Fundamental Theorem for Line Integrals — Question 2

Question 2

Let $f(x,y)=x^2+y^2$. A smooth curve $C$ travels from $A=(3,4)$ to $B=(0,5)$.

Tasks

1. Evaluate $\displaystyle\int_C\nabla f\cdot d\mathbf r$.

2. Interpret the result using level curves of $f$.

3. Decide whether a different smooth path from $A$ to $B$ could change the value.

Question 2 – Solution

Strategy. Compare the potential values at the endpoints before doing any curve calculation.

Step 1: Endpoint values $$f(A)=3^2+4^2=25, \qquad f(B)=0^2+5^2=25.$$ Hence $$\boxed{\int_C\nabla f\cdot d\mathbf r=f(B)-f(A)=0}.$$

Step 2: Interpret Both endpoints lie on the level circle $x^2+y^2=25$. The curve between them need not stay on that circle: $f$ may rise and fall along the way, but its total change is still zero.

Step 3: Compare paths Any other smooth path with the same orientation from $A$ to $B$ gives the same endpoint difference, so it cannot change the integral.

Verification The result is not claiming that $\nabla f$ vanishes. Indeed $\nabla f=\langle 2x,2y\rangle$ is nonzero at both endpoints; only the net change in $f$ is zero.