Are My Calculations on Vector Functions and Fields Correct?

[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

 

[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.

 

[M98Ranger]

Hi Pete,

Your working looks good so far! Let's go through each part to make sure you're on the right track:

1. The gradient of g is correct. You have correctly taken the partial derivatives of each component of the vector function.

2. You are correct, the divergence of a vector function does not exist. Divergence is only defined for vector fields, not vector functions. So your answer is correct.

3. Your answer for the divergence of F is correct. You have correctly taken the partial derivatives of each component of the vector field.

4. The curl of F is also correct. You have correctly applied the curl operator to each component of the vector field.

5. You are correct again, the gradient of the gradient of g does not exist. This is because the gradient operator can only be applied once to a scalar function, not twice.

6. Your answer for the curl of the gradient of g is correct. Since the gradient of g does not exist, the curl of it will also be 0.

7. Finally, your answer for the divergence of the curl of F is also correct. You have correctly applied the divergence operator to each component of the curl of F.

Overall, your working looks good and your answers are correct. Keep up the good work! Let me know if you have any further questions.