[Mathematical physics] - Integral problem

In summary, the author is not sure how to solve the problem and does not know of an integral theorem to help.
  • #1
Mulz
124
5

Homework Statement


Calculate the integral

[tex] \int_{S} (\frac{A}{r^2}\hat{r} + B\hat{z}) \cdot d\vec{S} [/tex]

Where S is the sphere with r = a.

2. The attempt at a solution

I have no clue how to solve this problem. I have thought of introducing spherical coordinates and somehow finding a connection but I don't think that works.

I tried breaking out [tex] d\vec{S} = \frac{\partial \vec{r}}{\partial u} \cdot \frac{\partial \vec{r}}{\partial v } dudv [/tex]

using the formula above but not sure on how the dot product works. What confuses me with the integrand with the z and r. The answer is [tex] 4πA [/tex].
 
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  • #2
Mulz said:
[tex] d\vec{S} = \frac{\partial \vec{r}}{\partial u} \cdot \frac{\partial \vec{r}}{\partial v } dudv [/tex]
This is not the correct expression for the surface element. The correct expression is
$$
d\vec S = \frac{\partial \vec r}{\partial u} \times \frac{\partial \vec r}{\partial v} du\, dv.
$$
This is equal to ##\vec n \, dS##, where ##\vec n## is a unit normal and ##dS## the area spanned by the tangent vectors on the sphere. What is the unit normal to the unit sphere?

Also, the second term does not contribute to the integral. Do you know of some integral theorem that you can invoke to come to this conclusion?
 
  • #3
Ask yourself, what is the vector ## d \vec S ##?
 
  • #4
Orodruin said:
This is not the correct expression for the surface element. The correct expression is
$$
d\vec S = \frac{\partial \vec r}{\partial u} \times \frac{\partial \vec r}{\partial v} du\, dv.
$$
This is equal to ##\vec n \, dS##, where ##\vec n## is a unit normal and ##dS## the area spanned by the tangent vectors on the sphere. What is the unit normal to the unit sphere?

Also, the second term does not contribute to the integral. Do you know of some integral theorem that you can invoke to come to this conclusion?

I might be completely lost but isn't the normal of a surface given by [tex] \nabla = \frac{\partial}{\partial x} + \frac{\partial}{\partial y} + \frac{\partial}{\partial z} [/tex]? It should be ⊥ to the tangentvector.

Or maybe the position vector [tex] \vec{r} = x\hat{x} + y\hat{y} + z\hat{z} [/tex]? I think this one simply desribes the coordinates of a sphere as opposed to a normal.

The don't know if the book covered the relevant integral theroem since I read it 4 times.
 
  • #5
The ##\nabla## operator in itself is a differential operator, not a vector. Acting on a scalar field, it becomes the gradient, which is normal to the level surfaces of the field.

The integral theorem is the divergence theorem. You can do without as well, but it helps.
 
  • #6
Orodruin said:
The ##\nabla## operator in itself is a differential operator, not a vector. Acting on a scalar field, it becomes the gradient, which is normal to the level surfaces of the field.

The integral theorem is the divergence theorem. You can do without as well, but it helps.
I don't think I'm supposed to use the divergence theorem since it's not discussed in this chapter.
 
  • #7
As I already said, it helps, but it is not necessary.
 
  • #8
Orodruin said:
As I already said, it helps, but it is not necessary.
How should I begin to solve it? I'm not sure how to start. Feels like I'm missing a lot of knowledge even when repeaditely reading the chapter.
 

FAQ: [Mathematical physics] - Integral problem

1. What is an integral in mathematical physics?

An integral is a mathematical concept that represents the accumulation of a quantity over a given interval. In mathematical physics, integrals are used to describe physical phenomena such as motion, energy, and forces.

2. How is an integral problem solved?

Solving an integral problem involves finding the anti-derivative of a given function and then evaluating it at the limits of integration. This can be done using various techniques such as substitution, integration by parts, and partial fractions.

3. What is the significance of integrals in mathematical physics?

Integrals play a crucial role in mathematical physics as they allow us to describe physical quantities that vary continuously over time or space. They are essential in the formulation of physical laws and equations, and in solving real-world problems.

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