x-y-z-e-cos-e-sin-0-0-2pi-R-r-x-i-y-j-z-k-a-r-a-r-a-r-a-

Question Number 1858 by 123456 last updated on 16/Oct/15

[\begin{matrix} x (ρ, θ, ξ) \\ y (ρ, θ, ξ) \\ z (ρ, θ, ξ) \end{matrix}] = [\begin{matrix} ρ e^{ξ} \cos θ \\ ρ e^{ξ} \sin θ \\ ξ \end{matrix}] {\begin{cases} ρ \in [0, + \infty) \\ θ \in [0, 2 π) \\ ξ \in R \end{cases}

r (ρ, θ, ξ) = x (ρ, θ, ξ) i + y (ρ, θ, ξ) j + z (ρ, θ, ξ) k

a_{ρ} = \frac{\partial r}{\partial ρ}

a_{θ} = \frac{\partial r}{\partial θ}

a_{ξ} = \frac{\partial r}{\partial ξ}

a_{ρ} \cdot a_{ρ} + a_{θ} \cdot a_{θ} + a_{ξ} \cdot a_{ξ} = ? ? ?

a_{ρ} \cdot a_{θ} + a_{ρ} \cdot a_{ξ} + a_{θ} \cdot a_{ξ} = ? ? ?

a_{ρ} \times a_{θ} + a_{ρ} \times a_{ξ} + a_{θ} \times a_{ξ} = ? ? ?

Answered by 112358 last updated on 16/Oct/15

r (ρ, θ, ξ) = (ρ e^{ξ} c o s θ) i + (ρ e^{ξ} s i n θ) j + ξ k

a_{ρ} = \frac{\partial r}{\partial ρ} = \frac{\partial}{\partial ρ} (ρ e^{ξ} c o s θ) i + \frac{\partial}{\partial ρ} (ρ e^{ξ} s i n θ) j + \frac{\partial}{\partial ρ} (ξ) k

a_{ρ} = (e^{ξ} c o s θ) i + (e^{ξ} s i n θ) j + 0 k

B y t h e s i m i l a r p r o c e s s i n v o l v i n g

p a r t i a l d i f f e r e n t i a t i o n w e g e t

a_{θ} = (- ρ e^{ξ} s i n θ) i + (ρ e^{ξ} c o s θ) j + 0 k

a_{ξ} = (ρ e^{ξ} c o s θ) i + (ρ e^{ξ} s i n θ) j + k

F o r a n y v e c t o r w e h a v e x . x =∣ x ∣^{2} .

∴ S_{1} = Σ a . a =∣ a_{θ} ∣^{2} + ∣ a_{ρ} ∣^{2} + ∣ a_{ξ} ∣^{2}

S_{1} = e^{2 ξ} ({s i n}^{2} θ + {c o s}^{2} θ) + 2 ρ^{2} e^{2 ξ} ({s i n}^{2} θ + {c o s}^{2} θ) + 1

S_{1} = e^{2 ξ} (2 ρ^{2} + 1) + 1

L e t S_{2} = a_{θ} . a_{ρ} + a_{ξ} . a_{ρ} + a_{θ} . a_{ξ}

a_{ρ} . a_{θ} = - ρ e^{2 ξ} c o s θ s i n θ + ρ e^{2 ξ} s i n θ c o s θ = 0

a_{ξ} . a_{ρ} = ρ e^{2 ξ} {c o s}^{2} θ + ρ e^{2 ξ} {s i n}^{2} θ = ρ e^{2 ξ}

a_{θ} . a_{ξ} = - ρ^{2} e^{2 ξ} s i n θ c o s θ + ρ^{2} e^{2 ξ} s i n θ c o s θ = 0

∴ S_{2} = 0 + ρ e^{2 ξ} + 0 = ρ e^{2 ξ}

a_{ρ} \times a_{θ} = i (0 - 0) - j (0 - 0) + k (e^{ξ} c o s θ \times ρ e^{ξ} c o s θ - e^{ξ} s i n θ (- ρ e^{ξ} s i n θ))

a_{ρ} \times a_{θ} = 0 i + 0 j + ρ e^{2 ξ} k

a_{ρ} \times a_{ξ} = i (e^{ξ} s i n θ \times 1 - 0) - j (1 \times e^{ξ} c o s θ - 0) + k (ρ e^{2 ξ} c o s θ s i n θ - ρ e^{2 ξ} c o s θ s i n θ)

a_{ρ} \times a_{ξ} = (e^{ξ} s i n θ) i - (e^{ξ} c o s θ) j + 0 k

a_{θ} \times a_{ξ} = i (ρ e^{ξ} c o s θ - 0) - j (- ρ e^{ξ} s i n θ - 0) + k (- ρ^{2} e^{2 ξ} {s i n}^{2} θ - ρ^{2} e^{2 ξ} {c o s}^{2} θ)

a_{θ} \times a_{ξ} = (ρ e^{ξ} c o s θ) i + (ρ e^{ξ} s i n θ) j - (ρ^{2} e^{2 ξ}) k

∴ a_{ρ} \times a_{θ} + a_{ρ} \times a_{ξ} + a_{θ} \times a_{ξ}

= e^{ξ} (ρ c o s θ + s i n θ) i + e^{ξ} (ρ s i n θ - c o s θ) j + ρ e^{2 ξ} (1 - ρ) k

Facebook Tweet Pin