Where Can I Find Proofs for Vector Calculus Identities?

In summary, Wikipedia provides a comprehensive list of vector calculus identities and a list of vector identities. There is a link that offers a new teaching strategy for deriving these identities using skew tensors and dyadic products instead of cross products. Some argue that proving these identities in cartesian coordinates is less efficient and intuitive, while others believe it is necessary to show that the identities hold in all coordinate systems.
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  • #2
I found a link that might be helpful:

Vector algebra is a powerful and needful tool for Physics but unfortunately, due to lack of mathematical skills, it becomes misleading for first undergraduate courses of science and engineering studies. Standard vector identities are usually proved using Cartesian components or geometrical arguments, accordingly. Instead, this work presents a new teaching strategy in order to derive symbolically vector identities without analytical expansions in components, either explicitly or using indicial notation. This strategy is mainly based on the correspondence between three-dimensional vectors and skew-symmetric second-rank tensors. Hence, the derivations are performed from skew tensors and dyadic products, rather than cross products. Some examples of skew-symmetric tensors in Physics are illustrated.
http://www.citeulike.org/user/pak/article/4524046
http://arxiv.org/PS_cache/arxiv/pdf/0904/0904.1814v1.pdf
 
  • #3
I don't think there is anything wrong with proving these identities in cartesian coordinates. If you have an identity such as [tex]\nabla (fg)=f\nabla g+g\nabla f[/tex], this just tells you that one vector is equal to another vector. If this is true in one coordinate system, it is true in all coordinate systems.
 
  • #4
daudaudaudau said:
I don't think there is anything wrong with proving these identities in cartesian coordinates. If you have an identity such as [tex]\nabla (fg)=f\nabla g+g\nabla f[/tex], this just tells you that one vector is equal to another vector. If this is true in one coordinate system, it is true in all coordinate systems.

If is a lot less efficient and less intuitive to prove vector relationships by resorting to expressing the vectors as individual components. People are much more likely to be able to see simplifications in vector algebra if they have some higher level algebra tools available to them.
 
  • #5
The identities of vector calculus are easily derived and proven by algebraic mean given a few lemmas. It would be bad to prove them in cartesian coordinates because it would be messy, lengthly, inelegant, and would exagerate the importance of coordinates.
Anyone who disagrees should post a cartesian coordinates proof of
[tex]\mathbf{(a\times\nabla)\times b+a\nabla\cdot b=a\times(\nabla\times b)+(a\cdot\nabla)b}[/tex]
from another thread
lurflurf said:
The trouble is commuting an opperator adds a commutator term
In single variable calculus
D(uv)=uDv+vDu not uDv
we can use partial opperators to avoid this
let an opperant in {} be fixed
D(uv)=D({u}v)+D(u{v})={u}Dv+{v}Du=uDv+vDu

∇(F . G )=∇({F} . G )+∇(F . {G} )
∇({F} . G )=Fx(∇xG)+(F.∇)G
∇(F . {G} )=Gx(∇xF)+(G.∇)F
∇(F . G )=∇({F} . G )+∇(F . {G} )=Fx(∇xG)+(F.∇)G+Gx(∇xF)+(G.∇)F
 

FAQ: Where Can I Find Proofs for Vector Calculus Identities?

What is the Vector Product Rule Deviation?

The Vector Product Rule Deviation is a mathematical concept that describes the difference between the cross product of two vectors and the cross product of their unit vectors. It is a measure of how much the direction of the cross product deviates from the expected direction based on the properties of vectors.

Why is the Vector Product Rule Deviation important?

The Vector Product Rule Deviation is important because it helps us understand the accuracy of vector operations and calculations. It tells us how much the cross product deviates from what we expect based on the properties of vectors, which is crucial in many scientific and engineering applications.

How is the Vector Product Rule Deviation calculated?

The Vector Product Rule Deviation is calculated by taking the magnitude of the cross product of the two vectors and dividing it by the product of the magnitudes of the two vectors. This value is then multiplied by the sine of the angle between the two vectors.

What does a high Vector Product Rule Deviation value indicate?

A high Vector Product Rule Deviation value indicates that the two vectors being crossed are not perpendicular to each other. This means that the cross product is not in the expected direction and the accuracy of the vector operation or calculation may be affected.

How can the Vector Product Rule Deviation be minimized?

The Vector Product Rule Deviation can be minimized by ensuring that the two vectors being crossed are perpendicular to each other. This can be achieved by carefully choosing the vectors or by using mathematical techniques such as orthogonalization to make them perpendicular.

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