar Funct i on
F = P ( x, y, z ) , Q ( x, y, z ) , R ( x, y, z )
f ( x, y, z )
grad ( f
G ra dient
∇f =
)
fx , f y , fz
D iv e rgence div ( F )
∇ ⋅F =
∂ ∂ ∂
∂P ∂Q ∂R
,
,
⋅ P, Q, R =
+
+
= Px + Q y + R z
∂x ∂y ∂z
∂x
∂y
∂z
C url curl ( F )
∇×F =
i
j
∂
∂x
P
∂
∂y
Q
k
∂
= ( R y − Q z ) i − ( R x − Pz ) j + ( Q x − Py ) k
∂z
R
∇ × F = R y − Q z , − ( R x − Pz ) , Q x − Py
9.7
F ( x, y, z ) = xe − z , 4 yz 2 ,3 ye − z
div ( F ) = ∇ ⋅ F =
∂ ∂ ∂
, ,
⋅ xe − z , 4 yz 2 ,3 ye − z = e − z + 4 z 2 − 3 ye − z
∂x ∂y ∂z
i
∂
curl ( F ) = ∇ × F =
∂x
xe − z
j
∂
∂y
4 yz 2
k
∂
= 3 e − z − 8 yz 2 , − 0 − ( − xe − z ) , 0
∂z
3 ye − z
(
)
3 e − z − 8 yz 2 , − xe − z , 0
9.7
2
grad ( scalar function ) = V ector F ield
div ( V ector F ield ) = scalar function
curl ( V ector F ield ) = V ector F ield
Which of the 9 ways to combine grad, div and curl by taking
one of each. Which of these combinations make sense?
grad ( grad
( f ) ) div ( grad ( f ) )
V ector F ield
V ector F ield
grad ( div ( F ) )
div ( div ( F ) )
scalar function
0
curl ( grad ( f ) ) vector
V ector F ield
curl ( div ( F ) )
scalar function
grad ( curl ( F ) )
V ector F ield
scalar function
0 curl ( curl ( F ) )
div ( curl ( F ) ) scalar
V ector F ield
2 of the above are always zero.
V ector F ield
9.7
Verify the given identity. Assume continuity of all partial derivatives.
curl ( grad ( f ) ) = 0.
grad ( f ) = f x , f y , f z
i
∂
curl ( grad ( f ) ) = ∇ × ∇f =
∂x
fx
j
∂
∂y
fy
k
∂
∂z
fz
∇ × ∇f = f zy − f yz , − ( f zx − f xz ) , f yx − f xy = 0
since mixed partial derivatives are equal.
Verify the given identity. Assume continuity of all partial derivatives.
div ( curl ( F ) ) = 0.
Let F = P ( x, y, z ) , Q ( x, y, z ) , R ( x, y, z )
curl ( F ) = Ry − Qz , Pz − Rx , Qx − Py
i
∂
curl ( F ) = ∇ × F =
∂x
P
div ( curl ( F ) ) = ( Ry − Qz ) x + ( Pz − Rx ) y + ( Qx − Py ) z
j
∂
∂y
Q
k
∂
∂z
R
div ( curl ( F ) ) = Ryx − Qzx + Pzy − Rxy + Qxz − Pyz
div ( curl ( F ) ) = Ryx − Qzx + Pzy − Rxy + Qxz − Pyz = 0
9.7
3