Partial Derivatives: Express vx from u,v in x,y

[mit_hacker]

Homework Statement

(Q) Express vx in terms of u an v if the equations x = v ln(u) and y = u ln(v) define u and v as functions of the independent variables x and y, and if vx exists. (Hint: Differentiate both equations with respect to x and solve for vx by eliminating ux.

Homework Equations

The Attempt at a Solution

I got to the following two equations but don't know how to proceed from there.


1= v/u (u_x )+ln⁡(u) (v_x)

∂y/∂x=u/v (v_x )+ln⁡(v) (u_x)

 

[mighty2000]

Hello, I am a scientist and I would like to help you with your question.

First, let's start by differentiating both equations with respect to x:

x = v ln(u)
x' = v'/u + v/u * u'

y = u ln(v)
y' = u'/v + u/v * v'

Now, we can rearrange these equations to solve for v' and u':

v' = (x' - v/u * u') * u
u' = (y' - u/v * v') * v

Next, we can substitute these expressions for v' and u' into our original equations:

1 = v/u * (x' - v/u * u') + ln(u) * (y' - u/v * v')
1 = x' - v^2/u * u' + ln(u) * y' - u/v * ln(u) * v'
1 = x' - u' * (v^2/u + ln(u) * v')
1 = x' - u' * (v^2 + u * ln(u) * v')/u

Now, we can solve for vx by eliminating u':

vx = (1 - x')/((v^2 + u * ln(u) * v')/u)

Finally, we can substitute our expression for u' into our equation for vx to get our final answer:

vx = (1 - x')/((v^2 + y' * v)/v)

I hope this helps and let me know if you have any further questions. Good luck!