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Today I was watching a series of online video lectures about electromagnetism. At some point of the lecture, the professor used this vector calculus identity: $$ \nabla\times\left(\mathbf{A}\times\mathbf{B}\right)=\mathbf{A}\left(\nabla\cdot\mathbf{B}\right)-\mathbf{B}\left(\nabla\cdot\mathbf{A}\right)+\left(\mathbf{B}\cdot\nabla\right)\mathbf{A}-\left(\mathbf{A}\cdot\nabla\right)\mathbf{B}$$

So, I tried to prove it by using the "bac-cab" identity and I kept getting wrong results. Later, I realized that the "bac-cab" identity for vector triple products doesn't hold anymore when vector operators like $\nabla$ are involved!

Is it normal that I don't know any vector analysis? I've taken multi-variable calculus, but they didn't teach us anything about such identities! Did my university cheat on us and taught us less than we should know or it's normal for pure math students not to know vector analysis identities as well as engineers and physicists know them?

And at last, I'll appreciate it if someone introduces a good introductory book about vector calculus that teaches me how to deal with such vector equations and identities.

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    $\begingroup$ Yes. The first time I did this stuff seriously was in a graduate PDE course. I've taken a lot of math classes and I honestly don't know what you mean by taking the cross product of a $\nabla$ and a vector. To answer your second question, don't bother. Learn it if and when you need to use it. $\endgroup$
    – Potato
    Jul 19, 2014 at 14:02
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    $\begingroup$ I've heard good things about Schey's Div Grad Curl And All That. $\endgroup$ Jul 19, 2014 at 14:04
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    $\begingroup$ Perfectly normal. Pure mathematicians use differential forms rather than vector calculus. $\endgroup$ Jul 19, 2014 at 14:06
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    $\begingroup$ @math.n00b From a theoretical standpoint, there is no need for vector calculus if you know differential forms. Practically, however, if you enter into areas where people habitually use vector calculus, you'd better be somewhat comfortable with that too. (Somewhat like, you don't need to know Italian to study literature, but if you want to become a Dante expert, you better learn Italian, pronto.) $\endgroup$ Jul 19, 2014 at 14:15
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    $\begingroup$ I do (I'm a theoretical physics grad student, actually). The essential place of application is in electromagnetism, especially in practical application. It also comes in for classical mechanics e.g. forces, torques. $\endgroup$ Jul 19, 2014 at 14:20

2 Answers 2

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The comments made thus far give excellent advice. I thought you might like to see the details.

We need the well-known identity $\sum_{j=1}^{3} \epsilon_{ikj}\epsilon_{lmj} = \delta_{il}\delta_{km}-\delta_{kl}\delta_{im}$. This is the dark heart of the BAC-CAB identity. \begin{align} \notag \nabla \times (\vec{A} \times \vec{B}) &= \sum_{i,j,k=1}^3 \epsilon_{ijk}\partial_i (\vec{A} \times \vec{B})_j\widehat{x}_k \\ &= \sum_{i,j,k=1}^3 \epsilon_{ijk}\partial_i \left(\sum_{l,m=1}^3A_lB_m\epsilon_{lmj} \right) \widehat{x}_k \\ &= \sum_{i,j,k=1}^3\sum_{l,m=1}^3 \epsilon_{ijk}\epsilon_{lmj}\partial_i \left(A_lB_m \right) \widehat{x}_k \\ &= -\sum_{i,j,k,l,m=1}^3 \color{red}{\epsilon_{ikj}\epsilon_{jlm}}\partial_i \left(A_lB_m \right) \widehat{x}_k \\ &= -\sum_{i,k,l,m=1}^3( \color{red}{\delta_{il}\delta_{km}-\delta_{im}\delta_{kl}})\partial_i \left(A_lB_m \right) \widehat{x}_k \\ &= -\sum_{i,k,l,m=1}^3\delta_{il}\delta_{km}\partial_i \left(A_lB_m \right) \widehat{x}_k+\sum_{i,k,l,m=1}^3\delta_{im}\delta_{kl}\partial_i \left(A_lB_m \right) \widehat{x}_k \\ &= -\sum_{i,k=1}^3\partial_i \left(A_iB_k \right) \widehat{x}_k+\sum_{i,k=1}^3\partial_i \left(A_kB_i \right) \widehat{x}_k \\ &= -\sum_{i,k=1}^3\left((\partial_i A_i)B_k+A_i\partial_i B_k \right) \widehat{x}_k+\sum_{i,k=1}^3 \left((\partial_iA_k)B_i+A_k\partial_iB_i \right) \widehat{x}_k \\ &= -\sum_{i,k=1}^3(\partial_i A_i)B_k\widehat{x}_k-\sum_{i,k=1}^3A_i\partial_i B_k\widehat{x}_k +\sum_{i,k=1}^3 B_i\partial_iA_k\widehat{x}_k+\sum_{i,k=1}^3(\partial_iB_i)A_k \widehat{x}_k \\ &= -(\nabla \cdot \vec{A})\vec{B}-(\vec{A} \cdot \nabla )\vec{B}+(\nabla \cdot \vec{B})\vec{A}+(\vec{B} \cdot \nabla )\vec{A} \end{align} If it's any consolation, I was a math & physics double major and this stuff escaped me until I had the good fortune in graduate school of studying with a student from Spain. I had pages and pages of stuff and he had three lines on a particular problem. It hit me, I probably should use $\epsilon_{ijk}$ for vector identity calculations. Moreover, there is a whole family of contracting levi-civita symbols as sums of antisymmetric kronecker deltas. The identity this post begins with is just a start. These are used in the tensor calculus of General Relativity.

