Directional derivatives and gradients

[Mdhiggenz]

1. Homework Statement [/b]
Let z=3x²-y². Find all points at which magnitude(nabla Z)=6

First things first

I took the partials dz/dx and dz/dy

dz/dx=6x

dx/dy=-2y

I know that √(36x²+4y²)=6 or (36x²+4y²)=36

Then using the above relation I solved for each variable getting
1.y=√(9-x²)

2.x=√(1-1/9y²)

We already have the relation from the partials

A.-2y=0

B.6x=0
So I plugged in 1 into A and 2 into B to get the following

x=+-1 and y=+-3

However the answers don't check out

Where did I go wrong?

 

[haruspex]

I took the partials dz/dx and dz/dy
dz/dx=6x
dx/dy=-2y

∂z/∂y

(36x²+4y²)=36
Then using the above relation I solved for each variable getting
1.y=√(9-x²)

No - try that again.

 

[Ray Vickson]

1. Homework Statement [/b]
Let z=3x²-y². Find all points at which magnitude(nabla Z)=6

First things first

I took the partials dz/dx and dz/dy

dz/dx=6x

dx/dy=-2y

I know that √(36x²+4y²)=6 or (36x²+4y²)=36

Then using the above relation I solved for each variable getting
1.y=√(9-x²)

2.x=√(1-1/9y²)

We already have the relation from the partials

A.-2y=0

B.6x=0
So I plugged in 1 into A and 2 into B to get the following

x=+-1 and y=+-3

However the answers don't check out

Where did I go wrong?

Why do you try to set dz/dx = 0, and/or dz/dy = 0?

RGV