Time Derivatives of Unit Vectors

[Jshroomer]

Homework Statement

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

Homework Equations

None really

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

 

[SammyS]

Homework Statement

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

Homework Equations

None really

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

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

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

 

[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