The document discusses gradient and divergence. Gradient is a vector quantity that points in the direction of maximum rate of increase of a scalar function. Divergence is a scalar quantity that measures the spread of a vector field from a point. It gives examples of gradient on a hillside and divergence for a faucet and whirlpool. It also provides mathematical expressions and geometric interpretations for gradient and divergence, and examples of problems calculating them.

1. 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.

2. 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.

3. 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.

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

5. i j k
∂ /∂x ∂ /∂y ∂ /∂z
Fx Fy Fz

6. Example:-
A Whirpool

7. 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