Stokes' Theorem — Question 2

Question 2 – Solution

Strategy. Use the triangular plane itself; its graph vector element is constant and points in the prescribed direction.

Step 1: Curl and surface element $$\nabla\times\mathbf F=\langle 1,1,1\rangle.$$ Writing $z=1-x-y$ over $D=\{x,y\ge 0:x+y\le 1\}$ gives $$\boxed{\mathbf n\,dS=\langle 1,1,1\rangle\,dx\,dy}.$$

Step 2: Apply Stokes $$\begin{align*} \oint_C\mathbf F\cdot d\mathbf r &=\iint_D\langle 1,1,1\rangle\cdot\langle 1,1,1\rangle\,dA\\ &=3\operatorname{area}(D)=\boxed{\frac 32}. \end{align*}$$

Step 3: Edge check The orientation follows $$(1,0,0)\to(0,1,0)\to(0,0,1)\to(1,0,0).$$ Parametrizing each edge by $0\le t\le 1$ gives integrands $1-t$, $1-t$, and $1-t$, respectively. Hence the three line integrals are each $1/2$, totaling $3/2$.

Verification Two successive directed edge vectors have cross product parallel to $\langle 1,1,1\rangle$, confirming the boundary orientation.