Question

a) Express the following surfaces in spherical coordinates
i)xz =3
ii) x2 +y2+z2=1​

Answer 1

Answer:

x=r\cdot sin\theta \cdot cos\phi\\y=r\cdot sin\theta \cdot sin\phi\\z=r\cdot cos\thetax=r⋅sinθ⋅cosϕ

y=r⋅sinθ⋅sinϕ

z=r⋅cosθ

The definition of spherical coordinates and their relation to Euclidean coordinates is shown in the figure

The surface (i) xz=3xz=3 in spherical coordinates will be x\cdot z=r\cdot sin\theta \cdot cos\phi \cdot r\cdot cos\theta=r^2\cdot cos\theta \cdot sin\theta\cdot cos\phi=\frac{1}{2}r^2 sin(2\theta)cos\phix⋅z=r⋅sinθ⋅cosϕ⋅r⋅cosθ=r

2

⋅cosθ⋅sinθ⋅cosϕ= 21 or 2sin(2θ)cosϕ

(i) r^2 sin(2\theta)cos\phi=6r 2

sin(2θ)cosϕ=6

The surface (ii) x^2+y^2-z^2=1x 2 +y 2 −z 2 =1 in spherical coordinates will be

x^2+y^2-z^2=(r\cdot sin\theta \cdot cos\phi)^2+(r\cdot sin\theta \cdot sin\phi)^2-(r\cdot cos\theta )^2=\\r^2\cdot(sin^2\theta\cdot(cos^2\phi+sin^2\phi)-cos^2\theta)=r^2\cdot(sin^2\theta- cos^2\theta)=-r^2\cdot cos(2\theta)x 2 +y 2 −z 2 =(r⋅sinθ⋅cosϕ) 2 +(r⋅sinθ⋅sinϕ) 2 −(r⋅cosθ) 2 =r 2 ⋅(sin

2 θ⋅(cos 2 ϕ+sin 2 ϕ)−cos 2 θ)=r 2 ⋅(sin 2 θ−cos θ)=−r 2 ⋅cos(2θ)

(ii) r^2\cdot cos(2\theta)=-1r

2

⋅cos(2θ)=−1

In deriving these formulas, we used relations for the trigonometric functions of the double angle. The last surface (ii) does not depend on the angle \phiϕ and thus is the surface of rotation (revolution) around the Z axis.

Answer: The surfaces is expressed in spherical coordinates as

(i) r^2 \cdot sin(2\theta)\cdot cos\phi=6r

2

⋅sin(2θ)⋅cosϕ=6

(ii) r^2\cdot cos(2\theta)=-1r

2

⋅cos(2θ)=−1