# Calculating Div and Curl for some arbitrary vector fields

• Vuldoraq
In summary, the task is to calculate the divergence and curl of three vector fields, given by (a) r^n*x, (b) r^n*a, and (c) r^n*(a x x), where r is the magnitude of the position vector x and a is a constant vector. The solution involves using spherical or Cartesian coordinates and vector identities to find the divergence and curl for each field.
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 .

?

## 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.

Vuldoraq said:
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.

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

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.

Vuldoraq said:

## 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 .

Vuldoraq said:
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.

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!).

## 1. What is the purpose of calculating div and curl for vector fields?

The purpose of calculating div and curl for vector fields is to gain a better understanding of the behavior and properties of the vector field. These calculations can provide information about the flow, rotation, and convergence of the vector field, which can be useful in various scientific and engineering applications.

## 2. How is the divergence of a vector field calculated?

The divergence of a vector field is calculated by taking the dot product of the gradient operator with the vector field. This operation results in a scalar value that represents the amount of flux flowing out of a point in the vector field per unit volume.

## 3. What does a positive or negative divergence value indicate?

A positive divergence value indicates that the vector field is diverging or spreading out from a point, while a negative value indicates that the field is converging or flowing towards a point. A divergence value of zero indicates that the vector field has no net flow at a given point.

## 4. How is the curl of a vector field calculated?

The curl of a vector field is calculated by taking the cross product of the gradient operator with the vector field. This operation results in a vector that represents the amount of rotation or circulation of the vector field at a given point.

## 5. Can the div and curl of a vector field be calculated for any arbitrary vector field?

Yes, the div and curl can be calculated for any arbitrary vector field as long as the vector field is continuous and differentiable at every point. However, the calculations may be more complex for certain vector fields, and some may require advanced mathematical techniques.

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