Understanding Vector Fields on a Sphere

[pizza_dude]

Homework Statement

find the values of the integral

$\int_{S} \vec A\cdot\ d\vec a$

where,

$\vec A\ = (x^2+y^2+z^2)(x\hat e_{1}+y\hat e_{2}+z\hat e_{3})$

and the surface S is defined by the sphere $R^2=x^2+y^2+z^2$

Homework Equations

first i must evaluate the integral directly, so i don't think there are any specific formulas other than ones you must derive from the geometry specific to the problem. i also have to calculate using gauss' theorem but for that there's a simple equation.

The Attempt at a Solution

really looking for an explanation on surface integrals. i know that $d\vec a$ is a small area on the surface of the sphere and equations must be derived from the geometry. I am having a hard time visualizing this and how it's suppose to work. for now, i would appreciate a good explanation of surface integrals to help me visualize the problem.

thanks in advance.

 

[Orodruin]

$d\vec a$ is the area element, which is an infinitesimal area multiplied by the unit normal of the surface. Try to answer the following: What is the unit normal of the sphere? What is the product of $\vec A$ with $d\vec a$? How can you simplify the integrand using this?

 

[BvU]

Hello Pizza, welcome to PF :)

You familiar with spherical coordinates ? Reason I ask is because after all this is all happening on a sphere. Can you transform $\vec A(x, y, z)$ to $\vec A(r,\theta, \phi)$ ?

[edit] Well, I see you got some good help already. Bedtime for me!

 

[pizza_dude]

$d\vec a$ is perpendicular to the surface of the area (2 possible directions).

 

[Orodruin]

$d\vec a$ is perpendicular to the surface of the area (2 possible directions).

Yes, this is what I said, but which vector is perpendicular to the sphere surface at a given point on the sphere?

 

[Chestermiller]

I think Orodruin is asking "which spherical coordinate system unit vector is perpendicular to the sphere surface at a given point on the sphere?"

What is your physical interpretation of the vector A at the surface of the sphere? In what direction is it pointing at each point over the surface?

Chet

 

[pizza_dude]

I think Orodruin is asking "which spherical coordinate system unit vector is perpendicular to the sphere surface at a given point on the sphere?"

What is your physical interpretation of the vector A at the surface of the sphere? In what direction is it pointing at each point over the surface?

Chet

$\hat r$ is the answer i believe you may be looking for. I am not quite sure how to physically interpret $\vec A$. i just know that it's some vector along the surface of the sphere. sorry if I am making this difficult.

 

[Orodruin]

$\hat r$ is the answer i believe you may be looking for. I am not quite sure how to physically interpret $\vec A$. i just know that it's some vector along the surface of the sphere. sorry if I am making this difficult.

Correct, so $d\vec a = \hat r\, da$ where da is the infinitesimal area. Now, can you write $\vec A$ in terms of the position vector $\vec r$?

 

[pizza_dude]

$\vec A = (x, y, z) \vec r$ which i believe is equal to$(x \hat e_{1} + y\hat e_{2} + z\hat e_{3})$ which is the second half of $\vec A$ mentioned in the problem

 

[Chestermiller]

What is your understanding of what (x²+y²+z²) represents at the surface of a sphere of radius r?

Chet

 

[BvU]

$\vec A = (x, y, z) \vec r$ which i believe is equal to$(x \hat e_{1} + y\hat e_{2} + z\hat e_{3})$ which is the second half of $\vec A$ mentioned in the problem

This $\vec A = (x, y, z) \vec r$ can't be right: what can possibly its components (if existent) ?

Dear dude, I am starting to worry if you understand how $(x \hat e_{1} + y\hat e_{2} + z\hat e_{3} )$ is to be interpreted. Can you express it in Cartesian coordinates ? And in Spherical coordinates ?