# Vector function/field problem

1. May 22, 2004

### galipop

Hi Folks,

I'm just working through a few exercises relating to vector functions and vector fields.

Can you look over my working and let me know if I'm on the right track?

vector function: g(x,y,z) = x^3 + y + z^2
vector field F = (2xz , sin y , e^y)

i need to evalute the following:
1. grad g = (3x^2 , 1 , 2z)

2. div g = does not exist. From what I've seen you can't find the div of a vector function. Is this correct?

3. div F = 2z + cos y + 0

4. curl F = e^y i + (2x) j + 0k

5. grad (grad g): does not exist as this operation can't be performed twice. correct?

6. curl (grad g) = 0

7. div ( curl F ) = 0

How does the above look?

Many Thanks,

Pete

2. May 22, 2004

### arildno

"2. div g = does not exist. From what I've seen you can't find the div of a vector function. Is this correct?"

If you meant to write "you can't find the div of a scalar function" function, then you are right.
The divergence operator acts on a vector and produces a scalar.

"5. grad (grad g): does not exist as this operation can't be performed twice. correct?"

I would think that in the beginning course you're taking, this would be correct (not too sure, though!)

The gradient operator is customarily introduced as an operator that takes a scalar function f into the vector function $$\nabla{f}$$

If this is basically what you've been told about the gradient operator, then you have the right answer.

However, it is extremely useful in maths to also be able to calculate the "gradient" of a vector function.
This object will be a matrix.
(If this is completely unknown to you, and the book you're reading makes no references to such matrices, then you should stick with your original answer)

Otherwise, it looks good.