Uncovering the Origins of Vector Analysis in Maxwell's Equations

The mathematical structures we see around us today were developed over the years, building on earlier work.In summary, Vector analysis, including Curl and Divergence and gradient operators, grew out of the relative complexity of using quaternions. Maxwell's original work on electromagnetism did not use vectors or quaternions, but modern vector analysis was developed later on to simplify dealing with quaternions. Maxwell himself did use quaternions in some of his later work, but they were not essential to his original development of the theory.
  • #1
cooev769
114
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I'm quite interested in the history behind vector analysis especially Curl and Divergence and gradient operators etc. When James Maxwell derived Maxwell's equations of electromagnetism where these sorts of operations well known and commonly used, or are they modern fabrications. Did Maxwell actually derive his equations using the methods we use today or some other methods?
 
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  • #3
There's a book of the subject:

http://en.wikipedia.org/wiki/A_History_of_Vector_Analysis

and at Amazon as a Dover publication:

https://www.amazon.com/dp/0486679101/?tag=pfamazon01-20

Vector analysis grew out of the relative complexity of using quaternions. Hamilton was trying to extend complex numbers into a new kind of number and that's where the i,j,k unit vector idea came from: a + bi + cj + dk

For the divergence there's a history of the divergence theorem:

http://www.sciencedirect.com/science/article/pii/0315086078902124

The curl must have been recognized and came out of similar theorems.
 
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  • #4
cooev769 said:
I'm quite interested in the history behind vector analysis especially Curl and Divergence and gradient operators etc. When James Maxwell derived Maxwell's equations of electromagnetism where these sorts of operations well known and commonly used, or are they modern fabrications. Did Maxwell actually derive his equations using the methods we use today or some other methods?

You can see for yourself. Maxwell's treatise on electricity and magnetism can be found at the
Internet Archive:

http://archive.org/search.php?query=creator:"Maxwell, James Clerk, 1831-1879"

Maxwell worked with differentials and integrals, as modern vector notation had not been developed. Most of what Maxwell used in his work were quaternions. Modern vector analysis was developed by Josiah Willard Gibbs in the 1870s to simplify having to deal with quaternions and to provide more facility for his students in understanding vector concepts (When was the last time a professor went out of his way to simplify things?)
 
  • #5
SteamKing said:
Maxwell worked with differentials and integrals, as modern vector notation had not been developed. Most of what Maxwell used in his work were quaternions.
The first sentence is correct, but the second is not. It has become an internet meme, favored by crackpots who like to imagine that the vector notation used at intermediate college level hides some of Tesla's key discoveries (whatever!).

Maxwell's original work, published in 1861 and 1862 as On Physical Lines of Force did not use vectors or quaternions. Nor did his Dynamical Theory of the Electromagnetic Field, published in 1864 and 1865. His original development of his theory followed the painstaking style of only using scalar equations that were in near universal use at the time. The mathematics to simplify the notation didn't exist.

He did mention quaternions in passing in his final book on the subject, A Treatise on Electricity and Magnetism, published in 1873. But that was in passing, where he discussed interesting developments by others in extending his work. Maxwell himself did not use quaternions in the development of his theory.
 
  • #6
Perhaps I was over generous in describing the extent of the use of quaternions by Maxwell in Electricity and Magnetism, but he did provide equivalent expressions for some of his results using quaternions.

That modern crackpots may have a bone to pick because Maxwell did or did not use quaternions in the manner they believe he should have used them is neither here nor there.
 
  • #7
Yes, Maxwell did demonstrate the use of quaternions on some of his results in A Treatise on Electricity and Magnetism, but that is not what he used to develop them. He published that book near the end of his life. He spent a good deal of his life developing electrodynamics, and he did that without the help of quaternions.

Aside: There is an even more elegant way to represent Maxwell's equations, which is with 4-vectors. That approach also wasn't available to Maxwell.To answer the question raised in the opening post, "Did Maxwell actually derive his equations using the methods we use today or some other methods?" He did it tediously, one scalar equation at a time. Most physics texts and journal articles written in the late 19th century and earlier are tedious to read because they didn't have the aid of higher mathematical structures.
 

1. What is vector analysis?

Vector analysis is a branch of mathematics that deals with the study and manipulation of vectors, which are quantities that have both magnitude and direction. It involves the use of mathematical operations such as addition, subtraction, and multiplication to analyze and solve problems involving vectors.

2. Who is credited with the development of vector analysis?

The history of vector analysis can be traced back to the 19th century, with the works of mathematicians such as Augustin-Louis Cauchy, William Rowan Hamilton, and Josiah Willard Gibbs. However, the concept of vector analysis as we know it today is primarily credited to the German mathematician Hermann Grassmann.

3. What are the applications of vector analysis?

Vector analysis has a wide range of applications in various fields, including physics, engineering, computer graphics, and even economics. It is used to describe and analyze physical quantities such as velocity, acceleration, and force, as well as to solve problems involving motion and forces in three-dimensional space.

4. How has vector analysis evolved over time?

The development of vector analysis has been a gradual process, with contributions from multiple mathematicians and scientists over the years. It has evolved from its early beginnings as a tool for studying mechanics and electromagnetism to a fundamental concept in modern mathematics and physics. The introduction of vector calculus by Gibbs and Oliver Heaviside in the late 19th century further expanded the applications of vector analysis.

5. What are some key concepts in vector analysis?

Some key concepts in vector analysis include vector addition and subtraction, dot and cross products, vector components, and vector equations. These concepts are used to perform calculations and solve problems involving vectors, as well as to visualize vector quantities in three-dimensional space.

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