Calculating Div and Curl for some arbitrary vector fields

[Vuldoraq]

Homework Statement

Calculate the (1) divergence and (2) curl of the following vector fields.
(a) $$\widehat{E}(\widehat{x}) = r^{n}\widehat{x}$$
(b) $$\widehat{E}(\widehat{x}) = r^{n}\widehat{a}$$
(c) $$\widehat{E}(\widehat{x}) = r^{n}*(\widehat{a} X \widehat{x}$$
where r = |$$\widehat{x}|$$ and $$\widehat{a}$$ is a constant vector .

Homework Equations

?

The Attempt at a Solution

I know how to find the divergence and curl of a vector or scalar field if told the equation for the field explicitly, but how would I solve the above? I don't know where to begin so any help would be greatly appreciated!

Also what is a constant vector? I thought to define a vector you have to give a direction and magnitude and in giving it those quantities you effectively make it constant? Unless it means that wherever in your coordiante system it is it has the same value (but wouldn't that just be a vector)? Please help.

 

[gabbagabbahey]

I know how to find the divergence and curl of a vector or scalar field if told the equation for the field explicitly, but how would I solve the above? I don't know where to begin so any help would be greatly appreciated!

You are given the explicit fields.

For the first one, you have two good options: (1) Use Spherical coordinates where $$\hat{x}$$ denotes the radial unit vector, and $$r$$ denotes the radius. (2) Use Cartesian coordinates, where $$r=\sqrt{x^2+y^2+z^2}$$ and $$\hat{x}=\frac{x\hat{i}+y\hat{j}+z\hat{k}}{\sqrt{x^2+y^2+z^2}}$$.

Also what is a constant vector? I thought to define a vector you have to give a direction and magnitude and in giving it those quantities you effectively make it constant? Unless it means that wherever in your coordiante system it is it has the same value (but wouldn't that just be a vector)? Please help.

The length or direction of some vectors depend on position. For example, the length of the vector $$\vec{u}=2x\hat{i}$$ depends on its position along the x-axis. A constant vector is a vector whose length and direction are position independent. For example; $$\vec{u}=4\hat{i}-3\hat{j}$$ always has length of 5 units, and always points in the same direction no matter where it is placed.

 

[Vuldoraq]

Okay, thankyou for your help.

For part (a) then, is this the right solution;

$$\nabla \cdot \widehat{E}=\frac{1}{r^{2}} \frac{d}{dr} r^{2}r^{n}$$
$$\frac{1}{r^{2}} \frac{d}{dr} r^{n+2}=(n+2)*r^{n-1}$$

$$\nabla \times \widehat{E}=0$$

For part (b) would both the divergence and the curl be zero because the constant vector $$\widehat{a}$$ has no dependence on the coordinate system?

For part c I'm not really sure how to cross $$\widehat{x}$$ and $$\widehat{a}$$. Is it right to say that (using spherical polars);

$$\widehat{x}=\widehat{r}*x$$
and
$$\widehat{a}=\widehat{r}*a_{1}+\widehat{\theta}*a_{2}+\widehat{\phi}*a_{3}$$?

 

[Vuldoraq]

I have made a cancerous error in the transcription of my problem from paper to pc. All the hats below, implying unit vectors, should be arrows, making them plain vectors.

Homework Statement

Calculate the (1) divergence and (2) curl of the following vector fields.
(a) $$\widehat{E}(\widehat{x}) = r^{n}\widehat{x}$$
(b) $$\widehat{E}(\widehat{x}) = r^{n}\widehat{a}$$
(c) $$\widehat{E}(\widehat{x}) = r^{n}*(\widehat{a} \times \widehat{x})$$
where r = |$$\widehat{x}|$$ and $$\widehat{a}$$ is a constant vector .

Thus we should have;

(a) $$\vec{E}(\vec{x}) = r^{n}\vec{x}$$
(b) $$\vec{E}(\vec{x}) = r^{n}\vec{a}$$
(c) $$\vec{E}(\vec{x}) = r^{n}*(\vec{a} \times \vec{x})$$
where r = |$$\vec{x}|$$ and $$\vec{a}$$ is a constant vector .

 

[gabbagabbahey]

I have made a cancerous error in the transcription of my problem from paper to pc. All the hats below, implying unit vectors, should be arrows, making them plain vectors.
Thus we should have;

(a) $$\vec{E}(\vec{x}) = r^{n}\vec{x}$$
(b) $$\vec{E}(\vec{x}) = r^{n}\vec{a}$$
(c) $$\vec{E}(\vec{x}) = r^{n}*(\vec{a} \times \vec{x})$$
where r = |$$\vec{x}|$$ and $$\vec{a}$$ is a constant vector .

Okay; then for (a)so the same thing as above but with $$\vec{x}=r\hat{x}$$

For (b) and (c), you will need to use a couple of vector identities. Be careful, $$\vec{a}$$ is a constant vector, so $$\vec{\nabla}\cdot\vec{a}=\vec{\nabla}\times\vec{a}=0$$...But that doesn't necessarily mean $$\vec{\nabla}\cdot(r^n\vec{a})=0$$ or $$\vec{\nabla}\times(r^n\vec{a})=0$$...you have vector identities for the div/curl of the product between a scalar function and a vector function don't you?...Use them.

 

[Vuldoraq]

I see. I think it was just the new notation my teacher is using this term that threw me off. It seems so simple now that you have explained it to me (a clear explanation is worth a hundred hours of hard work!).

Thanks for your help!