Need Good book on Vector Calculus

In summary, the conversation discusses the need for a good book or website that provides step-by-step proofs for vector calculus equations. The conversation includes the correct form of the equations, as well as a proof for them using cartesian tensors. It also mentions a more complex equation involving mixed vector products.
  • #1
newbie101
16
0
Hi All,

I need some suggestion on a good book for vector calculus/advanced vector calculus.
current book I am reading just give equations like

del x ( A x B ) = A del.B - Bdel.A + (B.del)A - (A.del)B

A x ( B x C ) = B(del.A) - C(A.B)

del x (f A) = f del x A + del f x A

etc

however they don't show the proof
Is there any book or maybe a website which gives the proof step by step

thanks
newbie101

* if there is a free book i could download .. it would be fantastic :smile:
 
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  • #2
They can be proven on components.All three of them are vector identities,so it suffices to prove only for one scalar component.The second one is really easy if u use cartesian tensors...The same goes for the 3-rd.

Daniel.
 
  • #3
dextercioby,

please show me how they are proven...

thanks
newbie
 
  • #4
The first u've written there is incorrect...

Come up with the correct form.

1

[tex] \vec{A}\times\left(\vec{B}\times\vec{C}\right)=...? [/tex]

[tex] \vec{B}\times\vec{C}=\epsilon_{ijk}B_{i}C_{j}\vec{e}_{k} [/tex]

[tex] \vec{A}\times\left(\vec{B}\times\vec{C}\right)=\epsilon_{lkn}A_{l}\left(\vec{B}\times\vec{C}\right)_{k}\vec{e}_{n}=\epsilon_{lkn}\epsilon_{ijk}A_{l}B_{i}C_{j}\vec{e}_{n} [/tex]

[tex] \epsilon_{lkn}\epsilon_{ijk}=-\epsilon_{lnk}\epsilon_{ijk}=-\left(\delta_{li}\delta_{nj}-\delta_{ni}\delta_{lj}\right)=\delta_{ni}\delta_{lj}-\delta_{li}\delta_{nj} [/tex]

Therefore,making the summations with the delta Kronecker

[tex] \vec{A}\times\left(\vec{B}\times\vec{C}\right)=B_{i}A_{l}C_{l}\vec{e}_{i}-A_{l}B_{l}C_{j}\vec{e}_{j}=\left(\vec{A}\cdot\vec{C}\right)\vec{B}-\left(\vec{A}\cdot\vec{B}\right)\vec{C} [/tex]

Q.e.d.




Daniel.
 
  • #5
2

[tex]\nabla\times\left(A\vec{B}\right)=\epsilon_{ijk}\partial_{i}\left(AB_{j}\right)\vec{e}_{k}=\epsilon_{ijk}\left(\partial_{i}A\right)B_{j}\vec{e}_{k}+\epsilon_{ijk}A\left(\partial_{i}B_{j}\right)\vec{e}_{k}=\left(\nabla A\right)\times\vec{B}+A\left(\nabla\times\vec{B}\right) [/tex]

Q.e.d.




Daniel.
 
  • #6
Okay.I'll make reference to post #4 in which the simply contracted tensor product of Levi-Civita tensor appears.

3

[tex]\nabla\times\left(\vec{A}\times\vec{B}\right)=\epsilon_{ijk}\partial_{i}\left(\epsilon_{lmj}A_{l}B_{m}\right)\vec{e}_{k}=\epsilon_{ijk}\epsilon_{lmj}\left[\left(\partial_{i}A_{l}\right)B_{m}+A_{l}\left(\partial_{i}B_{m}\right)\right]\vec{e}_{k} [/tex]

[tex]=-\epsilon_{ikj}\epsilon_{lmj}\left[\left(\partial_{i}A_{l}\right)B_{m}+A_{l}\left(\partial_{i}B_{m}\right)\right]\vec{e}_{k} =\left(\delta_{im}\delta_{kl}-\delta_{il}\delta_{km}\right)\left[\left(\partial_{i}A_{l}\right)B_{m}+A_{l}\left(\partial_{i}B_{m}\right)\right]\vec{e}_{k} [/tex]

[tex] =\left(\partial_{m}A_{k}\right)B_{m}\vec{e}_{k}+A_{k}\left(\partial_{m}B_{m}\right)\vec{e}_{k}-\left(\partial_{l}A_{l}\right)B_{k}\vec{e}_{k}-A_{i}\left(\partial_{i}B_{k}\right)\vec{e}_{k}[/tex]

[tex]=\left(\vec{B}\cdot\nabla\right)\vec{A}+\vec{A}\left(\nabla\cdot\vec{B}\right)-\vec{B}\left(\nabla\cdot\vec{A}\right)-\left(\vec{A}\cdot\nabla\right)\vec{B} [/tex]

Q.e.d.

