Question

For any vector field F⃗ and any scalar function f we define a new field 

a) Assuming that the appropriate partial derivatives are continuous, show the following formula:

b) Let ⃗x = x⃗i + y ⃗j + z ⃗k and the vector field 

Use the formula found in a) to answer
the following question: is there a number p such that F⃗ is incompressible (that is, its divergence is zero)?

f F)(x,y,z) = f(x,y,z)F(x,y, z)

Answer 1

div (fF) = div (FiİHFJ+P,k) div ø--+--+- . . definition of of of

Adding these eqns Ох (b) G - - and F-z

▽f-grad (x2+y2+z2) grad (x2+y2+z2) grad (x2+y+2 그-1 x +y +z 그-1 Эт

-P grad (хЧут 2, 2, 2 2 그-1 -p

X1 2( 2,22 2, 2 2 2 (x2+y2+22) (x+y2+z2)

Now since G is incompressible p+3 0 (x2+y2)