- #1

The Head

- 144

- 2

- Homework Statement:
- See Picture Attached

- Relevant Equations:
- fx=0, fy=0, fz=0

The beginning is straight forward and I found f=x^2-2yz, which satisfies grad(f)=F. Then I calculated W= f(x,y,z)-f(0,1,1) since it's conservative.

I get stuck when trying to find the max and mins. Given grad(f)=0 at extrema, we can see (0,0,0) is a point. On the boundary, I have to parameterize:

x=sqrt(2)cos(u)sin(v)

y=sqrt(2)sin(u)sin(v)

z=2cos(v)

Then I'm not exactly sure where to go. I feel like the options are either:

1) Substitute my parameters into f=x^2-2yz, and take the derivative. But then I'm not sure what to do because I'm used to just having one variable/parameter now.

2) Create the second partial derivative matrix (e.g., fxx, fxy, fxz in column one, fyx, fyy, fyz in column two, and fzx, fzy, fzz in column three. But I get a weird result there, with fxx>0, but fxx*fyy- fxy*fyx=0.

I usually at this point substitute the parameters and then take the derivative (with respect to a single parameter like t) and set equal to zero. Then I find all of the t's that work. I'm just not sure how to proceed here.

I get stuck when trying to find the max and mins. Given grad(f)=0 at extrema, we can see (0,0,0) is a point. On the boundary, I have to parameterize:

x=sqrt(2)cos(u)sin(v)

y=sqrt(2)sin(u)sin(v)

z=2cos(v)

Then I'm not exactly sure where to go. I feel like the options are either:

1) Substitute my parameters into f=x^2-2yz, and take the derivative. But then I'm not sure what to do because I'm used to just having one variable/parameter now.

2) Create the second partial derivative matrix (e.g., fxx, fxy, fxz in column one, fyx, fyy, fyz in column two, and fzx, fzy, fzz in column three. But I get a weird result there, with fxx>0, but fxx*fyy- fxy*fyx=0.

I usually at this point substitute the parameters and then take the derivative (with respect to a single parameter like t) and set equal to zero. Then I find all of the t's that work. I'm just not sure how to proceed here.