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Homework Statement:
 See Picture Attached
Relevant Equations:
 fx=0, fy=0, fz=0
The beginning is straight forward and I found f=x^22yz, 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^22yz, 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^22yz, 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.
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