Help w/ Seaborn's Mathematics for the Physical Sciences

In summary, the author follows a similar process to derive the other surfaces. They start by drawing a diagram representing the surface, and then derive the dimensions of the individual area elements.
  • #1
bcjochim07
374
0

Homework Statement


In spherical polar coordinates, a vector F is given by F=(r*costheta*cosphi)rhat + (r*costheta*sinphi)thetahat + (r*sintheta*cosphi)phihat

Check the validity of the divergence theorem for this vector and the volume that is one octant of a sphere radius b.


Homework Equations


[tex]\oint[/tex]F [tex]\cdot[/tex] dA = [tex]\int[/tex]Vgrad[tex]\cdot[/tex]Fd3r


The Attempt at a Solution

The solution for this problem is in my book, and I am having difficulty following it, I think because it is in polar coordinates, and I am still trying to get used to them.

The author proceeds to divide the surface into four area elements:

spherical surface : dA = rhat(b^2)*sintheta*dtheta*dphi

before I go on to any of the other elements, could someone please help me understand how the author derived this expression? Thanks.
 
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  • #2
bcjochim07 said:
spherical surface : dA = rhat(b^2)*sintheta*dtheta*dphi

before I go on to any of the other elements, could someone please help me understand how the author derived this expression? Thanks.

Hi bcjochim07! :smile:

Mark a "rectangle" on the surface, with change in latitude dθ and change in longitude dφ.

Then its width is bsinθ dθ, and its height is bdφ, so its area is b2sinθdθdφ. :wink:
 
  • #3
ok,

So I think I understand where the bdφ comes from; isn't that an arc length? Then I would think that the other dimension would be bdθ, so where does that sinθ come from?

Thanks.
 
  • #4
Just look at a globe of the Earth …

a circle of latitude has radius bsinθ

(while a circle of longitude has radius b). :smile:
 
  • #5
Thanks. Great explanation. Now I am trying to figure out how the book derives the other surfaces. For example, the area element in the xz plane is in the negative phi hat direction with dA = r dr dθ. Considering the area formula for a circular sector, I thought it should be (1/2)rdrdθ.
 
  • #6
Ok... I think I see it now. I just need to divide the area up into little "rectangles" just like I did for the spherical surface. Each of these rectangles has dimensions dr by rdθ.
 
  • #7
bcjochim07 said:
Ok... I think I see it now. I just need to divide the area up into little "rectangles" just like I did for the spherical surface. Each of these rectangles has dimensions dr by rdθ.

Yup! … it's all simple geometry …

just draw the right diagram, and it becomes obvious! :biggrin:
 

What is Seaborn's Mathematics for the Physical Sciences?

Seaborn's Mathematics for the Physical Sciences is a comprehensive textbook that covers various mathematical concepts and techniques used in the study of physical sciences such as physics, chemistry, and engineering.

Who is the author of Seaborn's Mathematics for the Physical Sciences?

The author of Seaborn's Mathematics for the Physical Sciences is Dr. Richard Seaborn, a renowned physicist and mathematician with years of experience in teaching and research in the field of physical sciences.

What topics are covered in Seaborn's Mathematics for the Physical Sciences?

Seaborn's Mathematics for the Physical Sciences covers a wide range of topics including calculus, linear algebra, differential equations, complex analysis, and vector calculus. It also includes special topics such as Fourier analysis, group theory, and tensors.

Is Seaborn's Mathematics for the Physical Sciences suitable for self-study?

Yes, Seaborn's Mathematics for the Physical Sciences is designed for both self-study and classroom use. It includes numerous examples, practice problems, and exercises to help readers understand and apply mathematical concepts in the physical sciences.

Are there any additional resources to accompany Seaborn's Mathematics for the Physical Sciences?

Yes, there is a companion website that provides additional resources such as solutions to selected problems, interactive quizzes, and extra practice problems. The website also includes links to useful online resources for further learning.

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