Answer "Find Divergence & Curl of Vector Field A

[gtfitzpatrick]

Homework Statement

find the divergence and curl of the vector field

A = (x/($$\sqrt{x^2 + y^2 + z^2}$$))i + (y/($$\sqrt{x^2 + y^2 + z^2}$$))j + (z/($$\sqrt{x^2 + y^2 + z^2}$$))k

Homework Statement

The Attempt at a Solution

Im not going to go through the whole lot but i have done the whole Differentiation but it would take for ever to input it into this.

and i got the curl to be 0i+0j+0k and the divergence to be 0, is this possible or likely?

 

[vela]

According to Mathematica, the curl is 0, but the divergence isn't.

 

[yungman]

Homework Statement

find the divergence and curl of the vector field

A = (x/($$\sqrt{x^2 + y^2 + z^2}$$))i + (y/($$\sqrt{x^2 + y^2 + z^2}$$))j + (z/($$\sqrt{x^2 + y^2 + z^2}$$))k

Homework Statement

The Attempt at a Solution

Im not going to go through the whole lot but i have done the whole Differentiation but it would take for ever to input it into this.

and i got the curl to be 0i+0j+0k and the divergence to be 0, is this possible or likely?

$$\nabla \cdot \vec{A} = \frac{\partial}{\partial x}(\frac{x}{\sqrt{x^2 + y^2 + z^2}}) + \frac{\partial}{\partial y}(\frac{y}{\sqrt{x^2 + y^2 + z^2}}) + \frac{\partial}{\partial z}(\frac{z}{\sqrt{x^2 + y^2 + z^2}})$$

All three terms are not zero.

 

[gtfitzpatrick]

I'm really really sorry, i inputed the question wrong here, the square root should be cubed like this

$$\nabla \cdot \vec{A} = \frac{\partial}{\partial x}(\frac{x}{(\sqrt{x^2 + y^2 + z^2})^3}) + \frac{\partial}{\partial y}(\frac{y}{(\sqrt{x^2 + y^2 + z^2})^3}) + \frac{\partial}{\partial z}(\frac{z}{(\sqrt{x^2 + y^2 + z^2})^3})$$

thanks a million for the replys,but could you tell if mathematica gets 0 and 0 for the curl and div now, thanks

 

[peanuts]

I'm really really sorry, i inputed the question wrong here, the square root should be cubed like this

$$\nabla \cdot \vec{A} = \frac{\partial}{\partial x}(\frac{x}{(\sqrt{x^2 + y^2 + z^2})^3}) + \frac{\partial}{\partial y}(\frac{y}{(\sqrt{x^2 + y^2 + z^2})^3}) + \frac{\partial}{\partial z}(\frac{z}{(\sqrt{x^2 + y^2 + z^2})^3})$$

thanks a million for the replys,but could you tell if mathematica gets 0 and 0 for the curl and div now, thanks

Hi why do you need mathematica to do the div for you. Just do some simple mental sums, apply product rule to each partial. Anyway its pretty clear that the div is zero.

 

[vela]

Yes, the div A and curl A both vanish, at least for $(x,y,z)\ne(0,0,0)$.

If you've taken a course on electromagnetism, you might have noticed that

$$\vec{A} = \frac{1}{r^2}\hat{r}$$

which is like the electric field around a point charge, so you'd expect the divergence to be zero away from the origin. Also, knowing the electric force is conservative, you would expect the curl to be zero as well.