Daniel.
 
  • #7
4

[tex] \vec{A}\times\left(\nabla\times\vec{B}\right)+\vec{B}\times\left(\nabla\times\vec{A}\right)+\left(\vec{B}\cdot\nabla\right)\vec{A}+\left(\vec{A}\cdot\nabla\right)\vec{B} [/tex]

[tex] =\epsilon_{ijk}A_{i}\left(\nabla\times\vec{B}\right)_{j}\vec{e}_{k}+\epsilon_{ijk}B_{i}\left(\nabla\times\vec{A}\right)_{j}\vec{e}_{k}+B_{m}\left(\partial_{m}A_{l}\right)\vec{e}_{l}+A_{m}\left(\partial_{m}B_{l}\right)\vec{e}_{l} [/tex]

[tex] =\epsilon_{ijk}A_{i}\left(\epsilon_{lmj}\partial_{l}B_{m}\right)\vec{e}_{k}+
\epsilon_{ijk}B_{i}\left(\epsilon_{lmj}\partial_{l}A_{m}\right)\vec{e}_{k}+B_{m}\left(\partial_{m}A_{l}\right)\vec{e}_{l}+A_{m}\left(\partial_{m}B_{l}\right)\vec{e}_{l} [/tex]

[tex] =\left(\delta_{im}\delta_{kl}-\delta_{il}\delta_{km}\right)A_{i}\left(\partial_{l}B_{m}\right)\vec{e}_{k}+\left(\delta_{im}\delta_{kl}-\delta_{il}\delta_{km}\right)B_{i}\left(\partial_{l}A_{m}\right)\vec{e}_{k}+B_{m}\left(\partial_{m}A_{l}\right)\vec{e}_{l}+A_{m}\left(\partial_{m}B_{l}\right)\vec{e}_{l} [/tex]

[tex] =A_{m}\left(\partial_{l}B_{m}\right)\vec{e}_{l}-A_{l}\left(\partial_{l}B_{m}\right)\vec{e}_{m}+B_{m}\left(\partial_{l}A_{m}\right)\vec{e}_{l}-B_{l}\left(\partial_{l}A_{m}\right)\vec{e}_{m}+B_{m}\left(\partial_{m}A_{l}\right)\vec{e}_{l}+A_{m}\left(\partial_{m}B_{l}\right)\vec{e}_{l} [/tex]

[tex] =B_{m}\left(\partial_{l}A_{m}\right)\vec{e}_{l}+A_{m}\left(\partial_{l}B_{m}\right)\vec{e}_{l}=\partial_{l}\left(\vec{A}\cdot\vec{B}\right)\vec{e}_{l} [/tex]

[tex] =\nabla\left(\vec{A}\cdot\vec{B}\right) [/tex]

Q.e.d.


Daniel.
 
Last edited:
  • #8
Thanks dextercioby :smile:

It will take a while for me to go through this... but you've been a great help!
 
  • #9
After that "while",if u become at ease with euclidean tensor calculus & its application to proving nasty vector identities,then u can deal with this one

[tex] \left(\vec{A}\times\vec{B}\right)\times\left(\vec{C}\times\vec{D}\right)=\left(\vec{A},\vec{C},\vec{D}\right)\vec{B}-\left(\vec{B},\vec{C},\vec{D}\right)\vec{A} [/tex]

,where the (...,...,...) stands for mixed vector product.

Daniel.
 

1. What is vector calculus?

Vector calculus is a branch of mathematics that deals with the study of vector fields, which are mathematical objects that represent quantities with both magnitude and direction. It involves the application of calculus concepts, such as derivatives and integrals, to vectors and functions of multiple variables.

2. Why is vector calculus important?

Vector calculus is important because it provides a framework for understanding and describing physical phenomena in fields such as physics, engineering, and economics. It is also essential for solving problems in fluid mechanics, electromagnetism, and other areas of science and engineering.

3. What are some key concepts in vector calculus?

Some key concepts in vector calculus include vector fields, gradient, divergence, curl, line integrals, surface integrals, and the fundamental theorem of calculus for line integrals. These concepts are used to describe and analyze vector quantities in both two- and three-dimensional space.

4. What are some practical applications of vector calculus?

Vector calculus has many practical applications, such as predicting the movement of fluids in engineering, modeling electromagnetic fields in physics, and analyzing economic trends in finance. It is also used in computer graphics and animation to create realistic 3D images and simulations.

5. How can I improve my understanding of vector calculus?

To improve your understanding of vector calculus, it is important to have a strong foundation in basic calculus, including derivatives and integrals. You can also practice solving problems and working with vector equations. Additionally, seeking out resources such as textbooks, online tutorials, and practice problems can also help improve your understanding of vector calculus.

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