Gradient divergence curl

gradient,divergence,curl

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│cos θ 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).( F x i+F y j+F z k ) .F = ∂ F x /∂x + ∂ F y /∂y +∂ F z /∂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. CURL It is operated on vecter function. We know, = (∂ /∂x) i +(∂ /∂y) j +(∂ /∂z) k curl xF = xF =(∂ F z /∂x-∂ F x /∂y) i -(∂ F z /∂∂ It is a vector quantity . GEOMETRICAL INTREPRETATION:- It is a measure of infinitesimal rotation/curl of a vector field from a given point in 3-D space.It’s direction can be given by Right hand screw rule. If a function have, xF =0,then it is called irrotational i j k ∂ /∂x ∂ /∂y ∂ /∂z F x F y F z

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 ) = 2x 2 +3y 3 +6z 6 b)f( x,y,z ) = 3x 2 y 6 z 7 3.Calculate the divergence of following vector function. a)F = x 3 i + 2xyz j + 3z 2 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