Answer:
f(2,2,4) = 8e^4 - 4
Step-by-step explanation:
Note that by integrating each entry of "f" with respect to the proper variable:
∇(yze^(x^2) + C) = {^(x^2), ze^(x^2), ye^(x^2)
}
Hence, f(x,y,z) = yze^(x^2) + C
So, the Fundamental Theorem of Calculus for line integral yields
∫c f(x,y,z) · dr, where C is a path from (0,0,0) to (2,2,4)
= (yze^(x^2) {from (0,0,0) to (2,2,4)}
= 8e^4.
Now, let's solve for C, using f(0,0,0) = -4:
-4= 0 + C ⇒ C = -4.
So, f(x,y,z) = yze^(x^2) - 4
⇒ f(2,2,4) = 8e^4 - 4.
