Calculate the gradient of the function

You have the notation wrong for gradient. You wrote div instead. You can't take the div of a scalar. V1, V2 and V3 are all scalars.

You also have the wrong expressions for del F in cartesian, cylindrical and spherical coordinates. Look them up.

 

You did not write "just del". You wrote del-dot. The dot was wrong. You can write del f or you can write grad f. Same thing. (But del-dot-F is OK whereas del F or grad F are not.)

Your expression for grad f is correct for Cartesian coordinates x,y,z. I use i, j , and k for the unit vectors. I also use bold type for all vectors.

When you get to V2 and V3 the expressions for grad V are very different from Cartesian. V2 is using cylindrical and V3 is using spherical.

 

Yes, ∂(-2xz)/∂y = 0.
No, you are not following your formula in post 3 for V1.
Don't keep changing your notation for the unit vectors. Use i j k why not.

 

Was what 6y i + k ?

 

I'm getting dizzy. Correct for what?

Tell you what, start over with your expression for grad V1 from your post 3, use i j k unit vectors, and let's see if we can't figure this one out. Remember to use V1, not f.

 

I said use V1, not f.
OK, one term at a time. Try ∂(V1)/∂x again?

 

What about ∂(-2xz)/∂x?

 

Better.
Now, can you do the rest of grad V1?