10) Divergence And Curl Of Vector Fields Flashcards
Divergence of vector fields and ROTATION of vector fields
DIVERGENCE:
Vectors are normal to curves and outwards gives potisitve circulation
Convergent field if inwards
ROTATION:
Has axis of rotation
Taking closed curves we can see properties 0 normal component.
A field can show both divergence and rotation
Line integrals showcase this circulation…
Line integrals showcase circulation and rotation
A= area enclosed by C.
Where ^T_ denotes the unit tangent to the field and ^n_ the outward normal vector.
I_1= (1/A) integralO(vector(v) • ^n_.ds)
Over c
I_2= (1/A) integralO(vector(v)• ^T_.ds)
Over c
- A divergent vector field: has I_1 larger than 0, I_2=0
- a field showing rotation only: has I_1=0 and I_2 larger than 0
Assuming traversed countercloxkwise.
To get a local measure of the divergence and rotation of a field at point P
We can take the limit of the integrals defined I_1 and I_2 as the enclosed area A tends to 0. Where the curve C is shrunk so that it always enclosed P.
These provide measures of local divergence and rotation of vector field.
div of a vector field v= (v_1, v_2, v_3)
If the components v_i for i=1,2,.. of a vector field v= (v_1, v_2, v_3) are CONTINUOUSLY DIFFERENTIABLE FUNCTIONS OF THE CARTESIAN COORDINATES x,y and z. Then the DIVERGENCE of a vector field is a SCALAR FIELD
div( vector(v)) = ∂v_1/∂x + ∂v_2/∂y+ ∂v_3/∂z
Curl of a vector field v= (v_1, v_2, v_3)
If the components v_i for i=1,2,.. of a vector field v= (v_1, v_2, v_3) are CONTINUOUSLY DIFFERENTIABLE FUNCTIONS OF THE CARTESIAN COORDINATES x,y and z. Then the CURL of a vector field is a VECTOR FIELD
Curl(vector(v)) = (∂v_3/∂y - ∂v_2/∂z, ∂v_1/∂z - ∂v_3/∂x, ∂v_2/∂x - ∂v_1/∂y)
= ( ∂v_3/∂y - ∂v_2/∂z) i_ +
( ∂v_1/∂y - ∂v_3/∂z) j_ +
( ∂v_2/∂y - ∂v_1/∂z) k_
where I,j,k unit vectors in the x,y,z directions.
From the linearity of differentiation curl and div..
Are also linear:
For any vectors v, v’ in R^3 and t,t’ in R.
div( tv + t’v’) = t div(v) + t’div(v’)
Curl( tv + t’v’) = t curl(v) + t’curl(v’)
If a potential function exists for a vector field then curl:
If v is a conservative field
If a CONTINUOUS DIFFERENTIABLE vector field vector(v) can be written as the gradient of a scalar field i.e. Has a potential function then
ASSUMING SECOND DERIVATIVES ARE CONTINUOUS Since vector(v)= grad φ = (φ_x, φ_y, φ_z)
Curl (v)= curl(grad φ) = ( MIXED SECOND PARTIAL DERIVATIVES OF φ )
= ( ∂^(2)φ/∂y∂z - ∂^(2)φ/∂z∂y, ..,..) = vecotr(0)
Due to equality of second derivatives
Irrotational
A vector field that has curl v = vector 0 in some region R of R^3 id called IRROTATIONAL on R. Hence if vector v is an irrotational vector field on a singly connected region R in R^3 then there exists a scalar field φ such that vector v=grad φ on R.
φ is called a scalar potential for the field vector(v) and is unique apart from an arbitrary additive constant.
If a vector field v has continuous second derivatives
Then div(curl(vector(v))) = 0.
Due to the equality of second derivatives.
Solenoidal on R
A vector field vector(v) for which div(vector(v)) =0 in some region R in R^3 is called solenoidal on R. It can be shown that if vector(v) is a solenoidal vector field on R then there exists a vector field A such that vector(v) = curl A on R
A is called a vector potential for vector(v)
If A is a vector potential for vector v then so Is A + grad( φ) for any scalar field φ
The del operator
Expresses grad div and curl.
∇ ≡ i_ ∂/∂x + j_ ∂/∂y + k_ ∂/∂z
≡ ( ∂/∂x, ∂/∂y, ∂/∂z )
Where each is an operator acting on a function.
For a scalar field φ and vector field v:
grad φ = ∇ φ
div v = ∇ • v. Dot product.
Curl v = ∇ X v. Cross product.
By the del operator we can remember curl v as
Det ( of rows (I,j,k) ,( ∂/∂x, ∂/∂y, ∂/∂z) , ( v_1, v_2, v_3) )
The del operator is linear
The laplacian operator
Similar laplacian operator ∇^2:
Defined by
∇^2 = ∇ • ∇ ≡ ∂^2 /∂x^2 + ∂^2 /∂y^2 + ∂^2 /∂z^2
Can act on both scalar and vector fields and is linear as the del operator is linear
∇^2 φ = div(grad φ)
∇^2 v = grad( div v) - curl(curl(v))
