Divergence and Curl "Del", ∇ - A defined operator ∂ ∂ ∂ ∇ = , , ∂x ∂ y ∂ z
The gradient of a function (at a point) is a vec tor that points in the direction in which the function increases most rapidly.
A vector field is a vector function that can be thou ght 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. 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 u nit 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 flu id 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. Vector Field Scalar Funct ion F = Pxyz()()(),,, Qxyz ,,, Rxyz ,, f( x, yz, )
Gra dient grad( f )
∇f = fffx, y , z Div ergence div (F ) ∂∂∂ ∂∂∂P Q R ∇⋅=F , , ⋅PQR, , =++=++ PQR ∂∂∂xyz ∂∂∂ xyz x y z Curl curl (F ) i j k ∂ ∂ ∂ ∇×=F =−−−+−()RQ ijk() RP() QP ∂x ∂ y ∂ z yz xz xy P Q R
∇×=F RQy − z, −() RPQP xzxy −, − F(x,, y z)= xe−z ,4 yz2 ,3 ye − z
∂ ∂ ∂ div()F=∇⋅= F ,, ⋅ xe−z ,4,3 yz2 ye − z = e−z+4 z2 − 3 ye − z ∂x ∂ y ∂ z
i j k ∂ ∂ ∂ curl ()F=∇×= F = 3e− z − 8 yz 2 , − ( 0 − ( − xe − z ) ) , 0 ∂x ∂ y ∂ z xe−z4 yz2 3 ye − z 3e−z− 8, yz2 − xe − z ,0 grad( scalar function ) = Vector Field div(Vector Field ) = scalarfunction curl (Vector Field) = Vector Field 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 )) curl( grad( f )) 0 Vector Field Vector Field Vector Field vector
grad( div (F )) div( div (F )) curl( div (F )) scalar function scalar function scalar function
grad( curl (F )) div( curl (F )) 0 curl( curl (F )) Vector Field Vector Field scalar Vector Field
2 of the above are always zero. Verify the given identity. Assume conti nuity of all partial derivatives. curl grad f = 0 ()() . i j k grad( f)= f, f , f ∂ ∂ ∂ x y z curl() grad() f=∇×∇ f = ∂x ∂ y ∂ z
fx f y f z
∇×∇=fffzy − yz, −( ffff zx − xz) , yx − xy = 0 since mixed partial derivatives are equal.
Verify the given identity. Assume conti nuity of all partial derivatives. div() curl ()F = 0.
Let F = Pxyz( ,,,) Qxyz( ,,,) Rxyz( ,, ) curl(F)=− Ry Q zz, P − R xx , Q − P y i j k ∂ ∂ ∂ curl ()F=∇× F = divcurl( (F))=−( Ryz Q) +−( PR zx) +−( Q xy P ) ∂x ∂ y ∂ z x y z P Q R div( curl(F))= Ryx − Q zx +− P zy R xy + Q xz − P yz
div( curl(F))= R yx − Qzx + Pzy − Rxy + Qxz − Pyz = 0