If x=rcosh 0, y = rsinh , then (), is equal to,
Answers
Step-by-step explanation:
Given : x=rcosθy=rsinθ
r
x
=cosθ
r
y
=sinθ
cosθ=
r
x
sinθ=
r
y
θ=cos
−1
(
r
x
)θ=sin
−1
(
r
y
)
Differentiate partially w.r.t
′
x
′
and
′
y
′
∂x
∂θ
=−
1−(
r
x
)
2
1
×
r
1
∂y
∂θ
=−
1−(
r
y
)
2
1
×
r
1
=
r
2
−x
2
−
r
2
×
r
1
=
r
2
−y
2
−
r
2
×
r
1
=
r
2
−x
2
−1
=
r
2
−y
2
1
Differentiate the equation below partially again with
′
x
′
and
′
y
′
∂x
∂θ
=
r
2
−x
2
−1
∂y
∂θ
=
r
2
−y
2
1
∂x
2
∂
2
θ
=
2
r
2
−x
2
−1
×−2x
∂y
2
∂
2
θ
=
r
2
−y
2
1
×−2y
=
r
2
−x
2
1
=
r
2
−y
2
−1
So the given statements are :
I:
∂x
2
∂
2
θ
+
∂y
2
∂
2
θ
=0
⇒
r
2
−x
2
1
−
r
2
−y
2
1
Substitute 'x' and 'y' with rcosθ and rsinθ
⇒
r
2
−r
2
cos
2
θ
1
−
r
2
−r
2
sin
2
θ
1
=
r
2
(1−cos
2
θ)
1
−
r
2
(1−sin
2
θ)
1
=
r
sin
2
θ
1
−
r
cos
2
θ
1
=
rsinθ
1
−
rcosθ
1
=0