prove that curl(r^n c^→ ×r^→ )=(n+2)r^n c^→ −nr^(n−2) (r^→ .c^→ ) . where c is the constant vector.


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Question Number 85865 by subhankar10 last updated on 25/Mar/20

$prove that$ $curl (r^{n} \vec{c} \times \vec{r}) = (n + 2) r^{n} \vec{c} - {nr}^{n - 2} (\vec{r} . \vec{c}) .$ $where c is the constant vector .$
	Answered by TANMAY PANACEA. last updated on 25/Mar/20
	$r^→ =ix+jy+kz→r^2 =x^2 +y^2 +z^2 r^n =(x^2 +y^2 +z^2 )^(n/2) c^→ =ia+ib+kd curlA^→ =▽^→ ×A^→ (i(∂/∂x)+j(∂/∂y)+k(∂/∂z))×A^→ now ▽^→ ×(c^→ ×r^n r^→ ) =(▽^→ .r^n r^→ )c^→ −(▽^→ .c^→ )r^n r^→ =c^→ (i(∂/∂x)+j(∂/∂y)+k(∂/∂z)).(x^2 +y^2 +z^2 )^(n/2) (ix+jy+kz) =c^→ [(∂/∂x){x.(x^2 +y^2 +z^2 )^(n/2) }+(∂/∂y){y(x^2 +y^2 +z^2 )}+(∂/∂z){z(x^2 +y^2 +z^2 )}] calculation of (∂/∂x){x(x^2 +y^2 +z^2 )^(n/2) } =(x^2 +y^2 +z^2 )^(n/2) ×1+x×(n/2)(x^2 +y^2 +z^2 )^((n/2)−1) ×2x =r^n +x^2 ×n(r^2 )^((n/2)−1) =r^n +x^2 ×n×r^(n−2) adding three 3r^n +nr^(n−2) (x^2 +y^2 +z^2 ) =3r^n +nr^n =r^n (n+3) so answer is r^n (n+3)c^→ timorrow i shall solve in paper .=.. c^→ ×r^→ ∣i j k ∣ ∣a b d ∣ ∣x y z ∣ =i(bz−yd)−j(az−xd)+k(ay−bx) now ∣i j$
	$\vec{r} = i x + j y + k z \to r^{2} = x^{2} + y^{2} + z^{2}$ $r^{n} = {(x^{2} + y^{2} + z^{2})}^{\frac{n}{2}}$ $\vec{c} = i a + i b + k d$ $c u r l \vec{A} = \vec{▽} \times \vec{A}$ $(i \frac{\partial}{\partial x} + j \frac{\partial}{\partial y} + k \frac{\partial}{\partial z}) \times \vec{A}$ $n o w$ $\vec{▽} \times (\vec{c} \times r^{n} \vec{r})$ $= (\vec{▽} . r^{n} \vec{r}) \vec{c} - (\vec{▽} . \vec{c}) r^{n} \vec{r}$ $= \vec{c} (i \frac{\partial}{\partial x} + j \frac{\partial}{\partial y} + k \frac{\partial}{\partial z}) . {(x^{2} + y^{2} + z^{2})}^{\frac{n}{2}} (i x + j y + k z)$ $= \vec{c} [\frac{\partial}{\partial x} {x . {(x^{2} + y^{2} + z^{2})}^{\frac{n}{2}}} + \frac{\partial}{\partial y} {y (x^{2} + y^{2} + z^{2})} + \frac{\partial}{\partial z} {z (x^{2} + y^{2} + z^{2})}]$ $c a l c u l a t i o n o f$ $\frac{\partial}{\partial x} {x {(x^{2} + y^{2} + z^{2})}^{\frac{n}{2}}}$ $= {(x^{2} + y^{2} + z^{2})}^{\frac{n}{2}} \times 1 + x \times \frac{n}{2} {(x^{2} + y^{2} + z^{2})}^{\frac{n}{2} - 1} \times 2 x$ $= r^{n} + x^{2} \times n {(r^{2})}^{\frac{n}{2} - 1}$ $= r^{n} + x^{2} \times n \times r^{n - 2}$ $a d d i n g t h r e e$ $3 r^{n} + {n r}^{n - 2} (x^{2} + y^{2} + z^{2})$ $= 3 r^{n} + {n r}^{n} = r^{n} (n + 3)$ $s o a n s w e r i s r^{n} (n + 3) \vec{c}$ $t i m o r r o w i s h a l l s o l v e i n p a p e r$ $. = . .$ $\vec{c} \times \vec{r}$ $∣ i j k ∣$ $∣ a b d ∣$ $∣ x y z ∣$ $= i (b z - y d) - j (a z - x d) + k (a y - b x)$ $n o w$ $∣ i j$
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