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Here is the all identities : http://en.wikipedia.org/wiki/Vector_calculus_identities

I need help concerning vector functions and indexing notations.

Let $\overrightarrow{a}$ be a (smooth) vector field and $\varphi$ be a (smooth) scalar function. Show $$ \overrightarrow {\nabla }\cdot \left( \varphi\,\overrightarrow {a}\right) = \varphi \left( \overrightarrow {\nabla }\cdot \overrightarrow {a}\right) +\overrightarrow {a} \cdot \overrightarrow {\nabla }\varphi.$$

I have to use this notation to prove this, but how?

I don't really understand.

My second identity is ; $$ \overrightarrow {\nabla }\times \left( \phi \cdot \overrightarrow {a}\right) $$

BySpecops.
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  • Is that thing on the left-hand side meant to be $\phi$? If so, $\phi \cdot a$ doesn't make sense. $\phi$ is a scalar field, $a$ is a vector (is it also a field)? Usually, the dot product of a scalar and a vector is not defined. – Muphrid Jan 14 '13 at 17:30
  • I edit the problem statement so that it makes sense. Please roll back/edit if I misinterpreted what was intended. – JohnD Jan 14 '13 at 17:52
  • John,thanks for edit.Thanks for comment,I'm looking here http://www.phys.ufl.edu/courses/phy3063/spring12/Lecture2-CovariantNot.pdf for basics.Then I will turn here again to see the answers.Becuase now,it's not make any sense on me.I don't understand,what does it mean? – BySpecops. Jan 14 '13 at 18:07
  • @Erbil: unfortunately, what's happened is that ordinary vector calculus is simply inadequate for some things, particularly when you get outside of 3d (for instance, in relativity, as that reference describes). So, what you're doing is converting dot and cross products into expressions with indices and learning how to work with those indexed expressions. Index notation is one way to do multivariable calculus outside of 3d in a way that makes sense. – Muphrid Jan 14 '13 at 18:23
  • @Muphrid : Get outside from 3d? Hmm.It's hard to imagine.But it is not important for now.It's enought to know how to express these multiplications.I will try to do what you did.And then I wil try for second identity.Thanks! Note : I apologies for my English. – BySpecops. Jan 14 '13 at 18:44
  • @Erbil: Just for your information, if you find answers helpful, please upvote them by clicking on the up arrows on the top left of an answer. If you find a "best" answer that resolves your original question, "accept" it by clicking on the check mark underneath the up/down arrows on the top left of that answer. – JohnD Jan 15 '13 at 00:42
  • @JohnD,http://math.stackexchange.com/questions/167754/proof-for-veca-times-vecb-times-vecc-veca-cdot-vecc-vec?rq=1 we use this method for prove.(answer 4,who used levi civita,einstein summation).I'm trouble with using these symbols. – BySpecops. Jan 15 '13 at 12:56
  • @Erbil: for this particular problem, you don't need the Levi-Civita symbol. You do need summation convention. What in particular about either do you have trouble with? Do you find neither of the two answers so far adequate? – Muphrid Jan 15 '13 at 14:01
  • @Muphrid,answers of John make a sense on me.Now I'm trying to do it with your method.Later I will come with another questions.I would like to thank you both ! – BySpecops. Jan 15 '13 at 16:10

4 Answers4

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Here's what's happening in $\mathbb{R}^3$ with rectangular coordinates. You can tweak as needed.

Let $g(x,y,z)$ be a smooth scalar function and $\mathbf{F}(x,y,z)=(F_1(x,y,z),F_2(x,y,z),F_3(x,y,z))$ be a smooth vector field. Then \begin{align} \nabla \cdot (g\,\mathbf{F})&=\nabla\cdot((gF_1,gF_2,gF_3))\\ &=(gF_1)_x+(gF_2)_y+(gF_3)_z\\ &=g_xF_1+g(F_1)_x+g_yF_2+g(F_2)_y+g_zF_3+g(F_3)_z, \end{align} while \begin{align} \nabla g\cdot \mathbf{F}&=g_x F_1+g_y F_2+g_z F_3,\\ g\,(\nabla\cdot \mathbf{F})&=g\,((F_1)_x+(F_2)_y+(F_3)_z)=g(F_1)_x+g(F_2)_y+g(F_3)_z. \end{align} Adding these last two yields the first.

