Question

Consider the function \(f(x,y) = x^3 + y^2\).

If \(\vec{F} = \nabla f\), then \(\displaystyle\int \vec{F} \cdot d\vec{r}\) evaluated from \((0,0)\) to \((1,2)\) is ______ (answer in integer).

Hint:

A gradient field is path independent, so the integral is just the potential difference between the endpoints.

Answer 1

Correct Answer: 5

Step 1: Recognize the field is conservative.
Since \(\vec{F} = \nabla f\), the vector field \(\vec{F}\) is a gradient field, so \(f\) itself is its potential function.
For a gradient field, the line integral between two points depends only on the endpoints, not on the path taken.

Step 2: Apply the fundamental theorem for line integrals.
\[ \int_{(0,0)}^{(1,2)} \vec{F} \cdot d\vec{r} = f(1,2) - f(0,0) \]

Step 3: Evaluate \(f\) at both points.
\(f(1,2) = 1^3 + 2^2 = 1 + 4 = 5\), and \(f(0,0) = 0^3 + 0^2 = 0\).

Final Answer:
The line integral equals \(5 - 0 = 5\).
\[ \boxed{5} \]