Gauss's Divergance Theorem and Stokes's Theorem

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In summary, this conversation discusses the concept of integral forms describing electric fields and the difficulty in using Gauss's and Stokes's theorems to transform these equations into differential forms. The conversation also includes questions about determining if div F or curl F are equal to zero, calculating flux and volume, and understanding the sideways derivative of the curl. An analogy involving water flow is used to explain the concept of curl.
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brentd49
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I've been reading the text: Electricity and Magnetism, by Purcell.

I understand about the integral forms describing the electric field, but when trying to answer questions at the end of the chapter on Gauss's and Stokes's therorems I have not been able two. These two theorems supposedly transform the integral equations that describe an electric field into differential equations, yet there are still integrals in the equations:

Gauss:
[tex]Integrate[F*da,over surface]=Integrate[div F*dv,over volume][/tex]
[tex]div F=4\pi\rho, del^2*potential=-4\pi\rho[/tex]

Stokes:
[tex]Integrate[F*ds,over circ.]=Integrate[curl F*da,over surface][/tex]
[tex]del X A[/tex]


Questions:

1. I'm having trouble with looking at field lines and judging if div F=0 (or not zero) or curl F=0 (or not zero)

2. How to calculate flux/volume for Gauss problems. I see in the book they use the midpoint. Or flux/area for Stokes problems. Why do the choose the midpoint?

3. Also would someone explain the sidways derivative of the curl, I don't understand why it is the way it is.
 
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Originally posted by brentd49
I've been reading the text: Electricity and Magnetism, by Purcell.

Questions:

1. I'm having trouble with looking at field lines and judging if div F=0 (or not zero) or curl F=0 (or not zero)


I found this helpful for curl. Imagine water flowing in a rectangular aquaduct. The water all moves at the same speed as you cross the aquaduct. This flow has no curl.

Now imagine an aquaduct where the water flows faster the further it is from the bank you're standing on. As you move perpendicular to the flow direction, the flow rate gets faster. This flow has a nonzero curl. Place a stick perpendicular to the flow, the far edge wants to move faster than the near end making the stick want to turn.

Now look at the math for curl. If the flow is in the x direction then the curl is given by the derivatives dv/dy and dv/dz. That is the velocity v is changing when you move perpendicular to the flow direction.
 
  • #3


Gauss's Divergence Theorem and Stokes's Theorem are two important mathematical tools in the study of electricity and magnetism. These theorems allow us to transform integral equations that describe an electric field into differential equations, which are easier to solve.

Gauss's Divergence Theorem relates the flux of a vector field through a closed surface to the divergence of that field over the enclosed volume. This is represented by the integral equations you mentioned, where F represents the vector field, da represents the surface element, and dv represents the volume element. The divergence of a vector field measures the net flow of that field out of a given point, and if it is zero, it means that the field is "sourcing" or "sinking" equally at all points. This is why div F=0 is often associated with field lines that are "spread out" or "diverging" from a point, while div F≠0 is associated with field lines that are "converging" or "sinking" towards a point.

Stokes's Theorem relates the circulation of a vector field around a closed curve to the curl of that field over the enclosed surface. This is represented by the integral equations you mentioned, where F represents the vector field, ds represents the curve element, and da represents the surface element. The curl of a vector field measures the tendency of that field to "rotate" or "curl" around a given point. If the curl is zero, it means that the field is "irrotational," or lacks this tendency to rotate. This is why curl F=0 is often associated with field lines that are "straight" or "line-like," while curl F≠0 is associated with field lines that are "curly" or "circular."

To calculate flux or circulation using these theorems, we use the midpoint of the element because it is the most accurate representation of the field at that point. This allows us to accurately measure the flow or rotation of the field through that element. As for the sideways derivative of the curl, this is simply a mathematical representation of the curl in a different coordinate system. It allows us to express the curl in terms of derivatives in the x, y, and z directions, making it easier to work with in some cases.

In conclusion, Gauss's Divergence Theorem and Stokes's Theorem are powerful tools that allow us to transform integral equations into differential equations, making it easier
 

1. What is Gauss's Divergence Theorem?

Gauss's Divergence Theorem, also known as the Gauss-Ostrogradsky Theorem, is a fundamental theorem in vector calculus that relates the surface integral of a vector field to the volume integral of its divergence. In simpler terms, it states that the net flow of a vector field through a closed surface is equal to the volume integral of the divergence of that vector field within the enclosed volume.

2. How is Gauss's Divergence Theorem used in physics?

Gauss's Divergence Theorem is used in physics to calculate flux, or the flow of a vector field through a surface. It is particularly useful in electrostatics and fluid dynamics, where it can be used to calculate the electric field or fluid flow through a closed surface.

3. What is Stokes's Theorem?

Stokes's Theorem is another fundamental theorem in vector calculus that relates the line integral of a vector field to the surface integral of its curl. In other words, it states that the circulation of a vector field around a closed curve is equal to the surface integral of its curl over the enclosed surface.

4. How is Stokes's Theorem used in physics?

Stokes's Theorem is used in physics to calculate the circulation of a vector field around a closed curve, which is important in understanding the behavior of fluids and electromagnetic fields. It can also be used to simplify complicated line integrals by converting them into surface integrals.

5. What is the relationship between Gauss's Divergence Theorem and Stokes's Theorem?

Gauss's Divergence Theorem and Stokes's Theorem are closely related, as they are both fundamental theorems in vector calculus that relate different types of integrals. They are often used together to simplify calculations and solve problems in physics and engineering.

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