Gradient divergence curl

GRADIENT
It is operated on scalar function.
From our derivative,We know
dF = (∂F/∂x)dx+(∂F/∂y)dy+(∂F/∂z)dz
dF = (∂F/∂xi +∂F/∂yj +∂F/∂zk).(dxi +dyj +dzk)
dF = F.dr
Where F = ∂F/∂xi +∂F/∂yj +∂F/∂zk
Fis the gradient of F.It is a vector quantity.
GEOMETRICAL INTERPETATION:-
dF = │ F││dr│cosθ
if dr = constant,then for maximum change in F,θ=0
so, F =dF/dr
F points is in the direction of maximum change in F.
F is slope at this maximum change direction.

Example:-
If we are standing on hillside,then the direction of
steepest ascent is direction of gradient&slope of
that direction is magnitude of gradient.

DIVERGENCE
It is operated on vector function.
Mathematical Expression
From defination of gradient,
= (∂ /∂x)i + (∂ /∂y)j +(∂ /∂z)k
.F =(∂/∂x i +∂/∂y j +∂/∂z k).(Fxi+Fy j+Fzk)
.F = ∂Fx /∂x + ∂Fy /∂y +∂Fz /∂z
It is a scalar quantity.
Geometrical Intrepretation:-
Gradient is a measure of spreadness (divergence)of a vector field
from a point.

Example:-
Faucet(Water is spread from it’s opening point.)

i j k
∂ /∂x ∂ /∂y ∂ /∂z
Fx Fy Fz

1.Find a function whose divergence &curl both are
zero.
2.Find gradient of following scalar function.
a) f(x,y,z) = 2x2 +3y3+6z6
b)f(x,y,z) = 3x2y6z7
3.Calculate the divergence of following vector
function.
a)F = x3 i + 2xyz j + 3z2 k
b) F= xy i + 2yz j + 3zx k
4.Calculate the curl of following vector function&also
direction.
a)F = x i+y j +z k
b)F = -y i+ x j

Gradient divergence curl

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