JohnD
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  • I noticed now! This is a great answer. – BySpecops. Jan 15 '13 at 16:07
  • Just a friendly reminder in case you don’t know how the site works: after you ask a question here, if you get an acceptable answer, you should "upvote" and/or "accept" the answer by clicking the up arrow and the check mark ✓ next to it. This scores points for you and for the person who answered your question and will encourage others to answer your other questions. You can find out more about accepting answers here: How do I accept an answer?, Why should we accept answers?. – JohnD Dec 15 '14 at 08:04
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If JohnD has interpreted the problem correctly, then here's how you would work it using index notation. Here, $i$ is an index running from 1 to 3 ($a^1$ might be the x-component of $a$, $a^2$ the y-component, and so on).

$$\nabla \cdot (\varphi a) = \nabla_i (\varphi a^i)$$

Since these are all components (not vectors), you can attack this with the product rule.

$$\nabla_i (\varphi a^i) = (\nabla_i \varphi) a^i + \varphi (\nabla_i a^i)$$

The first term is $a \cdot \nabla \varphi$ and the latter is $\varphi \nabla \cdot a$.

Muphrid
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You probably know the product rule $(uv)'=u'v +uv'$.

I have been taught to use such formula (which follows from derivative-like nature of $\nabla$):

$\nabla(\underline{uv}) = \nabla(\underline{u}v) + \nabla(u\underline{v})$

where derivatives work on the underlined part of the brackets. In your case you can factor $\varphi$ and $\vec{a}$ when they are not under effect of nabla, but in other cases (like $\nabla (\vec{a}\cdot\vec{b})$) it can't be done that easily, but you can derive a formula for dot product starting from formula with underlines.

Džuris
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Putting a smooth vector field U in the UNIT VECTORS, and a scalar variable ϕ $$\boldsymbol{U}=u_1\boldsymbol{i}+u_2\boldsymbol{j}+u_3\boldsymbol{k}$$ Now showing that

$$\boldsymbol{\nabla}\times(\boldsymbol{U}\phi)=\phi\boldsymbol{(\nabla\times U)}+\boldsymbol{\nabla}\phi\times\boldsymbol{U}$$

$$\boldsymbol{\nabla}\times(\boldsymbol{U}\phi)=\begin{vmatrix} \boldsymbol{i}&\boldsymbol{j}&\boldsymbol{k}\\ \frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\ u_1\phi&u_2\phi&u_3\phi \end{vmatrix}\\ =\left(\boldsymbol{i}\frac{\partial(u_3\phi)}{\partial y}+\boldsymbol{j}\frac{\partial(u_1\phi)}{\partial z}+\boldsymbol{k}\frac{\partial(u_2\phi)}{\partial x}\right)-\left(\boldsymbol{i}\frac{\partial(u_2\phi)}{\partial z}+\boldsymbol{j}\frac{\partial(u_3\phi)}{\partial x}+\boldsymbol{k}\frac{\partial(u_1\phi)}{\partial y}\right)\\ =\phi\left(\boldsymbol{i}\left(\frac{\partial u_3}{\partial y}-\frac{\partial u_2}{\partial z}\right)+\boldsymbol{j}\left(\frac{\partial u_1}{\partial z}-\frac{\partial u_3}{\partial x}\right)+\boldsymbol{k}\left(\frac{\partial u_2}{\partial x}-\frac{\partial u_1}{\partial y}\right)\right)+\left(\boldsymbol{i}\left(u_3\frac{\partial\phi}{\partial y}-u_2\frac{\partial\phi}{\partial z}\right)+\boldsymbol{j}\left(u_1\frac{\partial\phi}{\partial z}-u_3\frac{\partial\phi}{\partial x}\right)+\boldsymbol{k}\left(u_2\frac{\partial\phi}{\partial x}-u_1\frac{\partial\phi}{\partial y}\right)\right)\\ =\phi\begin{vmatrix} \boldsymbol{i}&\boldsymbol{j}&\boldsymbol{k}\\ \frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\ u_1&u_2&u_3 \end{vmatrix}+\begin{vmatrix} \boldsymbol{i}&\boldsymbol{j}&\boldsymbol{k}\\ \frac{\partial\phi}{\partial x}&\frac{\partial\phi}{\partial y}&\frac{\partial\phi}{\partial z}\\ u_1&u_2&u_3 \end{vmatrix}=\phi\boldsymbol{(\nabla\times U)}+\boldsymbol{\nabla}\phi\times\boldsymbol{U}$$ Having funs with the vector calaulus!! O:DDD

hd.scania
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