Flux through a box? And divergence as a limit?

In summary, the conversation discusses the calculation of div F in two ways: using the geometric definition and using partial derivatives. It also mentions finding the flux through a three-dimensional box with four corners at specific points, and solving for the limit and flux. The final result for div F is 2.
  • #1
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Let F=(7z+8)i+2zj+(2z+7)k, and let the point P=(abc), where a, b and c are constants. In this problem we will calculate div F in two different ways, first by using the geometric definition and second by using partial derivatives.

(a) Consider a (three-dimensional) box with four of its corners at (abc), (a+wbc), (ab+wc) and (abc+w), where w is a constant edge length. Find the flux through the box.

Thus, we have
div F(xyz)=lim/(w->0) = (BLANK/BLANK) = 2



I solved the div F to be 2... don't know how to solve for flux or the lim.

the lim in the textbook is written as lim ϵ-> 0 (3/4piϵ**3) o∫∫[F.NdS]

thanks for the help!
 
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  • #2
OK so might be confusing... trying to solve the fraction, i know the lim equals 2.

also trying to solve flux.
 

1. What is flux through a box?

Flux through a box is a measure of the flow of a vector field through a closed surface or boundary. It is calculated by taking the dot product of the vector field and the unit normal vector of the surface, and then integrating over the surface.

2. How is flux through a box related to divergence?

Flux through a box is closely related to divergence, as it is essentially a measure of the divergence of a vector field over a closed surface. When the divergence of a vector field is positive, more flux is going out of the surface than coming in, and when it is negative, more flux is going in than coming out.

3. What is the significance of flux through a box in physics?

Flux through a box is an important concept in physics, as it is used to calculate the rate of flow of various quantities, such as mass, energy, and electric charge. It is also used to understand and analyze the behavior of fluid flow, heat transfer, and electromagnetic fields.

4. How is divergence calculated as a limit?

Divergence is calculated as a limit by taking the divergence of a vector field at a point, and then shrinking the volume around that point to approach zero. This is represented mathematically as the limit of the flux through a closed surface divided by the volume enclosed by the surface, as the volume approaches zero.

5. What are some real-world applications of flux through a box and divergence?

Flux through a box and divergence have many real-world applications, such as in fluid dynamics to analyze the flow of fluids in pipes and channels, in weather forecasting to understand air movement and wind patterns, and in electromagnetism to study the behavior of electric and magnetic fields. They are also used in engineering and design, such as in the design of aerodynamic structures and heat transfer systems.

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