Hey there,
I stumbled upon this problem and im not sure really how to do it. Ill award points to anyone who can solve this. Thank you in advance
If F=∇f find f if:
F= <2xy^3+ 3yz + z cos xz + 2x, 3x^(2)y^(2)+ 3xz + z sin yz + 2y, 3xy + x cos xz + y sin yz + 2z>
-Rick
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Answers & Comments
Verified answer
What happen if we attempt to find f?
Here, set each function of F with the specific notation... E.g.
fx = 2xy³ + 3yz + zcos(xz) + 2x
fy = 3x²y² + 3xz + zsin(yz) + 2y
fz = 3xy + xcos(xz) + ysin(yz) + 2z
Take fx and integrate it with respect to x to get...
F = x²y³ + 3xyz + sin(xz) + x² + g(y,z)
Differentiate the function w/respect to y to get...
fy = 3x²y² + g_y(y,z) = 3x²y² + 3xz + zsin(yz) + 2y
Set g_y(y,z) = 3xz + zsin(yz) + 2y, and we have...
f_y = 3x²y² + 3xz + zsin(yz) + 2y
Integrate w/respect to y to get...
F = x²y³ + 3xyz - cos(yz) + y² + h(z)
However, this is not equivalent to the one I found for this problem, so we say that the field is not conservative. Why? Try another way.
∂P/∂y = ∂Q/∂x
So?
6xy² - z²sin(xz) + 2 ≠ 6xy² + 3z
Therefore, there is no existing function of f.
Good luck!