# Help w/ Seaborn's Mathematics for the Physical Sciences

• bcjochim07
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.
bcjochim07

## 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

$$\oint$$F $$\cdot$$ dA = $$\int$$Vgrad$$\cdot$$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.

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!

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.

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.

Just look at a globe of the Earth …

a circle of latitude has radius bsinθ

(while a circle of longitude has radius b).

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.

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.

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!

## 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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