Answer :
Final answer:
The function that satisfies grad f = (4yz, 4xz, 4xy - 6z) is f(x, y, z) = x^2y + xyz - 3z^3 + C. Using the fundamental theorem of line integrals, the evaluation of the given line integral on a straight line from (0,2,1) to (0,4,5) results in the value of 84.
Explanation:
To find a function f such that grad f = (4yz, 4xz, 4xy - 6z), we need to integrate each component to its corresponding variable, recognizing that this represents the gradient of f. Considering the given vector field, let's integrate each component to its corresponding variable while treating the other variables as constants.
The integral to x for the component 4yz would just be 4yzx (since y and z are constants to x).
Similarly, integrating 4xz to y gives 4xzy, and integrating 4xy - 6z to z gives 2xyz - 3z2.
To find the function f, we combine these integrated expressions, making sure not to double-count the terms that involve multiple integrations, and add an arbitrary constant C.
Hence, f(x, y, z) = x2y + xyz - 3z3 + C.
To evaluate the line integral of the given vector field along path C, we can use the fundamental theorem of line integrals.
This theorem states that if f is such that grad f equals our vector field, the line integral of the field over a curve from point A to point B is simply f(B) - f(A).
Substituting coordinates (0,4,5) for B and (0,2,1) for A into our function f, we get f(0,4,5) - f(0,2,1).
This calculation yields 84.
