Intuitive interpretation of some vector-dif-calc identities

In summary, the conversation discusses the difficulty in understanding certain formulas in electrodynamics and the search for resources that provide a more intuitive explanation of these formulas. One suggestion is to explore the use of differential forms instead of vector calculus, as advocated by Bill Burke. However, the individual is still struggling with visualizing the operations related to div, grad, and curl.
  • #1
Sevastjanoff
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Dear All,

I am studying electrodynamics and I am trying hard to clearly understand each and every formula. By "understand" I mean that I can "truly see its meaning in front of my eyes". Generally, I am not satisfied only by being able to prove or derive certain formula algebraically; I want to "see through it".

With this regards, there are three formulas from vector differential calculus, which I can happily derive or prove (by expanding and comparing individual vector components, etc.), however, I am unable to truly understand them.

The three formulas are famous: (1) curl(AXB), (2) div(AxB) and grad(A dot B).

Expansions of these formula consist of several terms, whose meaning I cannot grasp intuitively. By this I mean that I cannot "visualize" how all these terms together coherently join to form e.g. curl of the cross product, etc..

Is there any book or other resource, which does not deal with these formulas purely algebraically, and which does not simply states "expand the expression in term of individual components and be happy". I am looking for a resource which graphically or in any other way gives an intuitive explanation of these formulas.

Any hints about such a reference would be highly appreciated.

Thank you in advance and best regards
 
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  • #2
There are some interesting ideas on Bill Burke's page
http://www.ucolick.org/~burke/home.html

He was an advocate of using differential forms instead of vector calculus.

Along these lines, one might argue that
viewing or seeking patterns in "fields of vectors" might not be the right structure to use for visualizing your expressions.

Following Burke's approach (which likely came from Misner-Thorne-Wheeler's approach, which in turn likely came from Schouten's approach),
I can visualize vectors and covectors, bivectors and two-forms, and their algebraic operations...
but I don't yet have a good feel for visualizing the exterior derivative (which are related to div, grad, and curl).
 
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Likes Sevastjanoff
  • #3
Dear robphy,

thank you for the provided hint. I will examine the suggested web page with great interest.

Best regards
 

FAQ: Intuitive interpretation of some vector-dif-calc identities

What is the intuitive interpretation of the vector-dif-calc identity for the dot product?

The dot product of two vectors is equal to the product of their magnitudes and the cosine of the angle between them. This means that the dot product represents the projection of one vector onto the other, or the component of one vector that is parallel to the other.

How can the vector-dif-calc identity for the cross product be visualized?

The cross product of two vectors results in a third vector that is perpendicular to both of the original vectors. This can be visualized as a vector that points out of the plane formed by the two original vectors, with a magnitude equal to the product of their magnitudes and the sine of the angle between them.

What is the significance of the vector-dif-calc identity for the gradient of a scalar field?

The gradient of a scalar field represents the direction and magnitude of the steepest increase in the scalar quantity. This can be visualized as the direction a ball would roll if placed on the surface described by the scalar field, with the magnitude of the gradient representing the steepness of the slope.

How does the vector-dif-calc identity for the divergence of a vector field relate to the concept of flux?

The divergence of a vector field represents the tendency of the vector field to either converge or diverge at a given point. This is closely related to the concept of flux, which is the amount of a vector quantity (such as flow or electric field) flowing through a surface. The divergence theorem states that the flux of a vector field through a closed surface is equal to the volume integral of the divergence of the field over the enclosed volume.

Can the vector-dif-calc identity for the curl of a vector field be used to determine whether the field is conservative?

Yes, the curl of a vector field represents the tendency of the field to circulate or rotate around a given point. If the curl of a vector field is equal to zero, then the field is conservative, meaning that the work done by the field in moving a particle from one point to another is independent of the path taken. This is also known as the curl-free theorem.

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