Fundamental Theorem for Line Integrals — Question 1

Question 1

Let $$f(x,y,z)=x^2y+e^z,$$ and let $C$ be any smooth curve from $A=(1,0,0)$ to $B=(2,1,\ln 2)$. Evaluate $$\int_C \nabla f\cdot d\mathbf r.$$

Tasks

1. State the Fundamental Theorem for Line Integrals.

2. Evaluate the integral without parametrizing $C$.

3. Explain why the unspecified shape of $C$ is irrelevant.

Question 1 – Solution

Strategy. Since the field is explicitly $\nabla f$, apply the endpoint formula $\int_C\nabla f\cdot d\mathbf r=f(B)-f(A)$.

Step 1: Evaluate the endpoint values $$f(B)=2^2(1)+e^{\ln 2}=4+2=6, \qquad f(A)=1^2(0)+e^0=1.$$

Step 2: Apply the theorem $$\int_C\nabla f\cdot d\mathbf r=f(B)-f(A)=6-1=\boxed{5}.$$

Step 3: Explain path independence Along any smooth parametrization $\mathbf r(t)$, $\nabla f(\mathbf r(t))\cdot\mathbf r'(t)$ is the derivative of $f(\mathbf r(t))$. Its integral therefore records only the net change in $f$.

Verification The logarithmic coordinate was chosen so that $e^{\ln 2}=2$; using $\ln 2$ itself in $f(B)$ would be an endpoint-substitution error.