Find the divergence and curl of the given vector field

[chwala]

Homework Statement

This is my own question (set by myself). Refreshing on this area...

Find the divergence and curl of the given vector field;

$F = x \cos xi -e^y j+xyz k$

Relevant Equations

Vector calculus

Been long since i studied this area...time to go back.

$F = x \cos xi -e^y j+xyz k$

For divergence i have,

$∇⋅F = (\cos x -x\sin x)i -e^y j +xy k$

and for curl,

$∇× F = \left(\dfrac{∂}{∂y}(xyz)-\dfrac{∂}{∂z}(-e^y)\right) i -\left(\dfrac{∂}{∂x}(xyz)-\dfrac{∂}{∂z}(x \cos x)\right)j+\left(\dfrac{∂}{∂x}(-e^y)-\dfrac{∂}{∂y}(x\cos x)\right)k$

$∇× F = xz i -yzj$

cheers insight welcome.

 

[Hennessy]

Looks Good. One thing to watch out for is make sure your unit vectors have hats on them
like this. p.s this isn't pre calculus Maths as this is Vector Calculus !

$$\hat{i}\: \hat{j}\: \hat{k}$$

Just to avoid confusion. i thought you had i as the complex number and i was like but its derivative would be hyperbolic. Once i figured out i, j and k were your unit vectors all was good, you've differentiated each unit vector component correctly for the divergence and the curl :)

 

[MatinSAR]

$∇× F = xz i +yzj$

I think it is a typo but second component should be nagative.

Edit:
Now I think you made a mistake in calculating $∇× F$.
You can check your final answer using online calculators.

Check this link.

 

[Mark44]

p.s this isn't pre calculus Maths as this is Vector Calculus !

I moved the thread for exactly this reason.

 

[Orodruin]

Looks Good

It does not.

For divergence i have,

$∇⋅F = (\cos x -x\sin x)i -e^y j +xy k$

Divergence is a scalar, not a vector.

 

[chwala]

It does not.

Divergence is a scalar, not a vector.

I am aware of that ...let me amend..

$∇⋅F = (\cos x -x\sin x) -e^y +xy$

Cheers @Orodruin

 

[chwala]

I think it is a typo but second component should be nagative.

Edit:
Now I think you made a mistake in calculating $∇× F$.
You can check your final answer using online calculators.

Check this link.

correct- i will just amend my original post...cheers

 

[SammyS]

correct- i will just amend my original post...cheers

When you amend a post after several replies, especially in the case of the OP, it can be very confusing.
So, if you do amend later, please make that very clear in the amended post. - Mention what's amended and why.

 

[chwala]

When you amend a post after several replies, especially in the case of the OP, it can be very confusing.
So, if you do amend later, please make that very clear in the amended post. - Mention what's amended and why.

Noted @SammyS

 

[Mark44]

When you amend a post after several replies, especially in the case of the OP, it can be very confusing.
So, if you do amend later, please make that very clear in the amended post. - Mention what's amended and why.

Amen.

Using the strike BBCode strikeout capability is very helpful to let readers know what changed.

 

[Hennessy]

Looks Good. One thing to watch out for is make sure your unit vectors have hats on them
like this. p.s this isn't pre calculus Maths as this is Vector Calculus !

$$\hat{i}\: \hat{j}\: \hat{k}$$

Just to avoid confusion. i thought you had i as the complex number and i was like but its derivative would be hyperbolic. Once i figured out i, j and k were your unit vectors all was good, you've differentiated each unit vector component correctly for the divergence and the curl :) Edit: Other peeps have corrected me and it was also my mistake the answer should be scalar and not a vector , apologies for any confuison.