Why Does Implicit Differentiation Give Contradictory Results for u_z?

AI Thread Summary
The discussion revolves around the challenges of finding the partial derivative u_z using implicit differentiation on the given equations f and g. The user encounters contradictory results when applying implicit differentiation and is uncertain about the validity of their approach using the formula involving f_z, f_u, and g_z. There is a request for clarification on the differentiation process and guidance on the correct application of implicit differentiation. Additionally, there is a reminder to post homework-related queries in the appropriate section of the forum. The conversation highlights the complexities of implicit differentiation and the importance of proper problem categorization.
Icebreaker
f(x,y,z,u,v)=xe^y+uz-\cos v=2
g(x,y,z,u,v)=u\cos y+x^2v-yz^2=1

I need to find u_z. When I try to do it by implicitly differentiating and solving the equation, I get 2 contradictory answers. If I try the formula, i.e.

f_z + f_uu_z + f_vv_z = 0
g_z + g_uu_z + g_vv_z = 0

I get an answer, but I'm not sure if it's right, since it does not equal to the answer I get when I differentiate implicitly. Any help?

Also I'm not entirely sure if my "formula" is right. Maybe this formula is just implicit differentiation, I haven't looked into it.
 
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2 things:

1. Please show how you started, and where you got stuck.

2. Please stop posting homework problems in the Math section. They should go in the Homework section.

Thanks.
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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