Proving Characteristics of Partial Derivatives

  • #1

Homework Statement


Let F(x,y) be a twice differentiable function such that

4 * Fx2 + Fy2 = 0.

Set x = u2 - v2 and y = u*v. Show that

Fu2 + Fv2 = 0


The Attempt at a Solution


Fu = Fx*2u + Fy*v
Fv = Fx*-2v + Fy*u

Fu2 = 4v2Fx2-4FxFy*u*v+Fy2u2

Fv2 = 4u2Fx2+4FxFy*u*v+Fy2v2

Adding these two together gives

4Fx2*(u2v2) + 4Fy2*(u2v2),

Which from our first equation, can clearly equal zero if we factor out the (u2v2) and divide. Then, we see that, indeed, Fu2 + Fv2 = 0.

Sound about right?
 

Answers and Replies

  • #2
SammyS
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Homework Statement


Let F(x,y) be a twice differentiable function such that

4 * Fx2 + Fy2 = 0.

Set x = u2 - v2 and y = u*v. Show that

Fu2 + Fv2 = 0


The Attempt at a Solution


Fu = Fx*2u + Fy*v
Fv = Fx*-2v + Fy*u

[STRIKE]Fu2 [/STRIKE]= 4v2Fx2-4FxFy*u*v+Fy2u2 = Fv2

Fv2 = 4u2Fx2+4FxFy*u*v+Fy2v2 = Fu2

Adding these two together gives

4Fx2*(u2+v2) +[STRIKE] 4 [/STRIKE] Fy2*(u2+v2),

Which from our first equation, can clearly equal zero if we factor out the (u2+v2) and divide. Then, we see that, indeed, Fu2 + Fv2 = 0.

Sound about right?
You have some typos in your post.
 

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