# Integral form of Maxwell equations.

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In summary: The notation in the hyperphysics page is consistent with the notation used in Paul's Maths notes.In summary, the author of the hyperphysics page is using different notation for line integrals and surface integrals, which makes it difficult to tell which type of integrals are being used. The author suggests looking at the physical situation the equation is describing to make the correct association.
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I have been studying the Maxwell equations recently (namely the integral forms of them). Of course I had to study line integrals before that. Well, I went to a hyperphysics page to look up the equations:

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/maxeq.html

I noticed that the notations for those integrals do not match the notations for line integrals of vector fields that I learned on the following page:

http://tutorial.math.lamar.edu/Classes/CalcIII/CurlDivergence.aspx

(Look in the last two blue boxes on the page I just posted)

The integrals in the Maxwell equations on the hyperphysics page have differentials (dA and ds) that have the vector arrow head symbols over them.

http://tutorial.math.lamar.edu/Classes/CalcIII/CurlDivergence.aspx (last two blue boxes)

Your problem is unclear as you have described it - please be specific.

I'll explain:
I noticed that the notations for those integrals do not match the notations for line integrals of vector fields that I learned ...
It is very common for notation to differ between sources - you get used to it. You need to use context to make the correct associations.

This makes me question whether the integrals in the Maxwell equations are actually even line integrals or if they are surface integrals.
To tell the difference, use the physical situation they are describing and not the notation for clues.
Maxwell's equations are better understood in differential form though.

is the hyperphysics page using line integrals, surface integrals, or some other type of integral?
The hyperphysics page you link to spells out which they mean for each one in the following sections. Each description has a link to an example that also makes the matter clear.

Are Gauss' two laws the type of vector field line integrals that use the curl, while Faraday's law of induction and Ampere's law are the type that uses the divergence (or vice versa)?
... this is spelled out in the second section where the names of the equations are stated followed by the differential form as divergence and curl.
perhaps you will benefit from a refresher in nabla notation: https://en.wikipedia.org/wiki/Del

There is no Maxwell equation in integral form in the second web page.

Last edited:
I understood the second page (the Paul's Calculus notes page) was introduced to show notation learned for integrals of vector spaces.
However, the page only has two integrals in it - div and curl forms of Green's Theorem.

The page does not teach integral notation for vector fields - instead it teaches the basics of div and curl.
The notation in the hyperphysics page is consistent with the notation used in Paul's Maths notes.
Note: ##\vec k\mathrm{d}A = \mathrm{d}\vec A = \vec{\mathrm{d} A}## because Paul uses ##\vec k## as the standard unit normal to the surface element dA while hyperphysics combines the two in the same notation. Perhaps this is where the confusion arises?

## 1. What is the integral form of Maxwell's equations?

The integral form of Maxwell's equations is a set of four equations that describe the fundamental laws of electricity and magnetism. These equations relate the electric and magnetic fields to their sources, which can include charges and currents.

## 2. What are the advantages of using the integral form of Maxwell's equations?

One advantage of using the integral form of Maxwell's equations is that they are more general and can be applied to any type of source, not just point sources. Additionally, they are useful for analyzing systems with changing fields over time.

## 3. How are the integral form of Maxwell's equations related to the differential form?

The integral form of Maxwell's equations can be derived from the differential form by applying the divergence theorem and the curl theorem. Conversely, the differential form can be obtained from the integral form by taking derivatives of the integrals.

## 4. What are the physical implications of the integral form of Maxwell's equations?

The integral form of Maxwell's equations have many physical implications, such as the existence of electromagnetic waves, the relationship between electric and magnetic fields, and the conservation of charge and energy. These equations also form the basis for the theory of electromagnetism.

## 5. How are the integral form of Maxwell's equations used in practical applications?

The integral form of Maxwell's equations are used in a wide range of practical applications, including the design of electrical and electronic devices, the analysis of electromagnetic fields in materials, and the development of communication technologies such as radio and wireless networks.

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