�⃗�=�̂� 𝑟𝑛Consider the example of a vector field

Find out the divergence of the vector field, i.e.

=

n 3 2 1 0 -1 -2 -3 -4

4r 3 0 - -

EXERCISE

n 3 2 1 0 -1 -2 -3 -4

4r 3 0 - -

�⃗�=�̂� 𝑟𝑛

decreasing with r increasing with r

is a divergent vector field for all values of ‘n’ �⃗�=

�̂�𝑟2

An example of divergent field

Does it imply A is convergent field?

A vector-valued function F(x,y) can be visualized as a vector field. At a point (x, y), we plot the value of F(x,y) as a vector with tail anchored at (x,y), such as in the following figure

We repeat this over a set of points (x,y) so that we can realize the entire vector fieldFor example, consider the function F(x,y) = (y, -x). We calculate values of the function at a set of points, such as

F(1,0)=(0,-1)

F(0,1)=(1.0)

F(1,1)=(1,-1)

F(1,2)=(1,-2)

Visualization of Vector field

By plotting each of these vectors anchored at the corresponding points, we begin to see some of the structure of the vector field.

If we continued plotting such vectors at many points, they would begin to overlap and look quite messy. Hence, we typically scale the arrows in vector field plots. In the example, we drew the vectors at only 40% their actual length. By plotting this field of arrows, we see that the vector field F(x,y) = (y, -x) appears to rotate in a clockwise direction.

F(x,y) = (y, -x) F(x,y) = (x, y)