Consider the field, where is a vector constant. Using suffix notation, show that:
i)
ii)
Using suffix notation, we get
And then the dot product gives us
Just a little unsure where to go from here?
Thanks in advance
Consider the field, where is a vector constant. Using suffix notation, show that:
i)
ii)
Using suffix notation, we get
And then the dot product gives us
Just a little unsure where to go from here?
Thanks in advance
It should be
The dot product would then be
[EDIT]: This equation is incorrect. The derivative operator cannot go past the 1/r. See below for correction.
Can you see where this is going?
For the cross product, you'll have a double sum, and then you can reduce using your Levi-Civita to Kronecker Delta identity, that should reduce to the RHS. Make sense?
Ahh sorry a little typo in writing out the question. Not too sure how to evaluate the suffix partial differentiation though :\
Ahh that does make a little sense, I thought that you would end up using the result from the first part so hadn't really given this much of a go.
Will have a play around with it now, thankyou
Because it's a constant, when differentiated it just becomes zero, that's the one bit I knew already haha!
I know the fact about the Levi-Civita symbol being zero if any indices appear more than once, but do we 'sum' over the different s ?
Sorry if I'm not getting this, not had much chance at getting my head around the whole suffix notation thing :\
Hang on a bit, I think I made a mistake earlier. definitely depends on so the partial derivative can't go past that.
Ok, let me make the notation a little clearer:
(using the quotient rule)
Now if the Levi-Civita symbol cancels out the whole expression. So no need to examine that case. Assume Then how does this expression simplify? Recall that
Right.
Everyone learns basic things at some point! I would argue, though, that you already know this fact, you just don't recognize that you know it already.I've never heard that before, is this just basic knowledge that I should know haha?
Think about it for a second: I'm taking the partial derivative of one independent variable with respect to another independent variable. If the two variables happen to be the same, then I should get one, right? I mean, you definitely have
and similarly
That you know from Calculus I and Calculus III (or Differential Calculus and Multivariable Calculus, depending on where you took it).
However, let's suppose that and are both independent variables. What should
be?
Suppose I gave you a function and asked you to compute
What would you tell me? And then, if I gave you a function and asked you to compute
what would you tell me? But isn't
So what does that tell you?
Right I'm assuming that this is the obvious 0, treating the as a constant when we take the partial derivative wrt .
We know that both and are both zero, by the same principle as above?
So does this mean that the two partial derivatives in our previous equations are both zero. Hence the dot product is also zero?
Thanks again for the reply.
Ok, so it looks like you understand now why
Looking back at post # 6, and looking at the expression
I can simplify down to
But really, the is never going to get a chance to be because if then the Levi-Civita symbol at the beginning of the expression is zero. Thus, we are justified in simply getting rid of that term altogether, and so we now have the expression
Now, given that can you tell me what
is?
If all the indices are throwing you off, just think of
and compute
What do you get?
Your partial derivative of w.r.t. is correct, but your application of it to the expression at hand is incorrect. You need
where I've written the last expression in a (hopefully) suggestive manner. And I think I've answered your question as well.
Do you recognize the expression
Think geometrically here.
Ahh of course it is!!
Right, I don't recognise it exactly, but according to my lecture notes, .
Because of the cyclic property of the Levi-Cevita symbol, and the fact that is a vector constant, this means that it's equal to zero?
Either way, could you explain what the geometric interpretation is haha, could be useful...