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  • $\begingroup$ The only unfortunate thing is the profusion of dummy indices in such a calculation. It's why I always liked the idea of Penrose notation for such things, though since I don't do GR myself I've never needed their full power. $\endgroup$ Jul 19, 2014 at 18:23
  • $\begingroup$ @Semiclassical well, at first dummies are a bit unsettling (hence their absence in my calculation here intended for math audience) but, after a few hundred pages of calculation and omitting a few thousand $\sum$ symbols, you get numb to the abuse. In truth, it is very freeing, in combination with multi-index notation, you can say so much with so little. That said, I am not versed in en.wikipedia.org/wiki/Penrose_graphical_notation . Interesting. $\endgroup$ Jul 19, 2014 at 18:43
  • $\begingroup$ True enough. I more just mean how tedious those indices become, and the fact that in Penrose notation they just end up as lines. But again, not very practical for the typical physicist. $\endgroup$ Jul 19, 2014 at 18:45
  • $\begingroup$ @Semiclassical incidentally, you might like physicsoverflow.org I haven't said much of use there yet, but it seems like a good community to support. $\endgroup$ Jul 19, 2014 at 18:51
  • $\begingroup$ I see. Thanks. I guess I'll have to find some more vector calculus identities and try to prove them on my own. But wait, what is $\vec{A} \cdot \vec{\nabla}$? I mean, how does that even make sense? How should we interpret $(\vec{A} \cdot \vec{\nabla})\vec{B}$? $\endgroup$
    – math.n00b
    Jul 19, 2014 at 20:19
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I'm going to prove two other vector calculus identities to check if I've understood physicists notations well :D I'm going to use Einstein's summation notation as well. ;-)

1. $$\nabla \cdot (\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{A}) - \mathbf{A} \cdot (\nabla \times \mathbf{B})$$

$$\nabla \cdot (\mathbf{A} \times \mathbf{B}) = \partial_i(\mathbf{A}\times\mathbf{B})_i=\partial_i(\epsilon_{ijk}A_jB_k)=\epsilon_{ijk}\partial_i(A_jB_k)=\epsilon_{ijk}A_j\partial_iB_k+\epsilon_{ijk}B_k\partial_iA_j$$

$$\epsilon_{ijk}A_j\partial_iB_k+\epsilon_{ijk}B_k\partial_iA_j=-\epsilon_{jik}A_j\partial_iB_k+\epsilon_{kij}B_k\partial_iA_j=-A_j(\epsilon_{jik}\partial_iB_k)+B_k(\epsilon_{kij}\partial_iA_j)$$ $$-A_j(\epsilon_{jik}\partial_iB_k)+B_k(\epsilon_{kij}\partial_iA_j)=-A_j(\nabla \times \mathbf{B})_j+B_k(\nabla \times \mathbf{A})_k = -\mathbf{A}\cdot(\nabla \times \mathbf{B})+\mathbf{B}\cdot(\nabla \times \mathbf{A})$$

2. $$ \nabla \times \left( \nabla \times \mathbf{A} \right) = \nabla(\nabla \cdot \mathbf{A}) - \nabla^{2}\mathbf{A}$$

where $\nabla^{2}\mathbf{A}=\langle \nabla^{2}A_x, \nabla^{2}A_y, \nabla^{2}A_z\rangle$ and it's called vector Laplacian.

