What is the Differential of a Vector?

[Kalimaa23]

Greeting

A TA has got me very and utterly confused. He won't be avaible for a few days, so I'm asking you guys.

Consider the transformation to cilindrical coord.

x-->r.con[the]
y-->r.sin[the]
z-->z

I have the Jabobian (no problems here).
He then asks the differential da , where a is a vector.
Enter the first confusion. I know the differential of the transformation (the linear function given by the Jacobian matrix), but what the hell is the differential of a vector?

My guess is: da =dxx +dyy +dzz .

I have the unity vectors of the new system.

The trick is now what is dr in fuction of the new unity vectors?

Then answer : da =drr +rd[the][the] +dzz

How the hell is this determined?

 

[quantumdude]

Originally posted by Dimitri Terryn
Enter the first confusion. I know the differential of the transformation (the linear function given by the Jacobian matrix), but what the hell is the differential of a vector?

It's not a whole lot different from the differential of a real-valued function. Surely you've seen them before. In Physics I, you learn that velocity v is related to displacement r by:

v=dr/dt

The dr in the derivative is nothing more than the differential of the vector r.

My guess is: da =dxx +dyy +dzz .

That is true if a=xx+yy+zz. Is that what a is?

 

[Kalimaa23]

Yep, that's exactly right. I just got a little confused. The things was part of a introduction to tensors, covariance and contravariance, and between the sea of indices I somewhat lost sight.

It's all clear now, though.