taking the derivattive w/r to ...
same idea for
Hi folks, hopefully this is in the right forum. My textbook tells me I can use the product rule to go from the functions x = r cos phi and y = r sin phi to find expressions for dx and dy. I'm not clear whether this is partial differentiation (I thought I understood that!), or what I am differentiating with respect to! I have attached a jpg of the textbook section which should be clearer that typing it all out. I think I can get to the final expression, it's finding the middle two I don't understand. Can anyone help?
My jpg refers to equation 3.4, this is simply dl^2 = dx^2 + dy^2.
It's the "total differential". If f(x,y) is a function of two variables, and each of those variables is a function of the single variable t, x(t), y(t), (think of an object moving along some trajectory in the plane with t as time), then, by the chain rule
Since that is now a function of a single variable, we can write its "differential", :
(Which no longer has any dependence on t!)
In particular, if and , then . And if , then .
Squaring those,
and
Adding, the " " terms cancel while so that
Thanks very much for taking the time to answer chaps, having quoted one of your responses I can see the effort that goes into a properly formatted answer!
Skeeter: Your response was the same as I had seen in another textbook; whilst undoubtedly correct, unfortunately I am not able to make sense of it. I don;'t quite see why dx/dphi is not as simple as -r sin phi. Entirely my failing, not yours!
HallsofIvy: I have been able to follow quite a lot of your answer, but I still have a couple of questions about it. If you have time to explain further it would be much appreciated, if not it has at least got me to a working solution!
In my case I have x(r,phi) as one function of two variables, and y(r,phi) as a second function of two variables...
I can't quite see how this works in my case. Aren't r and phi two independant variables? What is the analogy to your t variable in my example?and each of those variables is a function of the single variable t, x(t), y(t), (think of an object moving along some trajectory in the plane with t as time),
I get this part! This is partial differentiation as I understand itthen, by the chain rule
Is this 'thing' df the "total differential"? And does the operation where you "multiply though" by dt have a name (I realise it is not as simple as multiplying through!)Since that is now a function of a single variable, we can write its "differential", :
(Which no longer has any dependence on t!)
That last bit all makes sense to me, I have never seen this type of differential manipulation before though which is why I am struggling!In particular, if and , then . And if , then .
Squaring those,
and
Adding, the " " terms cancel while so that
if were a constant, then ... however, is not a constant, it is an implicit function of .Skeeter: Your response was the same as I had seen in another textbook; whilst undoubtedly correct, unfortunately I am not able to make sense of it. I don;'t quite see why dx/dphi is not as simple as -r sin phi.