# Vector Identities: Calculate & Surface Integrals

• neworder1
In summary: We want to calculate the surface integral over the surface defined by \int \vec{r} (\vec{a} \cdot \vec{n}) dS\int \vec{n} (\vec{a} \cdot \vec{r}) dSwhere \vec{a} is a constant vector and \vec{n} is a unit vector normal to the surface.In summary, the textbook is asking us to calculate surface integrals.
neworder1

## Homework Statement

1.

Calculate:

$$\nabla \times (\frac{\vec{p} \times \vec{r}}{r^{3}})$$

in cartesian and spherical coordinates, where $$\vec{p}$$ is a constant vector.

2.

Calculate surface integrals:

$$\int \vec{r} (\vec{a} \cdot \vec{n}) dS$$
$$\int \vec{n} (\vec{a} \cdot \vec{r}) dS$$

where $$\vec{a}$$ is a constant vector and $$\vec{n}$$ is a unit vector normal to the surface.

## The Attempt at a Solution

I tried do the first by using some basic vector identities but I didn't get anywhere (the result wasn't by any means neat and short ;)). I was told that Dirac delta is supposed to show up somewhere, but I don't see it.

The second one is probably done using Stokes' Theorem but I don't see any simple fashion in which it can be applied.

Last edited:
? You haven't posted any question at all.

I'm not sure if I understand your questions. Your first one asks to "calculate" that mathematical expression presumably using vector identities involving curl but you didn't specify in what mathematical form do you want the answer to be expressed in? In other words, there's so many equivalent ways you can express that statement, but you didn't specify what vectors, mathematical notation the final answer should contain so we can eliminate the other possibilities. Is this a "Show that the following may be equivalently written as" type of question?

For your second question, you didn't say what vector r is, and you didn't tell us what surface you're integrating over, as such we can't tell if it's bounded by a closed loop or whether it's a closed surface (then we may apply Div theorem) ie. as such we can't tell (or at least I can't tell) how to help you.

Perhaps it's best if you post the exact question the textbook is asking here to clear up the confusion.

1. I mean something like this: we can calculate that
$$\nabla \cdot (\frac{\vec{r}}{r^3}) = 4 \pi \delta^{3}(\vec{r})$$ (Dirac delta),
and here we have to do something similar, i.e. express the rotation in a more "explicit" way (using Dirac delta etc.). Using vector identities I obtained $$\nabla \times (\frac{\vec{p} \times \vec{r}}{r^3}) = 4 \pi \vec{p} \delta^{3}(\vec{r}) + (\vec{p} \cdot \nabla)\frac{\vec{r}}{r^3}$$, and I wonder whether the second term (involving $$\vec{p} \cdot \nabla$$) can be simplified (expressed using Dirac delta etc.)?

2. We can assume that the surfaces are closed. $$\vec{r}$$ is the vector connecting the current point and the origin.

## 1. What are vector identities?

Vector identities are mathematical equations that describe relationships between vector quantities. They are used to simplify and solve problems involving vectors.

## 2. How do you calculate vector identities?

Vector identities can be calculated using algebraic manipulations, vector operations such as dot product and cross product, and the use of trigonometric functions.

## 3. What is the significance of vector identities in science?

Vector identities are essential in many fields of science, including physics, engineering, and mathematics. They are used to describe and understand the behavior of physical systems and to solve complex problems involving vectors.

## 4. What are surface integrals in vector identities?

A surface integral is a type of vector identity that calculates the flux or flow of a vector field through a surface. It is used to determine the amount of a vector quantity passing through a given surface.

## 5. How are surface integrals used in real-world applications?

Surface integrals are used in various real-world applications, such as determining the electric field around a charged object, calculating the flow of fluids in a pipe, and analyzing the stress distribution on a solid surface.

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