# Time Derivatives of Unit Vectors

1. Dec 14, 2011

### Jshroomer

1. The problem statement, all variables and given/known data

The Hyperbolic coordinate system is given by: u=2xy and v=x^2 - y^2

a.) Find the unit vectors u and v in terms of u,v,x(hat),y(hat)
b.)Find the time derivatives of u(hat) and v(hat), your answers will have du/dt and dv/dt in them

2. Relevant equations

None really

3. The attempt at a solution

I found the unit vectors by

u(hat) = (du/dx)i + (du/dy)j
v(hat) = (dv/dx)i + (dv/dy)j

then I substituted in for x and y to give equations in terms of v and u

u(hat) = (sqrt(2))sqrt(sqrt(u^2 + v^2) - v))x(hat) + (sqrt(2))sqrt(sqrt(u^2 + v^2) +v))y(hat)

v(hat) = (sqrt(2))sqrt(sqrt(u^2 + v^2) + v))x(hat) + (-sqrt(2))sqrt(sqrt(u^2 + v^2) - v))y(hat)

I just don't understand how to take the time derivatives of these unit vectors, any help would be greatly appreciated.

Thanks

2. Dec 14, 2011

### SammyS

Staff Emeritus
Your unit vectors do not appear to have the correct magnitude, which should be 1.

For $\hat{u}$ I get: $\displaystyle\hat{u}=\frac{\displaystyle \frac{\partial u}{\partial x}\hat{i}+\frac{\partial u}{\partial y}\hat{j}}{\displaystyle\sqrt{\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial u}{\partial y}\right)^2}}=\frac{y\hat{i}+x\hat{j}}{\sqrt{x^2+y^2}}$

3. Dec 14, 2011

### Jshroomer

Thank you for the correction sammy, I'm not very good at vector calculus, and my book is hard to get useful information from.

so, I have

u(hat) = (2yi + 2xj)/(sqrt(4x^2 + 4y^2))

and

v(hat) = (2xi -2yj)/(sqrt(4x^2 + 4y^2))

I still do not understand how to take the time derivatives though, thanks

Last edited: Dec 14, 2011