第一題
\(dr=-isint-jcost+k\)
\(F\cdot dr=-sint+cos^{2}t+1\)
\(\int_{c}^{}F\cdot dr=\int_{0}^{\pi}(-sint+cos^{2}t+1) dt=\pi-2\)
第二題
take \(g\left(x,y,z \right)=\frac{1}{x}-3y-z-10\)
\(\bigtriangledown g=\frac{-1}{x^{2}}i-3j-k\)
normal vector n=\(\frac{\bigtriangledown g}{\left|\bigtriangledown g \right | }\)
tangent plane \(n\cdot \left(x-1,y+3,z \right)=0 \)
第三題
\(0\leq x \leq 1\)
\(0\leq y \leq x\)
\(\int_{0}^{1}\int_{0}^{x}e^{x^{2}}dydx =\frac{e-1}{2}\)
