High School

We appreciate your visit to Find a function f such that nabla f 4yz 4xz 4xy 6z Use f to evaluate the line integral int C 4yz 4xz 4xy 6z. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!

Find a function \( f \) such that \(\nabla f = (4yz, 4xz, 4xy-6z)\).

Use \( f \) to evaluate the line integral:

\[
\int_C (4yz, 4xz, 4xy-6z) \cdot d\mathbf{r}
\]

where \( C \) is the straight line from \((0, 2, 1)\) to \((0, 4, 5)\).

A. 66
B. 72
C. 78
D. 84

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.

Thanks for taking the time to read Find a function f such that nabla f 4yz 4xz 4xy 6z Use f to evaluate the line integral int C 4yz 4xz 4xy 6z. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada