Find the divergence of the fields shown on the attached image.

Images:
This image has been Flagged as inappropriate Click to unflag
Image (1 of 1)

Expert Answers

An illustration of the letter 'A' in a speech bubbles

a) Divergence of a vector field is a scalar quantity that represents how the field spreads out, or "diverges", in different directions. It is usually denoted as

`vecgrad*vecF` and is calculated as

`vecgrad * vecF = (dF_x)/(dx) + (dF_y)/(dy) + (dF_z)/(dz)` .

In the given vector field, the components are

`F_x = x` , so `(dF_x)/(dx) = 1`

`F_y = y^3z^2` , so `(dF_y)/(dy) = 3y^2z^2`

`F_z = xz^3` , so `(dF_z)/(dz) = 3xz^2` .

Thus, the divergence of the given vector field is

`vec grad * vecF = 1 + 3y^2z^2 + 3xz^2` .

b) The divergence of this vector field can be calculated the same way. Here,

`(dF_x)/(dx) = cosy`

`(dF_y)/(dy) = 2xy`

and `(dF_z)/(dz) = 0`

So the divergence is

`vec grad * vecF = cosy + 2xy` .

 

Approved by eNotes Editorial Team

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial