0.0.0.3 解答

$\Vector{f}(\Vector{\varphi}(t))=\twovector{t^{3/2}}{t^2}$, $\Vector{\varphi}'(t)=\twovector{1}{\dfrac{3}{2}t^{1/2}}$ であるから、

$$\Vector{f}(\Vector{\varphi}(t))\cdot\Vector{\varphi}'(t) =t^{3/2}\cdot 1+t^2\cdot\frac{3}{2}t^{1/2}=t^{3/2}+\frac{3}{2}t^{5/2}.$$

ゆえに

$$\int_{C_1}\Vector{f}\cdot\D\Vector{r} =\int_0^1\left(t^{3/2}+\fra... ...right]_0^1 =\frac{2}{5}+\frac{3}{7}=\frac{2\cdot7+3\cdot 5}{35}=\frac{29}{35}.$$

一方、 $\Vector{f}(\Vector{\psi}(s))=\twovector{s^3}{(s^2)^2} =\twovector{s^3}{s^4}$, $\Vector{\psi}'(s)=\twovector{2s}{3s^2}$ であるから、

$$\Vector{f}(\Vector{\psi}(s))\cdot\Vector{\psi}'(s) =s^3\cdot 2s+s^4\cdot 3s^2=2s^4+3s^6.$$

ゆえに

$$\int_{C_2}\Vector{f}\cdot\D\Vector{r} =\int_0^1\left(2s^4+3s^6\ri... ...\right]_0^1 =\frac{2}{5}+\frac{3}{7}=\frac{2\cdot7+3\cdot 5}{35}=\frac{29}{35}.$$   (解答終り)