$$(\nabla \times \left( \nabla \times \mathbf{A} \right))_i = \epsilon_{ijk}\partial_j(\nabla\times \mathbf{A})_k=\epsilon_{ijk}\partial_j(\epsilon_{kmn}\partial_mA_n)=\epsilon_{ijk}\epsilon_{kmn}\partial_j(\partial_mA_n)$$ Now we use the famous equality: $$\epsilon_{ijk}\epsilon_{kmn}=\delta_{im}\delta_{jn}-\delta_{in}\delta_{jm}$$

$$(\nabla \times \left( \nabla \times \mathbf{A} \right))_i= \delta_{im}\delta_{jn}\partial_j(\partial_mA_n)-\delta_{in}\delta_{jm}\partial_j(\partial_mA_n)=\partial_j(\partial_iA_j)-\partial_j(\partial_jA_i)$$ $$\partial_j(\partial_iA_j)-\partial_j(\partial_jA_i)=\partial_i(\partial_jA_j)-(\partial_j\partial_j)A_i=\partial_i(\nabla \cdot\mathbf{A})-(\nabla^{2}\mathbf{A})_i=(\nabla(\nabla \cdot \mathbf{A})-\nabla^{2}\mathbf{A})_i$$

I'm sure that the last line needs some modifications because the LHS is the i-th component of $\nabla \times \left( \nabla \times \mathbf{A} \right)$ while in RHS a vector ($\nabla^{2}\mathbf{A}$) has showed up sooner than it should appear.

So, I've made the following assumptions in my calculations (Please verify them):

  1. Partials always act like differentials, regardless of the subscripts involved in our calculations. $\partial_iX_jY_k = (\partial_iX_j)Y_k + X_j(\partial_iY_k)$.

  2. If we assume the existence and continuity of second derivatives involved in our calculations, then since partial derivatives commute we have $\partial_i\partial_j= \partial_j\partial_i$.

  3. I've assumed that $\epsilon_{ijk}$ is a constant function and it can come out of partial derivatives with ease.

EDIT:

I want to prove that:

$$ \nabla(\mathbf{A} \cdot \mathbf{B}) = (\mathbf{A} \cdot \nabla)\mathbf{B} + (\mathbf{B} \cdot \nabla)\mathbf{A} + \mathbf{A} \times (\nabla \times \mathbf{B}) + \mathbf{B} \times (\nabla \times \mathbf{A}) $$

$$(\nabla(\mathbf{A} \cdot \mathbf{B}))_i = \partial_i (\mathbf{A}\cdot\mathbf{B})=\partial_i(A_jB_j)=\partial_i(A_j)B_j+\partial_i(B_j)A_j$$

I don't see how I should move forward. I'm stuck.

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    $\begingroup$ Nearly perfect, however, in the last line we should have $\partial_i (\nabla \cdot \vec{A})-\nabla^2 A_i = (\nabla(\nabla \cdot \vec{A})- \nabla^2 \vec{A})_i$. Which, is what you want as you calculated the $i$-th component of $\nabla \times (\nabla \times \vec{A})$. Nice work! $\endgroup$ Jul 20, 2014 at 4:01
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    $\begingroup$ In addition to that little note by Mr. Cook, I'd just like to add: 1) Make sure you realize why you can pull the $\epsilon_{ijk}$ in and out of the differential operators (so you know exactly when it's allowable). 2) I think you might want to be more careful about commuting factors. Technically the Leibniz rule states that $\partial_i(X_jY_k) = (\partial_iX_j) Y_k + X_j \partial_iY_k$. The factors do commute because they are scalars, but I'd just be more explicit. 3) I consider $\partial_j^2$ to be an abuse of notation. In that form it looks like there's only 1 index... $\endgroup$
    – user137731
    Jul 20, 2014 at 4:26
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    $\begingroup$ and so can fool you or your readers when using the Einstein summation convention. I prefer to just leave it as $\partial_j \partial_j$. If you can keep those things in mind, then yeah, I think you've got it. $\endgroup$
    – user137731
    Jul 20, 2014 at 4:28
  • $\begingroup$ @JamesS.Cook: Yes, I'd already noticed that but I was unsure whether $$\partial_j\partial_jA_i=\nabla^{2}A_i=(\nabla^{2}\mathbf{A})_i$$ was correct or not, later I verified it. Thanks for your help. I'm trying to prove that $$ \nabla(\mathbf{A} \cdot \mathbf{B}) = (\mathbf{A} \cdot \nabla)\mathbf{B} + (\mathbf{B} \cdot \nabla)\mathbf{A} + \mathbf{A} \times (\nabla \times \mathbf{B}) + \mathbf{B} \times (\nabla \times \mathbf{A}) $$ but I keep getting wrong results, I don't see how the cross product comes into the play. Would you please give me a hint? $\endgroup$
    – math.n00b
    Jul 20, 2014 at 5:57
  • $\begingroup$ @Bye_world: Yes, thanks for your notes. I can pull $\epsilon_{ijk}$ out because it's a constant, not a function. No? And now it makes more sense why writing $\partial^{2}_j$ is a bad idea.. because if there's only one $j$ index it won't be summed over. Right? $\endgroup$
    – math.n00b
    Jul 20, 2014 at 6:01

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