Homework Help: Two problems while reading Feynman lectures (vector field))

1. Jul 14, 2011

Proust

Question 1:

solved!

Question 2:

Why it's zero? I think we cannot get zero unless it's an exact differential form?

Many thanks.

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Last edited: Jul 14, 2011
2. Jul 14, 2011

EWH

Well, I don't know how to explain it better than Feynman, but the curl of the gradient of a scalar function is always zero.

3. Jul 15, 2011

Proust

In fact I just can't understand why we have

A X (AT) = (A X A) T

Why the same form ...

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4. Jul 15, 2011

Dick

5. Jul 15, 2011

Proust

6. Jul 15, 2011

cupid.callin

if i am getting your question right ... you are asking why A X (AT) = (A X A)T = 0
is that right?

7. Jul 15, 2011

SammyS

Staff Emeritus
I think he's asking why ᐁ X (ᐁT) = (ᐁ X ᐁ)T is of the same form as A X (AT) = (A X A)T.

8. Jul 16, 2011

Proust

Yes! And I'm still confused now!!

9. Jul 16, 2011

BruceW

These kinds of relations are easy when you use the levi-civita tensor instead of vector form.
Unfortunately, I guess tensors shouldn't be included in introductory physics?

Instead, you could write out all the components explicitly, and see that the equality holds.

10. Jul 16, 2011

cupid.callin

All i can understand and tell you is that, if A is a vector and T is some scalar constant then its kind of a basic rule of vectors that A X (AT) = (A X A)T because no matter f you multiply the scalar before of after solving cross product ... answer comes same.

and also $\vec{A} X \vec{B} \ = \ AB \ sin\theta \ \hat{n}$

where $\theta$ is and b/w $\vec{A} \ \ and \ \ \vec{B}$

so angle b/w $\vec{A} \ \ and \ \ \vec{A}$ is 0 and sin(0) = 0

thus (A X A)T = 0

11. Jul 16, 2011

tharinduuuu

i think post#10 explains it very well. just want to add that Feynman was probably trying to make you think of the del operator as just another vector and the scalar field(T) as just a scalar by showing the similarity between the two expressions. The fact that one of the expressions evaluate to zero should then help you guess that maybe the other one is zero too which in turn will help you in proving that it indeed is.And it will also help you remember and have an intuitive understanding of identities like this without having to memorize everything.