高専の数学解説

第6章 6.17

(1)

{x=12(u2−v2)y=uv\begin{cases} x = \dfrac{1}{2}(u^2 - v^2) \\ y = uv \end{cases}

∣u−vvu∣=u2+v2\begin{vmatrix} u & -v \\ v & u \end{vmatrix} = u^2 + v^2

(2)

{x=u2+v2y=uv\begin{cases} x = u^2 + v^2 \\ y = uv \end{cases}

xu=2uxv=2vyu=vyv=ux_u = 2u \qquad x_v = 2v \qquad y_u = v \qquad y_v = u

∣2u2vvu∣=2u2−2v2\begin{vmatrix} 2u & 2v \\ v & u \end{vmatrix} = 2u^2 - 2v^2

(3)

{x=log⁡u2+v2y=tan⁡−1vu\begin{cases} x = \log\sqrt{u^2 + v^2} \\ y = \tan^{-1}\dfrac{v}{u} \end{cases}

xu=uu2+v2xv=vu2+v2yu=−vu2+v2yv=uu2+v2x_u = \dfrac{u}{u^2+v^2} \qquad x_v = \dfrac{v}{u^2+v^2} \qquad y_u = \dfrac{-v}{u^2+v^2} \qquad y_v = \dfrac{u}{u^2+v^2}

∣uu2+v2vu2+v2−vu2+v2uu2+v2∣=u2+v2(u2+v2)2=1u2+v2\begin{vmatrix} \dfrac{u}{u^2+v^2} & \dfrac{v}{u^2+v^2} \\ \dfrac{-v}{u^2+v^2} & \dfrac{u}{u^2+v^2} \end{vmatrix} = \dfrac{u^2+v^2}{(u^2+v^2)^2} = \dfrac{1}{u^2+v^2}

(4)

{r=x2+y2θ=tan⁡−1yx\begin{cases} r = \sqrt{x^2+y^2} \\ \theta = \tan^{-1}\dfrac{y}{x} \end{cases}

rx=xx2+y2ry=yx2+y2θx=−yx2+y2θy=xx2+y2r_x = \dfrac{x}{\sqrt{x^2+y^2}} \qquad r_y = \dfrac{y}{\sqrt{x^2+y^2}} \qquad \theta_x = \dfrac{-y}{x^2+y^2} \qquad \theta_y = \dfrac{x}{x^2+y^2}

∣xx2+y2yx2+y2−yx2+y2xx2+y2∣=x2+y2(x2+y2)32=1x2+y2\begin{vmatrix} \dfrac{x}{\sqrt{x^2+y^2}} & \dfrac{y}{\sqrt{x^2+y^2}} \\ \dfrac{-y}{x^2+y^2} & \dfrac{x}{x^2+y^2} \end{vmatrix} = \dfrac{x^2+y^2}{(x^2+y^2)^{\frac{3}{2}}} = \dfrac{1}{\sqrt{x^2+y^2}}

(5)

{x=cos⁡uy=sin⁡u⋅cos⁡vz=sin⁡u⋅sin⁡v⋅cos⁡w\begin{cases} x = \cos u \\ y = \sin u \cdot \cos v \\ z = \sin u \cdot \sin v \cdot \cos w \end{cases}

xu=−sin⁡uxv=0xw=0yu=cos⁡u⋅cos⁡vyv=−sin⁡u⋅sin⁡vyw=0x_u = -\sin u \quad x_v = 0 \quad x_w = 0 \quad y_u = \cos u \cdot \cos v \quad y_v = -\sin u \cdot \sin v \quad y_w = 0

zu=cos⁡u⋅sin⁡v⋅cos⁡wzv=sin⁡u⋅cos⁡v⋅cos⁡wzw=−sin⁡u⋅sin⁡v⋅sin⁡wz_u = \cos u \cdot \sin v \cdot \cos w \quad z_v = \sin u \cdot \cos v \cdot \cos w \quad z_w = -\sin u \cdot \sin v \cdot \sin w

∣−sin⁡u00cos⁡u⋅cos⁡v−sin⁡u⋅sin⁡v0cos⁡u⋅sin⁡v⋅cos⁡wsin⁡u⋅cos⁡v⋅cos⁡w−sin⁡u⋅sin⁡v⋅sin⁡w∣=−sin⁡3u⋅sin⁡2v⋅sin⁡w\begin{vmatrix} -\sin u & 0 & 0 \\ \cos u \cdot \cos v & -\sin u \cdot \sin v & 0 \\ \cos u \cdot \sin v \cdot \cos w & \sin u \cdot \cos v \cdot \cos w & -\sin u \cdot \sin v \cdot \sin w \end{vmatrix} = -\sin^3 u \cdot \sin^2 v \cdot \sin w

(6)

{x=rsin⁡θ⋅cos⁡ϕy=rsin⁡θ⋅sin⁡ϕz=rcos⁡θ\begin{cases} x = r \sin\theta \cdot \cos\phi \\ y = r \sin\theta \cdot \sin\phi \\ z = r \cos\theta \end{cases}

xθ=rcos⁡θ⋅cos⁡ϕxϕ=−rsin⁡θ⋅sin⁡ϕxr=sin⁡θ⋅cos⁡ϕx_\theta = r\cos\theta \cdot \cos\phi \quad x_\phi = -r\sin\theta \cdot \sin\phi \quad x_r = \sin\theta \cdot \cos\phi

yθ=rcos⁡θ⋅sin⁡ϕyϕ=rsin⁡θ⋅cos⁡ϕyr=sin⁡θ⋅sin⁡ϕy_\theta = r\cos\theta \cdot \sin\phi \quad y_\phi = r\sin\theta \cdot \cos\phi \quad y_r = \sin\theta \cdot \sin\phi

zθ=−rsin⁡θzϕ=0zr=cos⁡θz_\theta = -r\sin\theta \quad z_\phi = 0 \quad z_r = \cos\theta

∣rcos⁡θ⋅cos⁡ϕ−rsin⁡θ⋅sin⁡ϕsin⁡θ⋅cos⁡ϕrcos⁡θ⋅sin⁡ϕrsin⁡θ⋅cos⁡ϕsin⁡θ⋅sin⁡ϕ−rsin⁡θ0cos⁡θ∣\begin{vmatrix} r\cos\theta \cdot \cos\phi & -r\sin\theta \cdot \sin\phi & \sin\theta \cdot \cos\phi \\ r\cos\theta \cdot \sin\phi & r\sin\theta \cdot \cos\phi & \sin\theta \cdot \sin\phi \\ -r\sin\theta & 0 & \cos\theta \end{vmatrix}

=r2cos⁡2θ⋅sin⁡θ⋅cos⁡2ϕ+r2sin⁡3θ⋅sin⁡2ϕ+r2sin⁡3θ⋅cos⁡2ϕ+r2cos⁡2θ⋅sin⁡θ⋅sin⁡2ϕ= r^2\cos^2\theta \cdot \sin\theta \cdot \cos^2\phi + r^2\sin^3\theta \cdot \sin^2\phi + r^2\sin^3\theta \cdot \cos^2\phi + r^2\cos^2\theta \cdot \sin\theta \cdot \sin^2\phi

=r2sin⁡θ= r^2\sin\theta

解説: r31bn1z